# LaTeX2HTML 2008 (1.71)
# Associate images original text with physical files.


$key = q/displaystyle>0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3148.png"
 ALT="$\displaystyle &gt; 0.$">|; 

$key = q/gamma,S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1080.png"
 ALT="$ \gamma S$">|; 

$key = q/displaystylev_c=(2,g,l)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4151.png"
 ALT="$\displaystyle v_c = (2 g l)^{1/2}.$">|; 

$key = q/0leq{rmMa}_inftyleq0.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5920.png"
 ALT="$ 0\leq {\rm Ma}_\infty \leq 0.8$">|; 

$key = q/displaystyle=frac{1}{pi}left[-ln(lambda)-1+lambda^{,2}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2519.png"
 ALT="$\displaystyle = \frac{1}{\pi}\left[-\ln(\lambda)-1+\lambda^{ 2}\right],$">|; 

$key = q/y=d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3574.png"
 ALT="$ y=d$">|; 

$key = q/1.40;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="$ 1.40$">|; 

$key = q/displaystyletantheta<2,s.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img745.png"
 ALT="$\displaystyle \tan\theta &lt; 2 s.$">|; 

$key = q/{mitTheta}_S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img527.png"
 ALT="$ {\mit\Theta}_S$">|; 

$key = q/displaystyle=deltaint_0^1(1-f),deta=deltaleft(frac{3}{10}-frac{{mitLambda}}{120}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3182.png"
 ALT="$\displaystyle =\delta\int_0^1(1-f) d\eta = \delta\left(\frac{3}{10}-\frac{{\mit\Lambda}}{120}\right),$">|; 

$key = q/displaystyle=-a_xsintheta+a_ycostheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6288.png"
 ALT="$\displaystyle = -a_x\sin\theta + a_y\cos\theta,$">|; 

$key = q/displaystylefrac{d}{dx}!left(y',frac{partialF}{partialy'}right)=y',frac{partialF}{partialy}+y'',frac{partialF}{partialy'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7322.png"
 ALT="$\displaystyle \frac{d}{dx}\!\left(y' \frac{\partial F}{\partial y'}\right) = y' \frac{\partial F}{\partial y} + y'' \frac{\partial F}{\partial y'}.$">|; 

$key = q/displaystyleV_x(X,infty)=widehat{U}(X),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2901.png"
 ALT="$\displaystyle V_x(X,\infty)= \widehat{U}(X),$">|; 

$key = q/lambdagg(nu^2slashg)^{1slash3}sim5times10^{-5},{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3901.png"
 ALT="$ \lambda\gg (\nu^2/g)^{1/3}\sim 5\times 10^{-5} {\rm m}$">|; 

$key = q/displaystylefrac{partialmbox{boldmathxi}}{partialt}={bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3934.png"
 ALT="$\displaystyle \frac{\partial\mbox{\boldmath$\xi$}}{\partial t} = {\bf v},$">|; 

$key = q/zeta={rmi}tmspace+thinmuskip{.1667em}{rme}^{tmspace+thinmuskip{.1667em}pitmspace+thinmuskip{.1667em}zslasha};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$ \zeta ={\rm i}\tmspace +\thinmuskip {.1667em}{\rm e}^{\tmspace +\thinmuskip {.1667em}\pi \tmspace +\thinmuskip {.1667em}z/a}$">|; 

$key = q/displaystyle=-nablacdot(d,{bfv}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5069.png"
 ALT="$\displaystyle = -\nabla\cdot (d {\bf v}),$">|; 

$key = q/displaystyle{bfv}_S(P)=frac{1}{2}left[{bfv}_S(P_+)+{bfv}_S(P_-)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3428.png"
 ALT="$\displaystyle {\bf v}_S(P)= \frac{1}{2}\left[{\bf v}_S(P_+)+ {\bf v}_S(P_-)\right].$">|; 

$key = q/displaystyleX+{rmi},Y={rmi},{rme}^{-{rmi},alpha},rho,V,{mitGamma}={rme}^{,{rmi},(pislash2-alpha)},rho,V,{mitGamma}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3287.png"
 ALT="$\displaystyle X+{\rm i} Y = {\rm i} {\rm e}^{-{\rm i} \alpha} \rho V {\mit\Gamma}= {\rm e}^{ {\rm i} (\pi/2-\alpha)} \rho V {\mit\Gamma}.$">|; 

$key = q/displaystylefrac{df}{dx}=frac{dq}{dxi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6031.png"
 ALT="$\displaystyle \frac{df}{dx} =\frac{dq}{d\xi}.$">|; 

$key = q/displaystylefrac{partial}{partialxi}left(frac{1}{varpi},frac{partialpsi}{partialeta}left(frac{1}{varpi},frac{partialpsi}{partialeta}right)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2700.png"
 ALT="$\displaystyle \frac{\partial}{\partial\xi}\left(\frac{1}{\varpi} \frac{\partia...
...partial\eta}\left(\frac{1}{\varpi} \frac{\partial\psi}{\partial\eta}\right)=0.$">|; 

$key = q/displaystyleleft.frac{dv_x}{dy}right|_{y=h}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3595.png"
 ALT="$\displaystyle \left.\frac{dv_x}{dy}\right\vert _{y=h} = 0.$">|; 

$key = q/displaystyle=frac{1}{v}left(frac{partialv}{partialT}right)_p,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5597.png"
 ALT="$\displaystyle = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_p,$">|; 

$key = q/displaystyleA_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7267.png"
 ALT="$\displaystyle A_i$">|; 

$key = q/[({bfb}cdot{nabla}){bfa}]_i=b_j,partiala_islashpartialx_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6992.png"
 ALT="$ [({\bf b}\cdot{\nabla}){\bf a}]_i= b_j \partial a_i/\partial x_j$">|; 

$key = q/y(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4037.png"
 ALT="$ y(x)$">|; 

$key = q/2,d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1214.png"
 ALT="$ 2 d$">|; 

$key = q/displaystyleleft(frac{partial}{partialt}+{bfV}cdotnablaright)phi=frac{p_1}{rho}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4078.png"
 ALT="$\displaystyle \left(\frac{\partial}{\partial t} + {\bf V}\cdot\nabla\right)\phi = \frac{p_1}{\rho}.$">|; 

$key = q/displaystyley_U(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5833.png"
 ALT="$\displaystyle y_U(x)$">|; 

$key = q/{bfS}=S,(cosalpha,,sinalpha,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6341.png"
 ALT="$ {\bf S} = S (\cos\alpha,  \sin\alpha,  0)$">|; 

$key = q/displaystyle(m+m'),frac{d{bfV}}{dt}={bfF}_{rmext}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1821.png"
 ALT="$\displaystyle (m+m') \frac{d{\bf V}}{dt} = {\bf F}_{\rm ext}.$">|; 

$key = q/a=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3156.png"
 ALT="$ a=0$">|; 

$key = q/_x)_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$ _x)_x$">|; 

$key = q/displaystylef_z=mpfrac{(alpha^{,2}-f^{,2})^{1slash2},(f^{,2}-beta^{,2})^{1slash2}}{f^{,2}+alpha,beta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1294.png"
 ALT="$\displaystyle f_z = \mp \frac{(\alpha^{ 2}-f^{ 2})^{1/2} (f^{ 2}-\beta^{ 2})^{1/2}}{f^{ 2}+\alpha \beta}.$">|; 

$key = q/displaystyle-frac{costheta}{sintheta},frac{partial{mitPhi}_r}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4846.png"
 ALT="$\displaystyle -\frac{\cos\theta}{\sin\theta} \frac{\partial{\mit\Phi}_r}{\partial\phi}$">|; 

$key = q/displaystyle(nablaphi)_i=frac{partialphi}{partialx_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6882.png"
 ALT="$\displaystyle (\nabla \phi)_i = \frac{\partial \phi}{\partial x_i}.$">|; 

$key = q/r-r_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img723.png"
 ALT="$ r-r_0$">|; 

$key = q/dS=dl,dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3448.png"
 ALT="$ dS = dl dz$">|; 

$key = q/k=m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6809.png"
 ALT="$ k=m$">|; 

$key = q/+(3slash2),sin^2epsilon,,zeta_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4546.png"
 ALT="$ +(3/2) \sin^2\epsilon  \zeta_E$">|; 

$key = q/displaystyle=frac{2,q_1}{sqrt{{rmMa}_1^{,2}-1}}int_{x_L}^{x_R}left[left(frac{dy_L}{dx}right)^2+left(frac{dy_U}{dx}right)^2right]dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5848.png"
 ALT="$\displaystyle = \frac{2 q_1}{\sqrt{{\rm Ma}_1^{ 2}-1}}\int_{x_L}^{x_R}\left[\left(\frac{dy_L}{dx}\right)^2+\left(\frac{dy_U}{dx}\right)^2\right]dx$">|; 

$key = q/overline{mu}slashmu=1slash10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$ \overline {\mu }/\mu =1/10$">|; 

$key = q/u(x,-y)=u(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3244.png"
 ALT="$ u(x,-y)=u(x,y)$">|; 

$key = q/displaystyle(a_1^{,2}-a_2^{,2})int_0^inftyleft[frac{a_1^{,2},a_2^{,2}}{(a_1^{,2}}+u)}-frac{a_3^{,2}}{(a_3^{,2}+u)}right]frac{du}{mitDelta}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img868.png"
 ALT="$\displaystyle (a_1^{ 2}-a_2^{ 2})\int_0^\infty\left[\frac{a_1^{ 2} a_2^{ 2...
...a_2^{ 2}+u)}- \frac{a_3^{ 2}}{(a_3^{ 2}+u)}\right]\frac{du}{\mit\Delta} = 0,$">|; 

$key = q/displaystylefrac{p_2}{p_1}=frac{1+gamma,{rmMa}_1^{,2}}{1+gamma,{rmMa}_2^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5477.png"
 ALT="$\displaystyle \frac{p_2}{p_1} = \frac{1+\gamma {\rm Ma}_1^{ 2}}{1+\gamma {\rm Ma}_2^{ 2}}.$">|; 

$key = q/displaystylepsi(r,theta)=-V_0,r,sin(theta-theta_0).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1666.png"
 ALT="$\displaystyle \psi(r,\theta)=-V_0 r \sin(\theta-\theta_0).$">|; 

$key = q/-Q';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2825.png"
 ALT="$ -Q'$">|; 

$key = q/lggd;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3648.png"
 ALT="$ l\gg d$">|; 

$key = q/{bfS}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6349.png"
 ALT="$ {\bf S} = {\bf0}$">|; 

$key = q/displaystylefrac{partialp}{partialr}-rho,omega^{,2},r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img818.png"
 ALT="$\displaystyle \frac{\partial p}{\partial r} - \rho \omega^{ 2} r$">|; 

$key = q/displaystyleK=-pi,rholeft.int_{eta=0}^{eta=pi}phi,dpsiright|_{xi=xi_0}=frac{1}{2ho,V^{,2}int_{-1}^1mu^{,2},dmu=frac{pi}{3},a^{,3},rho,V^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2749.png"
 ALT="$\displaystyle K= -\pi \rho\left.\int_{\eta=0}^{\eta=\pi} \phi d\psi\right\ver...
...rho V^{ 2}\int_{-1}^1 \mu^{ 2} d\mu= \frac{\pi}{3} a^{ 3} \rho V^{ 2}.$">|; 

$key = q/(partialhslashpartialy)_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6528.png"
 ALT="$ (\partial h/\partial y)_x$">|; 

$key = q/lambda,mu,nu=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6733.png"
 ALT="$ \lambda \mu \nu=-1$">|; 

$key = q/Q={mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4758.png"
 ALT="$ Q={\mit\Psi}$">|; 

$key = q/displaystylesimeq-p_0-frac{9}{2},mu,V,costheta,frac{a}{r^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3891.png"
 ALT="$\displaystyle \simeq -p_0 - \frac{9}{2} \mu V \cos\theta \frac{a}{r^{ 2}},$">|; 

$key = q/{rmMa}_1>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5503.png"
 ALT="$ {\rm Ma}_1&gt;1$">|; 

$key = q/tanh(x)rightarrow1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3927.png"
 ALT="$ \tanh(x)\rightarrow 1$">|; 

$key = q/displaystyle=frac{mitGamma}{2pi,r}+V_x(t),frac{a^{,2}}{r^{,2}},sintheta-V_y(t),frac{a^{,2}}{r^{,2}},costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1832.png"
 ALT="$\displaystyle =\frac{\mit\Gamma}{2\pi r}+V_x(t) \frac{a^{ 2}}{r^{ 2}} \sin\theta-V_y(t) \frac{a^{ 2}}{r^{ 2}} \cos\theta.$">|; 

$key = q/displaystylelambda_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img694.png"
 ALT="$\displaystyle \lambda_1$">|; 

$key = q/displaystyleV_1,,H_1G=V_2,,H_2G=left(frac{V_1,V_2}{V_1+V_2}right)H_1H_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img692.png"
 ALT="$\displaystyle V_1  H_1G = V_2  H_2G = \left(\frac{V_1 V_2}{V_1+V_2}\right)H_1H_2,$">|; 

$key = q/{rmRe}=U,aslashnu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3618.png"
 ALT="$ {\rm Re} = U a/\nu$">|; 

$key = q/e_{13}=singamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img887.png"
 ALT="$ e_{13}=\sin\gamma$">|; 

$key = q/h({rmmm})=13.5slasha({rmmm});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1147.png"
 ALT="$ h({\rm mm}) = 13.5/a({\rm mm})$">|; 

$key = q/{bfe}_i=h_i,nablau_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7064.png"
 ALT="$ {\bf e}_i = h_i \nabla u_i$">|; 

$key = q/displaystylefrac{partialphi(a,theta,varphi)}{partialr}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2640.png"
 ALT="$\displaystyle \frac{\partial\phi(a,\theta,\varphi)}{\partial r} = 0.$">|; 

$key = q/displaystyle{calL}(f,f_z)=2pi,gamma,f,(1+f_z^{,2})^{1slash2}-pi,p_0,f^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1249.png"
 ALT="$\displaystyle {\cal L}(f,f_z) = 2\pi \gamma f (1+f_z^{ 2})^{1/2} - \pi p_0 f^{ 2},$">|; 

$key = q/15.8,{rmcm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5040.png"
 ALT="$ 15.8 {\rm cm}$">|; 

$key = q/{bfv}=V_0,{rme}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1760.png"
 ALT="$ {\bf v}= V_0 {\rm e}_x$">|; 

$key = q/displaystyle({bfa}times{bfb})times{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6675.png"
 ALT="$\displaystyle ({\bf a}\times {\bf b})\times {\bf c}$">|; 

$key = q/displaystylep(x)-p_0=frac{6,mu,U}{alpha},frac{[d_1-d(x)],[d(x)-d_2]}{d^{,2}(x),(d_1+d_2)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3705.png"
 ALT="$\displaystyle p(x)-p_0 = \frac{6 \mu U}{\alpha}  \frac{[d_1-d(x)] [d(x)-d_2]}{d^{ 2}(x) (d_1+d_2)}.$">|; 

$key = q/partialv_islashpartialx_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="$ \partial v_i/\partial x_j$">|; 

$key = q/oint_C{bfA}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6746.png"
 ALT="$ \oint_C {\bf A}\cdot d{\bf r}$">|; 

$key = q/displaystylev_p=(g,d)^{1slash2}left[frac{tanh(k,d)}{k,d}right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3964.png"
 ALT="$\displaystyle v_p = (g d)^{1/2}\left[\frac{\tanh(k d)}{k d}\right]^{1/2}.$">|; 

$key = q/displaystyle=-frac{g}{a,sintheta},frac{partial}{partialphi}left[zeta-(1+k_2-h_2),skew{5}bar{zeta}_2-ointG_F,zeta,d{mitOmega}'right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4517.png"
 ALT="$\displaystyle = - \frac{g}{a \sin\theta} \frac{\partial}{\partial\phi}\left[\zeta-(1+k_2-h_2) \skew{5}\bar{\zeta}_2 -\oint G_F \zeta d{\mit\Omega}'\right],$">|; 

$key = q/displaystyle=lambda,delta_{ij},delta_{kl}+mu,delta_{ik},delta_{jl}+nu,delta_{il},delta_{jk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6961.png"
 ALT="$\displaystyle =\lambda \delta_{ij} \delta_{kl} + \mu \delta_{ik} \delta_{jl} +\nu \delta_{il} \delta_{jk},$">|; 

$key = q/displaystyle=-p+2,mu,frac{partialv_z}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="$\displaystyle = -p + 2 \mu \frac{\partial v_z}{\partial z},$">|; 

$key = q/l,(l+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2578.png"
 ALT="$ l (l+1)$">|; 

$key = q/F(z)={rmi},({mitGamma}slash2pi),ln(z-z_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1987.png"
 ALT="$ F(z)={\rm i} ({\mit\Gamma}/2\pi) \ln(z-z_0)$">|; 

$key = q/displaystyle=a_1,cosgamma,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img929.png"
 ALT="$\displaystyle =a_1 \cos\gamma,$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$ t$">|; 

$key = q/displaystyle=frac{partial}{partialt}+v_r,frac{partial}{partialr}+frac{v_theta}{r},frac{partial}{partialtheta}+v_z,frac{partial}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="$\displaystyle = \frac{\partial}{\partial t} + v_r \frac{\partial }{\partial r}...
...theta}{r} \frac{\partial}{\partial \theta} + v_z \frac{\partial}{\partial z},$">|; 

$key = q/duslashdz=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img791.png"
 ALT="$ du/dz=0$">|; 

$key = q/displaystyler^{,2},nabla^{,2}{calS}_n+n,(n+1),{calS}_n=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4313.png"
 ALT="$\displaystyle r^{ 2} \nabla^{ 2} {\cal S}_n + n (n+1) {\cal S}_n = 0$">|; 

$key = q/partialFslashpartialy=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7318.png"
 ALT="$ \partial F/\partial y = 0$">|; 

$key = q/displaystylep_c=K,rho_c^{4slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5201.png"
 ALT="$\displaystyle p_c = K \rho_c^{4/3},$">|; 

$key = q/f_{xx}=d^{,2}fslashdx^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1159.png"
 ALT="$ f_{xx}= d^{ 2}f/dx^{ 2}$">|; 

$key = q/displaystylefrac{partialrho}{partialt}+u,frac{partialrho}{partialx}+rho,frac{partialu}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5345.png"
 ALT="$\displaystyle \frac{\partial\rho}{\partial t} + u \frac{\partial \rho}{\partial x}+ \rho \frac{\partial u}{\partial x}$">|; 

$key = q/skew{3}bar{psi}=-skew{3}bar{psi}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2188.png"
 ALT="$ \skew{3}\bar{\psi}=-\skew{3}\bar{\psi}_1$">|; 

$key = q/displaystyley'^{,2}=left(lambda'+frac{y}{h}right)^2-1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7347.png"
 ALT="$\displaystyle y'^{ 2} = \left(\lambda' + \frac{y}{h}\right)^2 - 1,$">|; 

$key = q/displaystyle{rmFr}=frac{v}{sqrt{g,h}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1441.png"
 ALT="$\displaystyle {\rm Fr} = \frac{v}{\sqrt{g h}}.$">|; 

$key = q/displaystyleoint_C{bfF}cdot{bft},dr=int_Snablatimes{bfF}cdot{bfn},dS.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1094.png"
 ALT="$\displaystyle \oint_C {\bf F}\cdot {\bf t} dr= \int_S \nabla\times {\bf F}\cdot {\bf n} dS.$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter12slashphasem2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5051.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter12/phasem2.eps}}
\end{figure}">|; 

$key = q/displaystyle(nablatimes{bfa})_i=epsilon_{ijk},frac{partiala_k}{partialx_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6884.png"
 ALT="$\displaystyle (\nabla\times {\bf a})_i = \epsilon_{ijk} \frac{\partial a_k}{\partial x_j}.$">|; 

$key = q/(c,coth,eta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2475.png"
 ALT="$ (c \coth \eta_0$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter02slashfig3.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="373" BORDER="0"
 SRC="|."$dir".q|img707.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter02/fig3.02.eps}}
\end{figure}">|; 

$key = q/Vsimeq-u_1+sqrt{g,h};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1518.png"
 ALT="$ V\simeq -u_1+\sqrt{g h}$">|; 

$key = q/displaystylefrac{U}{nu},delta_2,frac{ddelta_2}{dx}+frac{delta_2^{,2}}{nu},frac{dt)=frac{delta_2}{U}left.frac{partialu}{partialy}right|_{y=0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3186.png"
 ALT="$\displaystyle \frac{U}{\nu} \delta_2 \frac{d\delta_2}{dx} +\frac{\delta_2^{ ...
...ight) =\frac{\delta_2}{U}\left.\frac{\partial u}{\partial y}\right\vert _{y=0}.$">|; 

$key = q/displaystyle=frac{1}{2},X,sinbeta,cosbeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4039.png"
 ALT="$\displaystyle =\frac{1}{2} X \sin\beta \cos\beta,$">|; 

$key = q/rho_0simeq10^3,{rmkg,m^{-3}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img570.png"
 ALT="$ \rho_0\simeq 10^3 {\rm kg m^{-3}}$">|; 

$key = q/{rmMa}_s=V_sslashc_1=u_1slashc_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5535.png"
 ALT="$ {\rm Ma}_s=V_s/c_1=u_1/c_1$">|; 

$key = q/m_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5183.png"
 ALT="$ m_p$">|; 

$key = q/S_2,(p_2+rho,v_2^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1396.png"
 ALT="$ S_2 (p_2+\rho v_2^{ 2})$">|; 

$key = q/displaystyleh=frac{int_{A_0}int_0^deltaz,dx,dy,dz}{V_0}=frac{1}{2,V_0},int_{A_0}delta^{,2}(x,y),dx,dy=frac{delta_0^{,2},A_0}{2,V_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="459" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img633.png"
 ALT="$\displaystyle h = \frac{\int_{A_0}\int_0^\delta z dx dy dz}{V_0}=\frac{1}{2 V_0} \int_{A_0} \delta^{ 2}(x,y) dx dy=\frac{\delta_0^{ 2} A_0}{2 V_0},$">|; 

$key = q/_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1539.png"
 ALT="$ _1$">|; 

$key = q/displaystylefrac{omega^{,2}}{2pi,G,rho};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img878.png"
 ALT="$\displaystyle \frac{\omega^{ 2}}{2\pi G \rho}$">|; 

$key = q/displaystyleoint{bfA}cdotd{bfS}={bfA}cdotointd{bfS}={bfA}cdot{bfS}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6576.png"
 ALT="$\displaystyle \oint {\bf A}\cdot d{\bf S} = {\bf A}\cdot \oint d{\bf S} = {\bf A} \cdot {\bf S}=0,$">|; 

$key = q/displaystylea_i'={calR}_{ij},a_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6825.png"
 ALT="$\displaystyle a_i' = {\cal R}_{ij} a_j,$">|; 

$key = q/displaystylev_x(x,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2873.png"
 ALT="$\displaystyle v_x(x,0)$">|; 

$key = q/theta_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3370.png"
 ALT="$ \theta_-$">|; 

$key = q/p_e<p_{rmatm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5435.png"
 ALT="$ p_e&lt;p_{\rm atm}$">|; 

$key = q/displaystylex;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1939.png"
 ALT="$\displaystyle x$">|; 

$key = q/displaystylefrac{T}{T_0}=frac{1}{1+beta,z},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5261.png"
 ALT="$\displaystyle \frac{T}{T_0} = \frac{1}{1+\beta z},
$">|; 

$key = q/s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img601.png"
 ALT="$ s$">|; 

$key = q/displaystyle(widetilde{nabla{bfA}})_{phiphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7189.png"
 ALT="$\displaystyle (\widetilde{\nabla {\bf A}})_{\phi\phi}$">|; 

$key = q/displaystylelambda=frac{G,M'}{R^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img988.png"
 ALT="$\displaystyle \lambda = \frac{G M'}{R^{ 3}},$">|; 

$key = q/displaystyle=Q^{,3}+R^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6097.png"
 ALT="$\displaystyle = Q^{ 3}+R^{ 2},$">|; 

$key = q/displaystyle_S,dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3439.png"
 ALT="$\displaystyle _S dS,$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.23.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="482" HEIGHT="196" BORDER="0"
 SRC="|."$dir".q|img6593.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.23.eps}}
\end{figure}">|; 

$key = q/displaystyle=frac{sin^2theta}{a_2^{,2}}+frac{cos^2theta}{a_3^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7262.png"
 ALT="$\displaystyle =\frac{\sin^2\theta}{a_2^{ 2}}+\frac{\cos^2\theta}{a_3^{ 2}}.$">|; 

$key = q/displaystyleh_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7173.png"
 ALT="$\displaystyle h_\phi$">|; 

$key = q/displaystyle{bfv}=-frac{Q}{2pi,r},{bfe}_r,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1678.png"
 ALT="$\displaystyle {\bf v} = - \frac{Q}{2\pi r} {\bf e}_r,$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter15slashrefl.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="444" HEIGHT="257" BORDER="0"
 SRC="|."$dir".q|img6147.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter15/refl.eps}}
\end{figure}">|; 

$key = q/p_{-r}rightarrow{rmi}^{,n+1},p_{-n}^{,-m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4993.png"
 ALT="$ p_{-r}\rightarrow {\rm i}^{ n+1} p_{-n}^{ -m}$">|; 

$key = q/ACP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1616.png"
 ALT="$ ACP$">|; 

$key = q/p_0={mitDelta}p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1238.png"
 ALT="$ p_0={\mit\Delta} p$">|; 

$key = q/displaystyley_i',frac{partialF}{partialy_i'}-F={rmconst},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7368.png"
 ALT="$\displaystyle y_i' \frac{\partial F}{\partial y_i'} - F = {\rm const},$">|; 

$key = q/{mitPhi}({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4352.png"
 ALT="$ {\mit\Phi}({\bf r})$">|; 

$key = q/displaystyle=frac{v,v_x}{h}-g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3559.png"
 ALT="$\displaystyle = \frac{v v_x}{h} - g,$">|; 

$key = q/d{bfS}=(0,,dS_y,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6638.png"
 ALT="$ d{\bf S} = (0, dS_y, 0)$">|; 

$key = q/displaystylefrac{D{bfv}}{Dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="$\displaystyle \frac{D{\bf v}}{Dt}$">|; 

$key = q/displaystyle{bfv}=-nablaphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img545.png"
 ALT="$\displaystyle {\bf v}= -\nabla\phi,
$">|; 

$key = q/displaystylefrac{5}{2},frac{pi,a^{,2}}{S}left(frac{h}{2,g}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1597.png"
 ALT="$\displaystyle \frac{5}{2} \frac{\pi a^{ 2}}{S}\left(\frac{h}{2 g}\right)^{1/2}.
$">|; 

$key = q/displaystylefrac{p}{p_0}=expleft[frac{a^{,2}}{H}left(frac{1}{r}-frac{1}{a}right)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5264.png"
 ALT="$\displaystyle \frac{p}{p_0} = \exp\left[\frac{a^{ 2}}{H}\left(\frac{1}{r}-\frac{1}{a}\right)\right],
$">|; 

$key = q/N_A=6.022times10^{23};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5293.png"
 ALT="$ N_A= 6.022\times 10^{23}$">|; 

$key = q/360^circslash{mitPi}_E=20940,{rmyears};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4604.png"
 ALT="$ 360^\circ/{\mit\Pi}_E = 20940 {\rm years}$">|; 

$key = q/displaystylex=left[c_1+frac{1}{2},(gamma+1),uright]t.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5567.png"
 ALT="$\displaystyle x =\left[c_1+\frac{1}{2} (\gamma+1) u\right]t.$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.14.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="200" BORDER="0"
 SRC="|."$dir".q|img6452.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.14.eps}}
\end{figure}">|; 

$key = q/zetaleqzeta_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1283.png"
 ALT="$ \zeta\leq \zeta_c$">|; 

$key = q/displaystyleint!int!int_Vf(x,y,z),dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6503.png"
 ALT="$\displaystyle \int\!\int\!\int_V f(x,y,z) dV,$">|; 

$key = q/displaystyleF_y=-int_{-infty}^inftydeltap(x,0),dx=frac{rho,Q^{,2}}{2pi^2,a}int_{ac{zeta^{,2}}{(1+zeta^{,2})^2},dzeta=frac{rho,Q^{,2}}{4pi,a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2030.png"
 ALT="$\displaystyle F_y = -\int_{-\infty}^\infty \delta p(x,0) dx = \frac{\rho Q^{\...
...y \frac{\zeta^{ 2}}{(1+\zeta^{ 2})^2} d\zeta= \frac{\rho Q^{ 2}}{4\pi a}.$">|; 

$key = q/displaystylefrac{dy}{dx}=tanbeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4036.png"
 ALT="$\displaystyle \frac{dy}{dx} = \tan\beta,$">|; 

$key = q/int_0^1y^{,2},dx=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7371.png"
 ALT="$ \int_0^1 y^{ 2} dx = 2$">|; 

$key = q/BQ';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img735.png"
 ALT="$ BQ'$">|; 

$key = q/nablatimes{bfv}_1={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4076.png"
 ALT="$ \nabla\times{\bf v}_1={\bf0}$">|; 

$key = q/displaystyle{rmMa}_{1,2}=frac{u_{1,2}}{sqrt{gamma,{calR},T_{1,2}}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5473.png"
 ALT="$\displaystyle {\rm Ma}_{1,2} = \frac{u_{1,2}}{\sqrt{\gamma {\cal R} T_{1,2}}},$">|; 

$key = q/displaystyleL=f,cosalpha=left(frac{2pi,sinalpha,cosalpha}{pi,sinalpha+4}right)rho,V^{,2},b.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2338.png"
 ALT="$\displaystyle L = f \cos\alpha = \left(\frac{2\pi \sin\alpha \cos\alpha}{\pi \sin\alpha+4}\right)\rho V^{ 2} b.$">|; 

$key = q/displaystyle=-left[frac{1+k_2-h_2}{1-(1+k_2'-h_2'),alpha_2}right](zeta_{mitOmega}+zeta_M+zeta_E),P_2(costheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="434" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4641.png"
 ALT="$\displaystyle = - \left[\frac{1+k_2-h_2}{1-(1+k_2'-h_2') \alpha_2}\right](\zeta_{\mit\Omega}+\zeta_M+\zeta_E) P_2(\cos\theta).$">|; 

$key = q/0leqxleqc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6067.png"
 ALT="$ 0\leq x\leq c$">|; 

$key = q/delta(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img628.png"
 ALT="$ \delta(x,y)$">|; 

$key = q/displaystylexi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2162.png"
 ALT="$\displaystyle \xi$">|; 

$key = q/displaystylefrac{T}{W}=g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3533.png"
 ALT="$\displaystyle \frac{T}{W} = g.$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter10slashstokes.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3772.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter10/stokes.eps}}
\end{figure}">|; 

$key = q/displaystyleC_{pU}=C_{p0},pi^{-1},cos^{-1}left(1-2sqrt{{rmMa}_infty^{,2}-1},frac{y}{x}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6066.png"
 ALT="$\displaystyle C_{pU} = C_{p0} \pi^{-1} \cos^{-1}\left(1-2\sqrt{{\rm Ma}_\infty^{ 2}-1} \frac{y}{x}\right)$">|; 

$key = q/displaystyle{rmdet}(A);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7012.png"
 ALT="$\displaystyle {\rm det}(A)$">|; 

$key = q/displaystylefrac{p}{p_infty}=left[frac{2+(gamma-1),{rmMa}_infty^{,2}}{2+(gamma-1),{rmMa}^{,2}}right]^{,gammaslash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5927.png"
 ALT="$\displaystyle \frac{p}{p_\infty} = \left[\frac{2+(\gamma-1) {\rm Ma}_\infty^{ 2}}{2+(\gamma-1) {\rm Ma}^{ 2}}\right]^{ \gamma/(\gamma-1)},$">|; 

$key = q/j=7;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$ j=7$">|; 

$key = q/displaystyleint!int{bfv}cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6502.png"
 ALT="$\displaystyle \int\!\int {\bf v}\cdot d{\bf S}$">|; 

$key = q/displaystyle=left(frac{partial{calE}}{partial{calS}}right)_v,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5578.png"
 ALT="$\displaystyle = \left(\frac{\partial {\cal E}}{\partial {\cal S}}\right)_v,$">|; 

$key = q/displaystyle(1-mu^{,2}),frac{dP_n^{,m}}{dmu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4691.png"
 ALT="$\displaystyle (1-\mu^{ 2}) \frac{dP_n^{ m}}{d\mu}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter15slashexp.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="498" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img5761.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter15/exp.eps}}
\end{figure}">|; 

$key = q/nablacdot(nablaT);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6611.png"
 ALT="$ \nabla\cdot (\nabla T)$">|; 

$key = q/displaystylefrac{2}{gamma-1}left(frac{partialc}{partialt}+u,frac{partialc}{partialx}right)+c,frac{partialu}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5656.png"
 ALT="$\displaystyle \frac{2}{\gamma-1}\left(\frac{\partial c}{\partial t} + u \frac{\partial c}{\partial x}\right)+c \frac{\partial u}{\partial x}$">|; 

$key = q/8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4564.png"
 ALT="$ 8$">|; 

$key = q/displaystyle=cotmu=sqrt{{rmMa}_infty^{,2}-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6081.png"
 ALT="$\displaystyle = \cot\mu = \sqrt{{\rm Ma}_\infty^{ 2}-1},$">|; 

$key = q/z_0+dz'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2065.png"
 ALT="$ z_0+dz''$">|; 

$key = q/displaystyle=frac{2}{pi},V,a,coseta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2790.png"
 ALT="$\displaystyle = \frac{2}{\pi} V a \cos\eta,$">|; 

$key = q/displaystylerho,frac{D{bfv}}{Dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3571.png"
 ALT="$\displaystyle \rho \frac{D{\bf v}}{Dt}$">|; 

$key = q/dUslashdxneq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3198.png"
 ALT="$ dU/dx\neq 0$">|; 

$key = q/delta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="$ \delta_{ij}$">|; 

$key = q/displaystyleF(x+{rmi},y)=(x+{rmi},y)^2=(x^{,2}-y^{,2})+2,{rmi},x,y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1952.png"
 ALT="$\displaystyle F(x + {\rm i} y) = (x+{\rm i} y)^2 = (x^{ 2}-y^{ 2}) + 2 {\rm i}  x y,$">|; 

$key = q/displaystylepsi(r,alpha,pislash2)=psi(r,2pi-alpha,pislash2)=psi(r,pi)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1869.png"
 ALT="$\displaystyle \psi(r,\alpha \pi/2) =\psi(r,2\pi-\alpha \pi/2)=\psi(r,\pi)=0.$">|; 

$key = q/d(x)=d_1-alpha,x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3700.png"
 ALT="$ d(x)=d_1-\alpha x$">|; 

$key = q/displaystyle=-f_x=frac{mu,U,l^{,2}}{d_0^{,2}},frac{3}{2,k^2}left[lnleft(frac{1+k}{1-k}right)-2,kright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3713.png"
 ALT="$\displaystyle = -f_x = \frac{\mu U l^{ 2}}{d_0^{ 2}} \frac{3}{2 k^2}\left[\ln\left(\frac{1+k}{1-k}\right)-2 k\right],$">|; 

$key = q/displaystyleV=int!intD(x,y),dS.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6487.png"
 ALT="$\displaystyle V= \int\!\int D(x,y) dS.$">|; 

$key = q/displaystyleG(z)=F({rmi},{rme}^{,pi,zslasha})={rmi},frac{mitGamma}{2pi}ln,tanhleft(frac{pi,z}{2,a}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2108.png"
 ALT="$\displaystyle G(z)=F({\rm i} {\rm e}^{ \pi z/a}) = {\rm i} \frac{\mit\Gamma}{2\pi} \ln \tanh\left(\frac{\pi z}{2 a}\right),$">|; 

$key = q/displaystyle=-frac{partialpsi}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img535.png"
 ALT="$\displaystyle =-\frac{\partial \psi}{\partial y},$">|; 

$key = q/displaystyle=frac{rho,D_1,D_2}{pi,x^{,3}},sin(alpha_1+alpha_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2458.png"
 ALT="$\displaystyle =\frac{\rho D_1 D_2}{\pi x^{ 3}} \sin(\alpha_1+\alpha_2),$">|; 

$key = q/overline{rho}slashrhosimeq2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3816.png"
 ALT="$ \overline{\rho}/\rho\simeq 2$">|; 

$key = q/displaystyle=mu_sint_{mitOmega}{mitPsi}_r'',{mitPsi}_s,d{mitOmega}=-int_{mitOmega}{mitPsi}_r'',D,{mitPsi}_s,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4876.png"
 ALT="$\displaystyle =\mu_s\int_{\mit\Omega} {\mit\Psi}_r'' {\mit\Psi}_s d{\mit\Omega} = -\int_{\mit\Omega} {\mit\Psi}_r'' D {\mit\Psi}_s d{\mit\Omega}$">|; 

$key = q/displaystylefrac{dm}{dr}=4pi,r^{,2},rho,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5171.png"
 ALT="$\displaystyle \frac{dm}{dr}= 4\pi r^{ 2} \rho,$">|; 

$key = q/3,H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5258.png"
 ALT="$ 3 H$">|; 

$key = q/displaystyle-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1580.png"
 ALT="$\displaystyle -$">|; 

$key = q/displaystylepsi=psi(x,y,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1617.png"
 ALT="$\displaystyle \psi= \psi(x,y,t).$">|; 

$key = q/displaystylesimeq2.9times10^{-3},V_0({rmm,s^{-1}}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="$\displaystyle \simeq 2.9\times 10^{-3} V_0({\rm m s^{-1}}).$">|; 

$key = q/displaystylechi_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6105.png"
 ALT="$\displaystyle \chi_s$">|; 

$key = q/displaystyleH=frac{R,T}{g,M};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5124.png"
 ALT="$\displaystyle H = \frac{R T}{g M}$">|; 

$key = q/displaystylembox{if{j=m{and{k=lneqj{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6807.png"
 ALT="$\displaystyle \mbox{if $j=m$ and $k=l\neq j$}$">|; 

$key = q/displaystyle{calG}={calH}-T,{calS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5590.png"
 ALT="$\displaystyle {\cal G} = {\cal H} - T {\cal S}.
$">|; 

$key = q/displaystyle=-alpha(x)-frac{dh}{dx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5844.png"
 ALT="$\displaystyle = -\alpha(x)-\frac{dh}{dx}.$">|; 

$key = q/y=f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6026.png"
 ALT="$ y=f(x)$">|; 

$key = q/sigma_{ij}({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4373.png"
 ALT="$ \sigma_{ij}({\bf r})$">|; 

$key = q/displaystyle={rmRe}left(-frac{1}{2},{rmi},rhoointleft[V^{,2},R^{,2},{rme}^{,2,{r{V,{mitGamma},R,{rme}^{,{rmi},(theta+alpha)}}{pi}right.right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3290.png"
 ALT="$\displaystyle ={\rm Re}\left(- \frac{1}{2} {\rm i} \rho\oint\left[V^{ 2} R^...
...ac{V {\mit\Gamma} R {\rm e}^{ {\rm i} (\theta+\alpha)}}{\pi}\right.\right.$">|; 

$key = q/(dhslashdr)_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6525.png"
 ALT="$ (dh/dr)_{\rm max}$">|; 

$key = q/r=(x_1^{,2}+x_2^{,2}+x_3^{,2})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7289.png"
 ALT="$ r=(x_1^{ 2}+x_2^{ 2}+x_3^{ 2})^{1/2}$">|; 

$key = q/a_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img837.png"
 ALT="$ a_2$">|; 

$key = q/displaystyle=left(frac{gamma}{gamma-1}right){calR},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5614.png"
 ALT="$\displaystyle =\left(\frac{\gamma}{\gamma-1}\right){\cal R},$">|; 

$key = q/|1-v_xslashU_0|=|u|slashU_0ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3066.png"
 ALT="$ \vert 1-v_x/U_0\vert=\vert u\vert/U_0\ll 1$">|; 

$key = q/displaystyle=nu({rmMa}_1)+tan^{-1}left[({rmMa}_1^{,2}-1)^{1slash2}right]-tan^{-1t[({rmMa}_1^{,2}-1)^{1slash2}frac{tanz}{tanz_1}right]nonumber;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="522" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6135.png"
 ALT="$\displaystyle = \nu({\rm Ma}_1) +\tan^{-1}\left[({\rm Ma}_1^{ 2}-1)^{1/2}\righ...
...tan^{-1}\left[({\rm Ma}_1^{ 2}-1)^{1/2}\frac{\tan z}{\tan z_1}\right]\nonumber$">|; 

$key = q/displaystyle=int_0^adzint_{-(a-z)slash2}^{(a-z)slash2}dyint_{-(a-z)slash2}^{(a-z)slash2}dx=int_0^a(a-z)^2,dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6510.png"
 ALT="$\displaystyle = \int_0^a dz \int_{-(a-z)/2}^{(a-z)/2} dy\int_{-(a-z)/2}^{(a-z)/2} dx= \int_0^a (a-z)^2 dz$">|; 

$key = q/displaystyletilde{E}(phi,k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1176.png"
 ALT="$\displaystyle \tilde{E}(\phi,k)$">|; 

$key = q/displaystyleleft(frac{x}{a}right)^2+left(frac{y}{b}right)^2=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2126.png"
 ALT="$\displaystyle \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 = 1$">|; 

$key = q/displaystylep_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5190.png"
 ALT="$\displaystyle p_r$">|; 

$key = q/{rmMa}_infty>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5917.png"
 ALT="$ {\rm Ma}_\infty &gt; 1$">|; 

$key = q/displaystyle{bfcurl},{bfA}=left(frac{partialA_z}{partialy}-frac{partialA_y}{part,frac{partialA_y}{partialx}-frac{partialA_x}{partialy}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6645.png"
 ALT="$\displaystyle {\bf curl} {\bf A} = \left(\frac{\partial A_z}{\partial y}- \fra...
...x},  \frac{\partial A_y}{\partial x}- \frac{\partial A_x} {\partial y}\right),$">|; 

$key = q/displaystyle{bfa}+({bfb}+{bfc})=({bfa}+{bfb})+{bfc}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6236.png"
 ALT="$\displaystyle {\bf a} + ({\bf b} + {\bf c}) = ({\bf a} + {\bf b}) + {\bf c}.$">|; 

$key = q/displaystyle=-frac{3,mu,V}{2,a},4pi,a^{,2}+2pi,a^{,2}int_0^pileft(-p_0+rho,g,a,costhetaright)costheta,sintheta,dtheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="491" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3807.png"
 ALT="$\displaystyle = -\frac{3 \mu V}{2 a} 4\pi a^{ 2} + 2\pi a^{ 2}\int_0^\pi\left(-p_0+\rho g a \cos\theta\right)\cos\theta \sin\theta d\theta,$">|; 

$key = q/displaystyle=-frac{1}{2},{mitOmega}^{,2},r^{,2},sin^2theta=chi_0(r)+chi_2(r,theta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4378.png"
 ALT="$\displaystyle = - \frac{1}{2} {\mit\Omega}^{ 2} r^{ 2} \sin^2\theta =\chi_0(r) + \chi_2(r,\theta),$">|; 

$key = q/displaystyleU=-frac{3}{5-n},frac{G,M_ast^{,2}}{R},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5283.png"
 ALT="$\displaystyle U= - \frac{3}{5-n} \frac{G M_\ast^{ 2}}{R},
$">|; 

$key = q/displaystylealpha_0=frac{y_U(x_L)-y_U(x_R)}{x_R-x_L}=frac{y_L(x_L)-y_L(x_R)}{x_R-x_L}=frac{y_C(x_L)-y_C(x_R)}{x_R-x_L}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5827.png"
 ALT="$\displaystyle \alpha_0 = \frac{y_U(x_L)-y_U(x_R)}{x_R-x_L}= \frac{y_L(x_L)-y_L(x_R)}{x_R-x_L}=\frac{y_C(x_L)-y_C(x_R)}{x_R-x_L}.$">|; 

$key = q/displaystyle=frac{2}{a_1^{,2},sin^3gamma}left[-frac{E(gamma,alpha)}{cos^2alpha}+frac{cosbeta,singamma}{cos^2alpha,cosgamma}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1005.png"
 ALT="$\displaystyle = \frac{2}{a_1^{ 2} \sin^3\gamma}\left[-\frac{E(\gamma,\alpha)}{\cos^2\alpha} + \frac{\cos\beta \sin\gamma}{\cos^2\alpha \cos\gamma}\right],$">|; 

$key = q/displaystylephi(x,y)=f(x-lambda,y)+g(x+lambda,y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5987.png"
 ALT="$\displaystyle \phi(x,y)= f(x-\lambda y) + g(x+\lambda y),$">|; 

$key = q/displaystylefrac{f}{(1+f_z^{,2})^{1slash2}}=frac{f^{,2}}{2,a}+C,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1239.png"
 ALT="$\displaystyle \frac{f}{(1+f_z^{ 2})^{1/2}} = \frac{f^{ 2}}{2 a} + C,$">|; 

$key = q/theta=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4287.png"
 ALT="$ \theta=\pi/2$">|; 

$key = q/-v,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5679.png"
 ALT="$ -v {\bf e}_y$">|; 

$key = q/displaystylepsi(x,y)=-left[frac{2,nu,U_0,x^{,m+1}}{m+1}right]^{1slash2}f(eta)=-U,m}left[frac{2,nu}{(m+1),U_0,x^{,m-1}}right]^{1slash2}f(eta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="592" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2918.png"
 ALT="$\displaystyle \psi(x,y) = -\left[\frac{2 \nu U_0 x^{ m+1}}{m+1}\right]^{1/2...
...)=-U_0 x^{ m}\left[\frac{2 \nu}{(m+1) U_0 x^{ m-1}}\right]^{1/2} f(\eta),$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter07slashconformal.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="505" HEIGHT="278" BORDER="0"
 SRC="|."$dir".q|img2701.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter07/conformal.eps}}
\end{figure}">|; 

$key = q/displaystyle=({bfa}times{bfb}cdot{bfd}),{bfc}-({bfa}times{bfb}cdot{bfc}),{bfd},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6680.png"
 ALT="$\displaystyle = ({\bf a}\times {\bf b}\cdot{\bf d}) {\bf c} - ({\bf a}\times {\bf b}\cdot{\bf c}) {\bf d},$">|; 

$key = q/xslashLstackrel{_{normalsize>}}{_{normalsizesim}}9times10^4slash{rmRe};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3070.png"
 ALT="$ x/L \stackrel {_{\normalsize &gt;}}{_{\normalsize\sim}}9\times 10^4/{\rm Re}$">|; 

$key = q/{bfS}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1537.png"
 ALT="$ {\bf S}_1$">|; 

$key = q/displaystyleC_psimeq-left[frac{2,u'}{U}+(1-{rmMa}_infty^{,2})left(frac{u'}{U}right)^{,2}+left(frac{v'}{U}right)^2right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5932.png"
 ALT="$\displaystyle C_p \simeq -\left[\frac{2 u'}{U}+(1-{\rm Ma}_\infty^{ 2})\left(\frac{u'}{U}\right)^{ 2}+ \left(\frac{v'}{U}\right)^2\right].$">|; 

$key = q/displaystyle-left(frac{partial}{partialt},D+2,{mitOmega},frac{partial}{partialphntheta,frac{partial}{partialtheta}right)skew{5}dot{mitPhi}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4669.png"
 ALT="$\displaystyle -\left(\frac{\partial}{\partial t} D +2 {\mit\Omega} \frac{\pa...
...,D-\sin\theta \frac{\partial}{\partial\theta}\right)\skew{5}\dot{\mit\Phi} =0.$">|; 

$key = q/displaystylefrac{A}{A_1}=frac{1}{{rmMa}}left[1+left(frac{gamma-1}{gamma+1}right)eft({rmMa}^{,2}-1right)right]^{,(gamma+1)slash[2,(gamma-1)]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="412" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5415.png"
 ALT="$\displaystyle \frac{A}{A_1} = \frac{1}{{\rm Ma}}\left[1+\left(\frac{\gamma-1}{\...
...a+1}\right)\left({\rm Ma}^{ 2}-1\right)\right]^{ (\gamma+1)/[2 (\gamma-1)]}.$">|; 

$key = q/x<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5444.png"
 ALT="$ x&lt;0$">|; 

$key = q/displaystyle({bfv}cdotnabla),{bfv}=nabla(v^{,2}slash2)-{bfv}times;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1563.png"
 ALT="$\displaystyle ({\bf v}\cdot\nabla) {\bf v} =\nabla(v^{ 2}/2) -{\bf v}\times$">|; 

$key = q/lambdarightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2226.png"
 ALT="$ \lambda\rightarrow 0$">|; 

$key = q/yslashx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3234.png"
 ALT="$ y/x$">|; 

$key = q/displaystylel=left[frac{gamma}{rho,g,(s-1)}right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4181.png"
 ALT="$\displaystyle l = \left[\frac{\gamma}{\rho g (s-1)}\right]^{1/2}.$">|; 

$key = q/displaystylea_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6285.png"
 ALT="$\displaystyle a_{x'}$">|; 

$key = q/displaystylewidehat{L};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img897.png"
 ALT="$\displaystyle \widehat{L}$">|; 

$key = q/displaystylep_0=rho,U_0^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2890.png"
 ALT="$\displaystyle p_0 = \rho U_0^{ 2}.$">|; 

$key = q/Q,drightarrowD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2450.png"
 ALT="$ Q d\rightarrow D$">|; 

$key = q/d=ctmspace+thinmuskip{.1667em}qopnamerelaxo{tan}mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$ d=c\tmspace +\thinmuskip {.1667em}\qopname \relax o{tan}\mu $">|; 

$key = q/lrightarrowl+{calO}(deltay^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7305.png"
 ALT="$ l\rightarrow l + {\cal O}(\delta y^{ 2})$">|; 

$key = q/displaystyle=-frac{1}{r^{,2},sintheta},frac{partialpsi}{partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3733.png"
 ALT="$\displaystyle = - \frac{1}{r^{ 2} \sin\theta} \frac{\partial\psi}{\partial \theta},$">|; 

$key = q/displaystyle=c,coshxi,coseta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2796.png"
 ALT="$\displaystyle = c \cosh\xi \cos\eta,$">|; 

$key = q/displaystyle=frac{g,sinalpha}{2,nu},y,(2,h-y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3871.png"
 ALT="$\displaystyle =\frac{g \sin\alpha}{2 \nu} y (2 h-y),$">|; 

$key = q/displaystylefrac{u(x,y)}{U(x)}=f(eta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3152.png"
 ALT="$\displaystyle \frac{u(x,y)}{U(x)} = f(\eta),$">|; 

$key = q/displaystyleh_T(x,,y)=h(x_0,,y_0)+alpha,(x-x_0)+beta,(y-y_0).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="382" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6531.png"
 ALT="$\displaystyle h_T(x, y)= h(x_0, y_0) + \alpha (x-x_0)+\beta (y-y_0).$">|; 

$key = q/displaystyle{bfF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3442.png"
 ALT="$\displaystyle {\bf F}$">|; 

$key = q/displaystylef(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3162.png"
 ALT="$\displaystyle f(1)$">|; 

$key = q/p=p_0-(1slash6),rho,V^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2823.png"
 ALT="$ p=p_0-(1/6) \rho V^{ 2}$">|; 

$key = q/displaystyle{mitPsi}(x_1,x_2,x_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7290.png"
 ALT="$\displaystyle {\mit\Psi}(x_1,x_2,x_3)$">|; 

$key = q/nablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7066.png"
 ALT="$ \nabla\phi$">|; 

$key = q/displaystyle=-frac{b}{2},cosphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3483.png"
 ALT="$\displaystyle =-\frac{b}{2} \cos\phi,$">|; 

$key = q/ngg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5052.png"
 ALT="$ n\gg 1$">|; 

$key = q/mu=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2577.png"
 ALT="$ \mu=\cos\theta$">|; 

$key = q/displaystyle=nabla^{,2}v_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="$\displaystyle = \nabla^{ 2} v_i.$">|; 

$key = q/displaystylephi(x,y)=frac{U,epsilon}{m},cos(alpha,x),frac{cosh[m,alpha,(b-y)]}{sinh(m,alpha,b)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6218.png"
 ALT="$\displaystyle \phi(x,y) = \frac{U \epsilon}{m} \cos(\alpha x) \frac{\cosh[m \alpha (b-y)]}{\sinh(m \alpha b)}.
$">|; 

$key = q/omega_varphi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2698.png"
 ALT="$ \omega_\varphi=0$">|; 

$key = q/displaystyle={rmi}^{,n+n'},delta_{n,,n'}^{,m,,m'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4997.png"
 ALT="$\displaystyle = {\rm i}^{ n+n'} \delta_{n, n'}^{ m, m'},$">|; 

$key = q/displaystyleD,{mitPhi}_r+lambda_r,{mitPhi}_r=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4805.png"
 ALT="$\displaystyle D {\mit\Phi}_r +\lambda_r {\mit\Phi}_r=0,$">|; 

$key = q/H(k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3722.png"
 ALT="$ H(k)$">|; 

$key = q/chi({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4452.png"
 ALT="$ \chi({\bf r})$">|; 

$key = q/displaystyleleft[frac{partial}{partialt}+(u+c),frac{partial}{partialx}right](u+c)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5665.png"
 ALT="$\displaystyle \left[\frac{\partial}{\partial t} + (u+c) \frac{\partial}{\partial x}\right](u+c) = 0.
$">|; 

$key = q/displaystyleI_n=pi,frac{sin(n,phi)}{sinphi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3543.png"
 ALT="$\displaystyle I_n = \pi \frac{\sin(n \phi)}{\sin\phi}.
$">|; 

$key = q/(+);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$ (+)$">|; 

$key = q/displaystyleeta_1-eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6158.png"
 ALT="$\displaystyle \eta_1-\eta$">|; 

$key = q/2,{mitOmega}-3,omega_1+omega_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4534.png"
 ALT="$ 2 {\mit\Omega}-3 \omega_1+\omega_2$">|; 

$key = q/displaystylefrac{b}{a}>left(1-frac{1}{s}right)^{1slash3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1040.png"
 ALT="$\displaystyle \frac{b}{a}&gt; \left(1-\frac{1}{s}\right)^{1/3}.
$">|; 

$key = q/displaystyleH,V=(H+h-d),v.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1454.png"
 ALT="$\displaystyle H V = (H+h-d) v.$">|; 

$key = q/({bfA}cdotnabla),{bfB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7080.png"
 ALT="$ ({\bf A}\cdot\nabla) {\bf B}$">|; 

$key = q/displaystylephantom{=}+frac{1}{2}left(frac{1}{r},frac{partialv_r}{partialtheta}+hi}+frac{partialv_varphi}{partialr}-frac{v_varphi}{r}right)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="506" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img515.png"
 ALT="$\displaystyle \phantom{=}+\frac{1}{2}\left(\frac{1}{r} \frac{\partial v_r}{\pa...
...ial \varphi}+\frac{\partial v_\varphi}{\partial r}-\frac{v_\varphi}{r}\right)^2$">|; 

$key = q/u=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img936.png"
 ALT="$ u=0$">|; 

$key = q/2,Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2167.png"
 ALT="$ 2 Q$">|; 

$key = q/v_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4009.png"
 ALT="$ v_p$">|; 

$key = q/displaystyle2,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1289.png"
 ALT="$\displaystyle 2 a$">|; 

$key = q/0<gammaslash(p_0,alpha)leq0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1326.png"
 ALT="$ 0&lt;\gamma/(p_0 \alpha)\leq 0.5$">|; 

$key = q/displaystyle=frac{k,g,(sigma-rho)}{sigma,coth(k,d)+rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4211.png"
 ALT="$\displaystyle = \frac{k g (\sigma-\rho)}{\sigma \coth(k d)+\rho},$">|; 

$key = q/displaystyle=chi+frac{kappa,M}R,nabla^{,2}left(frac{p}{rho}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="$\displaystyle =\chi + \frac{\kappa M}R \nabla^{ 2}\left(\frac{p}{\rho}\right),$">|; 

$key = q/{bfn}=left.{bfe}_rright|_{r=R};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1114.png"
 ALT="$ {\bf n}= \left.{\bf e}_r\right\vert _{r=R}$">|; 

$key = q/displaystylev_x(x,infty)=U(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2905.png"
 ALT="$\displaystyle v_x(x,\infty)= U(x)$">|; 

$key = q/E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5295.png"
 ALT="$ E$">|; 

$key = q/displaystyle{bfv}=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6650.png"
 ALT="$\displaystyle {\bf v} =$">|; 

$key = q/A,kappa^{,2},sslashV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1047.png"
 ALT="$ A \kappa^{ 2} s/V$">|; 

$key = q/displaystylephi(r,theta,t)=V_x(t),frac{a^{,2}}{r},costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1803.png"
 ALT="$\displaystyle \phi(r,\theta,t) = V_x(t) \frac{a^{ 2}}{r} \cos\theta.$">|; 

$key = q/displaystylep(r,z)=p_0+rholeft(frac{1}{2},omega^{,2},r^{,2}-g,zright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img820.png"
 ALT="$\displaystyle p(r,z) = p_0 + \rho\left(\frac{1}{2} \omega^{ 2} r^{ 2}- g z\right),$">|; 

$key = q/v_r=a,k,r^{,n},exp[-k,(n+1),theta];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1919.png"
 ALT="$ v_r=a k r^{ n} \exp[-k (n+1) \theta]$">|; 

$key = q/displaystyleX';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4741.png"
 ALT="$\displaystyle X'$">|; 

$key = q/displaystylefrac{dF}{dz}=frac{partialphi}{partialx}+{rmi},frac{partialpsi}{partialx}=-v_x+{rmi},v_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1991.png"
 ALT="$\displaystyle \frac{dF}{dz} = \frac{\partial\phi}{\partial x} + {\rm i} \frac{\partial\psi}{\partial x} = -v_x+{\rm i} v_y.$">|; 

$key = q/|bar{F}|rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2290.png"
 ALT="$ \vert\bar{F}\vert\rightarrow \infty$">|; 

$key = q/(b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2037.png"
 ALT="$ (b$">|; 

$key = q/displaystyleC_D=frac{f}{rho,V^{,2},b}=frac{2pi}{pi+4}=0.88.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2285.png"
 ALT="$\displaystyle C_D=\frac{f}{\rho V^{ 2} b} = \frac{2\pi}{\pi+4}= 0.88.$">|; 

$key = q/displaystyle=muleft(frac{1}{r,sintheta},frac{partialv_theta}{partialvarphi}+fractialv_varphi}{partialtheta}-frac{cottheta,v_varphi}{r}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img501.png"
 ALT="$\displaystyle = \mu\left(\frac{1}{r \sin\theta} \frac{\partial v_\theta}{\par...
...rac{\partial v_\varphi}{\partial\theta}-\frac{\cot\theta v_\varphi}{r}\right),$">|; 

$key = q/overline{mu}slashmu=10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$ \overline {\mu }/\mu =10$">|; 

$key = q/P_-P_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3419.png"
 ALT="$ P_-P_+$">|; 

$key = q/A=(V-c),zeta_0slashtanh(k,d);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4091.png"
 ALT="$ A=(V-c) \zeta_0/\tanh(k d)$">|; 

$key = q/N_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4563.png"
 ALT="$ N_2$">|; 

$key = q/displaystylefrac{partialxi}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6011.png"
 ALT="$\displaystyle \frac{\partial\xi}{\partial y}$">|; 

$key = q/gamma>2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1775.png"
 ALT="$ \gamma&gt;2$">|; 

$key = q/displaystylelambda=-left(frac{k,V}{bar{F}}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2259.png"
 ALT="$\displaystyle \lambda =-\left(\frac{k V}{\bar{F}}\right)^{1/2},$">|; 

$key = q/displaystyleC_L=frac{L}{q_1,c},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5797.png"
 ALT="$\displaystyle C_L = \frac{L}{q_1 c},$">|; 

$key = q/z_G=cslash2-s,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img671.png"
 ALT="$ z_G= c/2-s c$">|; 

$key = q/displaystylefrac{p}{p_c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5207.png"
 ALT="$\displaystyle \frac{p}{p_c}$">|; 

$key = q/A_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img609.png"
 ALT="$ A_0$">|; 

$key = q/displaystylefrac{1}{xi^{,2}}frac{d}{dxi}!left(xi^{,2},frac{dtheta}{dxi}right)=-theta^{,n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5273.png"
 ALT="$\displaystyle \frac{1}{\xi^{ 2}}\frac{d}{d\xi}\!\left(\xi^{ 2} \frac{d\theta}{d\xi}\right)=-\theta^{ n},
$">|; 

$key = q/{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6334.png"
 ALT="$ {\bf S}$">|; 

$key = q/alpha'=1-alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1882.png"
 ALT="$ \alpha'=1-\alpha$">|; 

$key = q/mgeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1874.png"
 ALT="$ m\geq 0$">|; 

$key = q/rho,{mitGamma},{mitGamma}'slash(2pi,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2438.png"
 ALT="$ \rho {\mit\Gamma} {\mit\Gamma}'/(2\pi r)$">|; 

$key = q/omega_varphi=omega_varphi(varpi,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2565.png"
 ALT="$ \omega_\varphi=\omega_\varphi(\varpi,z)$">|; 

$key = q/displaystylerightarrowbeta_{-n,,-n'}^{,-m,,-m'}=-int_{mitOmega}frac{costheta}{sic{partial{mitPsi}_{n'}^{,m'}}{partialtheta}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="485" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4947.png"
 ALT="$\displaystyle \rightarrow \beta_{-n, -n'}^{ -m, -m'}= -\int_{\mit\Omega}\fra...
...}  \frac{\partial{\mit\Psi}_{n'}^{ m'}}{\partial\theta}\right) d{\mit\Omega},$">|; 

$key = q/Dslash(q_1,c);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5857.png"
 ALT="$ D/(q_1 c)$">|; 

$key = q/nablacdot{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6582.png"
 ALT="$ \nabla\cdot{\bf A}$">|; 

$key = q/h_n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4445.png"
 ALT="$ h_n'$">|; 

$key = q/displaystyle=frac{1}{2}left(frac{partialA_theta}{partialz}+frac{1}{r}frac{partialA_z}{partialtheta}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7158.png"
 ALT="$\displaystyle =\frac{1}{2}\left(\frac{\partial A_\theta}{\partial z} + \frac{1}{r}\frac{\partial A_z}{\partial \theta}\right).$">|; 

$key = q/displaystylebeta_{-r,,s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4870.png"
 ALT="$\displaystyle \beta_{-r, s}$">|; 

$key = q/displaystyletan(beta-theta)=frac{u_2}{v}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5703.png"
 ALT="$\displaystyle \tan(\beta-\theta)= \frac{u_2}{v}.$">|; 

$key = q/displaystyle=left(frac{1}{gamma-1}right){calR},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5385.png"
 ALT="$\displaystyle =\left(\frac{1}{\gamma-1}\right){\cal R},$">|; 

$key = q/V_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3102.png"
 ALT="$ V_y$">|; 

$key = q/F_{21}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1475.png"
 ALT="$ F_{21}=0$">|; 

$key = q/displaystyleoint;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1541.png"
 ALT="$\displaystyle \oint$">|; 

$key = q/displaystyle={calR},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5612.png"
 ALT="$\displaystyle = {\cal R},$">|; 

$key = q/displaystylephantom{=}+{rmMa}_infty^{,2}left[(gamma-1),frac{u'}{U}+left(frac{gamright)left(frac{u'}{U}right)^2right]frac{partialv'}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="535" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5905.png"
 ALT="$\displaystyle \phantom{=}+ {\rm Ma}_\infty^{ 2}\left[(\gamma-1) \frac{u'}{U} ...
...ma-1}{2}\right)\left(\frac{u'}{U}\right)^2\right]\frac{\partial v'}{\partial y}$">|; 

$key = q/nabla^{,2}(pslashrho)rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5114.png"
 ALT="$ \nabla^{ 2}(p/\rho)\rightarrow 0$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter15slashdetach.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="565" HEIGHT="257" BORDER="0"
 SRC="|."$dir".q|img5779.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter15/detach.eps}}
\end{figure}">|; 

$key = q/displaystyleU,frac{dq}{dxi}simeq-frac{partialskew{3}bar{phi}(xi,0)}{partialeta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6029.png"
 ALT="$\displaystyle U \frac{dq}{d\xi}\simeq- \frac{\partial\skew{3}\bar{\phi}(\xi,0)}{\partial \eta}.$">|; 

$key = q/omega=sqrt{2},gslashv_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3570.png"
 ALT="$ \omega= \sqrt{2} g/v_0$">|; 

$key = q/displaystyle{bfV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3297.png"
 ALT="$\displaystyle {\bf V}$">|; 

$key = q/displaystyleF(lambda);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3195.png"
 ALT="$\displaystyle F(\lambda)$">|; 

$key = q/2,{mitOmega}-2,omega_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4569.png"
 ALT="$ 2 {\mit\Omega}-2 \omega_E$">|; 

$key = q/displaystylealpha_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img860.png"
 ALT="$\displaystyle \alpha_i$">|; 

$key = q/displaystylep(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3872.png"
 ALT="$\displaystyle p(y)$">|; 

$key = q/displaystylex_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5677.png"
 ALT="$\displaystyle x_\ast$">|; 

$key = q/displaystyleskew{3}ddot{r}=-omega^{,2},(r_H-r_G)<0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img831.png"
 ALT="$\displaystyle \skew{3}\ddot{r} = -\omega^{ 2} (r_H-r_G) &lt;0.$">|; 

$key = q/displaystyley_C(x)=y_C(x_L)-alpha_0,(x-x_L)-alpha_C(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5828.png"
 ALT="$\displaystyle y_C(x) = y_C(x_L) -\alpha_0 (x-x_L) -\alpha_C(x),$">|; 

$key = q/F'(-z)=F'(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3255.png"
 ALT="$ F'(-z)=F'(z)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter09slashair.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="339" BORDER="0"
 SRC="|."$dir".q|img3302.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/air.eps}}
\end{figure}">|; 

$key = q/displaystyleH=h+frac{v^{,2}}{2,g}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1425.png"
 ALT="$\displaystyle H = h + \frac{v^{ 2}}{2 g}.$">|; 

$key = q/={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1569.png"
 ALT="$ ={\bf0}$">|; 

$key = q/(1-e),ln[1slash(1-e)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2814.png"
 ALT="$ (1-e) \ln[1/(1-e)]$">|; 

$key = q/pi>theta_+>theta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3365.png"
 ALT="$ \pi&gt;\theta_+&gt;\theta_0$">|; 

$key = q/displaystyleint_{mitOmega}{mitPhi}_r,D,{mitPhi}_r,d{mitOmega}=-lambda_rint_{mitOfrac{partial{mitPhi}_r}{partialphi}right)^2right]d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="585" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4810.png"
 ALT="$\displaystyle \int_{\mit\Omega} {\mit\Phi}_r D {\mit\Phi}_r d{\mit\Omega} = ...
...heta} \frac{\partial {\mit\Phi}_r}{\partial\phi}\right)^2\right]d{\mit\Omega},$">|; 

$key = q/displaystylefrac{dF}{dz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1961.png"
 ALT="$\displaystyle \frac{dF}{dz}$">|; 

$key = q/displaystyle=frac{partialpsi}{partialr},{bfe}_r+frac{1}{r}frac{partialpsi}{partialtheta},{bfe}_theta+frac{partialpsi}{partialz},{bfe}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7129.png"
 ALT="$\displaystyle = \frac{\partial \psi}{\partial r} {\bf e}_r + \frac{1}{r}\frac{...
...{\partial\theta} {\bf e}_\theta + \frac{\partial \psi}{\partial z} {\bf e}_z,$">|; 

$key = q/tau=W,lambda,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img624.png"
 ALT="$ \tau=W \lambda \sin\theta$">|; 

$key = q/dzeta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2079.png"
 ALT="$ d\zeta'$">|; 

$key = q/displaystylekappa_{calS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5600.png"
 ALT="$\displaystyle \kappa_{\cal S}$">|; 

$key = q/displaystyle=(1-omega,z^{-1})^2+(1-omega,z)^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img785.png"
 ALT="$\displaystyle =(1-\omega z^{-1})^2 + (1-\omega z)^2,$">|; 

$key = q/displaystyle=-r,{mitOmega}^{,2},{bfe}_r,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3632.png"
 ALT="$\displaystyle =-r {\mit\Omega}^{ 2} {\bf e}_r,$">|; 

$key = q/displaystyle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2639.png"
 ALT="$\displaystyle ,$">|; 

$key = q/displaystylea_{rmmin}simeqleft(frac{rho}{p_infty}right)^{1slash2}left(frac{mitGamma}{2pi}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1726.png"
 ALT="$\displaystyle a_{\rm min}\simeq \left(\frac{\rho}{p_\infty}\right)^{1/2}\left(\frac{\mit\Gamma}{2\pi}\right).$">|; 

$key = q/displaystyle{bf0}=nablap+rho,nabla({mitPsi}+{mitPsi}'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img812.png"
 ALT="$\displaystyle {\bf0} = \nabla p + \rho \nabla({\mit\Psi}+{\mit\Psi}') ,$">|; 

$key = q/displaystylefrac{dzeta}{dz}=K',(zeta-b),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2173.png"
 ALT="$\displaystyle \frac{d\zeta}{dz} = K' (\zeta-b),$">|; 

$key = q/displaystyleL=I_{33},omega=frac{1}{5},M,(a_1^{,2}+a_2^{,2}),omega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img853.png"
 ALT="$\displaystyle L = I_{33} \omega=\frac{1}{5} M (a_1^{ 2}+a_2^{ 2}) \omega,$">|; 

$key = q/displaystylefrac{dF}{dz}={rmi}left[V,sin(alpha+theta)+frac{{mitGamma}}{pi,(c+del]frac{(c+delta)}{left(delta,costheta+{rmi},c,sinthetaright)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3313.png"
 ALT="$\displaystyle \frac{dF}{dz} = {\rm i}\left[V \sin(\alpha+\theta) + \frac{{\mit...
...ight]\frac{(c+\delta)}{\left(\delta \cos\theta+{\rm i} c \sin\theta\right)}.$">|; 

$key = q/displaystylepsi(x,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2909.png"
 ALT="$\displaystyle \psi(x,0)$">|; 

$key = q/dpsi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2406.png"
 ALT="$ d\psi=0$">|; 

$key = q/C_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5999.png"
 ALT="$ C_p$">|; 

$key = q/sinalpha_0simeqalpha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5804.png"
 ALT="$ \sin\alpha_0\simeq \alpha_0$">|; 

$key = q/lambda_c=2pi,l=1.7times10^{-2},{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4150.png"
 ALT="$ \lambda_c=2\pi l=1.7\times 10^{-2} {\rm m}$">|; 

$key = q/displaystylefrac{x}{l}=int_phi^{pislash2}frac{k,cos(2,phi)}{(1-k^{,2},sin^2phi)^{1slash2}},dphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1173.png"
 ALT="$\displaystyle \frac{x}{l} = \int_\phi^{\pi/2} \frac{k \cos(2 \phi)}{(1-k^{ 2} \sin^2\phi)^{1/2}} d\phi,$">|; 

$key = q/1<{rmMa}_1<[4slash(3-gamma)]^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6171.png"
 ALT="$ 1&lt;{\rm Ma}_1&lt;
[4/(3-\gamma)]^{1/2}$">|; 

$key = q/displaystyleint_P^Qleft(y,dx+x^{,3},dyright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6456.png"
 ALT="$\displaystyle \int_P^Q \left( y dx + x^{ 3} dy\right)$">|; 

$key = q/displaystyle=c,sinhxi,sineta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2797.png"
 ALT="$\displaystyle = c \sinh\xi \sin\eta.$">|; 

$key = q/psi({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7132.png"
 ALT="$ \psi({\bf r})$">|; 

$key = q/{bfv}_{mitSigma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3414.png"
 ALT="$ {\bf v}_{\mit\Sigma}$">|; 

$key = q/displaystyle=int_{mitOmega}left(frac{partial{mitPsi}_r''}{partialtheta},frac{parialphi},frac{partial{mitPsi}_s}{partialphi}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4877.png"
 ALT="$\displaystyle =\int_{\mit\Omega} \left(\frac{\partial {\mit\Psi}_r''}{\partial\...
...}{\partial\phi} \frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega},$">|; 

$key = q/displaystyleF(z)=-frac{Q}{2pi},lnz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1976.png"
 ALT="$\displaystyle F(z)=- \frac{Q}{2\pi} \ln z,$">|; 

$key = q/lambda=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3213.png"
 ALT="$ \lambda=0$">|; 

$key = q/displaystylep_2-p_3=frac{4,epsilon}{sqrt{{rmMa}_1^{,2}-1}},q_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5812.png"
 ALT="$\displaystyle p_2-p_3 = \frac{4 \epsilon}{\sqrt{{\rm Ma}_1^{ 2}-1}} q_1,$">|; 

$key = q/displaystylelambda=cosalpha+zeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2317.png"
 ALT="$\displaystyle \lambda =\cos\alpha+\zeta,$">|; 

$key = q/displaystylerightarrowbeta_{-n,,n'}^{,-m,,m'}=-int_{mitOmega}costhetaleft(frac{prac{partial{mitPhi}_{n'}^{,m'}}{partialphi}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="519" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4945.png"
 ALT="$\displaystyle \rightarrow \beta_{-n, n'}^{ -m, m'}= -\int_{\mit\Omega}\cos\t...
...hi}  \frac{\partial{\mit\Phi}_{n'}^{ m'}}{\partial\phi}\right) d{\mit\Omega},$">|; 

$key = q/displaystylelanglecdotsrangle=int_0^{2pi}(cdots),frac{d(k,x)}{2pi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3974.png"
 ALT="$\displaystyle \langle \cdots\rangle = \int_0^{2\pi} (\cdots) \frac{d(k x)}{2\pi}$">|; 

$key = q/displaystylelambda_n^{,m}=n,(n+1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4938.png"
 ALT="$\displaystyle \lambda_n^{ m} = n (n+1),$">|; 

$key = q/alpha_x=pislash2-alpha_y=alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6340.png"
 ALT="$ \alpha_x = \pi/2-\alpha_y=\alpha$">|; 

$key = q/displaystyleQ=frac{pi,G}{4,mu},frac{a^{,3},b^{,3}}{a^{,2}+b^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3882.png"
 ALT="$\displaystyle Q = \frac{\pi G}{4 \mu} \frac{a^{ 3} b^{ 3}}{a^{ 2} + b^{ 2}}.
$">|; 

$key = q/displaystylefrac{d^{,2}delta_1}{dt^{,2}}=-frac{g}{delta_0},delta_1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img616.png"
 ALT="$\displaystyle \frac{d^{ 2} \delta_1}{dt^{ 2}} = - \frac{g}{\delta_0} \delta_1.$">|; 

$key = q/displaystyle=int_{mitOmega}left(frac{partial{mitPhi}_r'}{partialtheta},frac{partialphi},frac{partial{mitPhi}_s}{partialphi}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4866.png"
 ALT="$\displaystyle =\int_{\mit\Omega} \left(\frac{\partial {\mit\Phi}_r'}{\partial\t...
...}{\partial\phi} \frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega},$">|; 

$key = q/displaystylephi(r,theta)=-V,r,P_1(costheta)+beta_1,r^{,-2},P_1(costheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2642.png"
 ALT="$\displaystyle \phi(r,\theta) =-V r P_1(\cos\theta) + \beta_1 r^{ -2} P_1(\cos\theta).$">|; 

$key = q/overline{rho}slashrho>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3813.png"
 ALT="$ \overline{\rho}/\rho&gt;1$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{AppendixAslashfigA.06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="167" BORDER="0"
 SRC="|."$dir".q|img6323.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{AppendixA/figA.06.eps}}
\end{figure}">|; 

$key = q/displaystyle{bfr}'=r',sintheta',cosvarphi',{bfe}_r+r',sintheta',sinvarphi',{bfe}_theta+r',costheta',{bfe}_varphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4255.png"
 ALT="$\displaystyle {\bf r}' = r' \sin\theta' \cos\varphi' {\bf e}_r + r' \sin\theta' \sin\varphi' {\bf e}_\theta + r' \cos\theta' {\bf e}_\varphi.$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter11slashwedge.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img4058.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter11/wedge.eps}}
\end{figure}">|; 

$key = q/displaystyle=2,muleft[left(frac{partialv_r}{partialr}right)^2+left(frac{1}{r},frlvarphi}+frac{v_r}{r}+frac{cottheta,v_theta}{r}right)^2right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="558" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img514.png"
 ALT="$\displaystyle =2 \mu\left[\left(\frac{\partial v_r}{\partial r}\right)^2+\left...
...{\partial \varphi}+\frac{v_r}{r}+\frac{\cot\theta v_\theta}{r}\right)^2\right.$">|; 

$key = q/displaystyle=cos^2gammaleft[F(gamma,alpha)-2,E(gamma,alpha)+E(gamma,alpha),frac{ha}-frac{sinalpha,sinbeta,cosbeta,cosgamma}{cos^2alpha}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="585" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1010.png"
 ALT="$\displaystyle =\cos^2\gamma\left[F(\gamma,\alpha)-2 E(\gamma,\alpha) + E(\gamm...
...pha} - \frac{\sin\alpha \sin\beta \cos\beta \cos\gamma}{\cos^2\alpha}\right]$">|; 

$key = q/displaystylea_i';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6848.png"
 ALT="$\displaystyle a_i'$">|; 

$key = q/displaystylephi(r,theta)=frac{D}{2pi},frac{cos(theta-theta_0)}{r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1701.png"
 ALT="$\displaystyle \phi(r,\theta) = \frac{D}{2\pi} \frac{\cos(\theta-\theta_0)}{r}.$">|; 

$key = q/displaystylefrac{dF}{dz},dbar{F}=frac{dF}{dz},dF=left(frac{dF}{dz}right)^{,2}dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2408.png"
 ALT="$\displaystyle \frac{dF}{dz} d\bar{F} = \frac{dF}{dz} dF = \left(\frac{dF}{dz}\right)^{ 2}dz,$">|; 

$key = q/displaystylefrac{partialz}{partialxi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2684.png"
 ALT="$\displaystyle \frac{\partial z}{\partial\xi}$">|; 

$key = q/displaystylefrac{partialv}{partialt}+f,u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5065.png"
 ALT="$\displaystyle \frac{\partial v}{\partial t} +f u$">|; 

$key = q/stackrel{displaystylerightarrow}{AC};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="30" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6701.png"
 ALT="$ \stackrel{\displaystyle \rightarrow}{AC}$">|; 

$key = q/displaystylepsileft(r,[3-alpha],pislash2right)=psileft(r,-[1-alpha],pislash2right)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1878.png"
 ALT="$\displaystyle \psi\left(r,[3-\alpha] \pi/2\right) =\psi\left(r,-[1-\alpha] \pi/2\right)=0.$">|; 

$key = q/v_y=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2872.png"
 ALT="$ v_y=0$">|; 

$key = q/displaystyle=sum_{i=1,3}A_i,B_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7085.png"
 ALT="$\displaystyle =\sum_{i=1,3} A_i B_i,$">|; 

$key = q/beta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2596.png"
 ALT="$ \beta_l$">|; 

$key = q/displaystyle=-nablaP+mu,nabla^{,2}{bfv}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3645.png"
 ALT="$\displaystyle = - \nabla P + \mu \nabla^{ 2}{\bf v}.$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{Chapter11slashrt.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="546" HEIGHT="226" BORDER="0"
 SRC="|."$dir".q|img4087.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{Chapter11/rt.eps}}
\end{figure}">|; 

$key = q/d{bfv}slashdt=D{bfv}slashDt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1556.png"
 ALT="$ d{\bf v}/dt = D{\bf v}/Dt$">|; 

$key = q/displaystylec_1^{,2}=frac{1}{2}left(frac{lambda}{lambda_c}+frac{lambda_c}{lambda}right)c_0^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4192.png"
 ALT="$\displaystyle c_1^{ 2} = \frac{1}{2}\left(\frac{\lambda}{\lambda_c}+\frac{\lambda_c}{\lambda}\right)c_0^{ 2}$">|; 

$key = q/displaystyleD,{mitPhi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4759.png"
 ALT="$\displaystyle D {\mit\Phi}$">|; 

$key = q/f'(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3178.png"
 ALT="$ f'(0)=0$">|; 

$key = q/displaystylefrac{v_s}{U_0}ggfrac{1}{{rmRe}^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3271.png"
 ALT="$\displaystyle \frac{v_s}{U_0}\gg \frac{1}{{\rm Re}^{1/2}},
$">|; 

$key = q/gammageqbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img930.png"
 ALT="$ \gamma\geq \beta$">|; 

$key = q/(partialuslashpartialy)_{y=0}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3144.png"
 ALT="$ (\partial u/\partial y)_{y=0}=0$">|; 

$key = q/displaystylek=left[frac{sinh(2,k,d)-2,k,d}{sinh(2,k,d)+2,k,d}right]^{1slash2}k_c,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4214.png"
 ALT="$\displaystyle k = \left[\frac{\sinh(2 k d)-2 k d}{\sinh(2 k d)+2 k d}\right]^{1/2} k_c,
$">|; 

$key = q/displaystyle{mitPhi}_{n>0}(r>a,theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4368.png"
 ALT="$\displaystyle {\mit\Phi}_{n&gt;0}(r&gt;a,\theta,\phi)$">|; 

$key = q/nabla^{,2}T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6616.png"
 ALT="$ \nabla^{ 2} T$">|; 

$key = q/displaystylep(r,theta)=p_0-rho,g,r,costheta+frac{3,mu,V,a}{2,r^{,2}},costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3793.png"
 ALT="$\displaystyle p(r,\theta) = p_0- \rho g r \cos\theta + \frac{3 \mu V a}{2 r^{ 2}} \cos\theta.$">|; 

$key = q/displaystylev(x,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3150.png"
 ALT="$\displaystyle v(x,0)$">|; 

$key = q/displaystyle=-2,costhetaleft(frac{A}{r^{,3}}+frac{B}{r}+C+D,r^{,2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3835.png"
 ALT="$\displaystyle = -2 \cos\theta\left(\frac{A}{r^{ 3}}+\frac{B}{r} + C+D r^{ 2}\right),$">|; 

$key = q/j=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$ j=3$">|; 

$key = q/lambda=e_{13}^{,2}slashz^{,2}-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img877.png"
 ALT="$ \lambda=e_{13}^{ 2}/z^{ 2}-1$">|; 

$key = q/partialslashpartialzsimd^{,-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4496.png"
 ALT="$ \partial/\partial z\sim d^{ -1}$">|; 

$key = q/+(3slash2),sin^2epsilon,,zeta_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4552.png"
 ALT="$ +(3/2) \sin^2\epsilon  \zeta_M$">|; 

$key = q/displaystyleK_{rmfluid}=int_{rmfluid}frac{1}{2},rho,v^{,2},dV=2piint_a^inftyfrac,V_x^{,2}left(frac{a}{r}right)^4,r,dr=frac{1}{2},m',V_x^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="510" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1824.png"
 ALT="$\displaystyle K_{\rm fluid} = \int_{\rm fluid}\frac{1}{2} \rho v^{ 2} dV =2...
...,\rho V_x^{ 2}\left(\frac{a}{r}\right)^4 r dr = \frac{1}{2} m' V_x^{ 2}.$">|; 

$key = q/nablaxicdotnablaeta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2688.png"
 ALT="$ \nabla\xi\cdot\nabla\eta=0$">|; 

$key = q/S_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4545.png"
 ALT="$ S_0$">|; 

$key = q/displaystylea_{ij}=lambda,delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6907.png"
 ALT="$\displaystyle a_{ij } =\lambda \delta_{ij}.$">|; 

$key = q/displaystyler=b,cosh(|z-z_0|slashb),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1258.png"
 ALT="$\displaystyle r = b \cosh(\vert z-z_0\vert/b),$">|; 

$key = q/{rmMa}_1,sinbetageq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5694.png"
 ALT="$ {\rm Ma}_1 \sin\beta\geq 1$">|; 

$key = q/{bfa}(t+deltat);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6405.png"
 ALT="$ {\bf a}(t+\delta t)$">|; 

$key = q/displaystylepsi(r,theta)=alpha_0+beta_0,lnr+sum_{m>0}(alpha_m,r^{,m}+beta_m,r^{-m}),sin[m,(theta-theta_m)],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="511" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1746.png"
 ALT="$\displaystyle \psi(r,\theta) = \alpha_0+ \beta_0 \ln r + \sum_{m&gt;0} (\alpha_m r^{ m}+\beta_m r^{-m})  \sin[m (\theta-\theta_m)],$">|; 

$key = q/displaystylesimeq3.2times10^{-1},V_0({rmm,s^{-1}})slash[L({rmm})]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="$\displaystyle \simeq 3.2\times 10^{-1} V_0({\rm m s^{-1}})/[L({\rm m})]^{1/2}.$">|; 

$key = q/0.92995;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img922.png"
 ALT="$ 0.92995$">|; 

$key = q/a_y=+g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1863.png"
 ALT="$ a_y=+g$">|; 

$key = q/displaystyle={mitPsi}_r';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4853.png"
 ALT="$\displaystyle = {\mit\Psi}_r'$">|; 

$key = q/displaystyle{bfa}cdot{bfb}=a_i,b_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6773.png"
 ALT="$\displaystyle {\bf a}\cdot{\bf b} = a_i b_i.$">|; 

$key = q/displaystyle=m^{,2},R,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1733.png"
 ALT="$\displaystyle =m^{ 2} R,$">|; 

$key = q/displaystylephi(r,theta)=frac{Q}{2pi},lnr.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1680.png"
 ALT="$\displaystyle \phi(r,\theta) = \frac{Q}{2\pi} \ln r.$">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="$ p$">|; 

$key = q/displaystylecdotd{bfS}=intnablacdot;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1542.png"
 ALT="$\displaystyle \cdot d{\bf S} = \int \nabla\cdot$">|; 

$key = q/c_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6858.png"
 ALT="$ c_{ij}$">|; 

$key = q/0.5leqgammaslash(p_0,alpha)<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1316.png"
 ALT="$ 0.5\leq \gamma/(p_0 \alpha) &lt; 1$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter09slashaircraft.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="465" HEIGHT="339" BORDER="0"
 SRC="|."$dir".q|img3519.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter09/aircraft.eps}}
\end{figure}">|; 

$key = q/displaystyle=int_{mitOmega}costhetaleft(frac{partial{mitPhi}_r}{partialtheta},frtialphi},frac{partial{mitPsi}_s}{partialphi}right)d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4878.png"
 ALT="$\displaystyle = \int_{\mit\Omega}\cos\theta\left(\frac{\partial{\mit\Phi}_r}{\p...
...r}{\partial\phi} \frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}$">|; 

$key = q/displaystyleleft.frac{p}{rho^{,gamma}}right|_{z_2}<left.frac{p}{rho^{,gamma}}right|_{z_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5168.png"
 ALT="$\displaystyle \left.\frac{p}{\rho^{ \gamma}}\right\vert _{z_2} &lt; \left.\frac{p}{\rho^{ \gamma}}\right\vert _{z_1},$">|; 

$key = q/int{bfA}cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6585.png"
 ALT="$ \int {\bf A}\cdot d{\bf S}$">|; 

$key = q/{rmMa}_inftygg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5918.png"
 ALT="$ {\rm Ma}_\infty \gg 1$">|; 

$key = q/displaystylep(r)=frac{kappa^{,2},r^{,2},rho}{2,a^{,4}}+p_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1910.png"
 ALT="$\displaystyle p(r) = \frac{\kappa^{ 2} r^{ 2} \rho}{2 a^{ 4}}+ p_0,
$">|; 

$key = q/phi=phi(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2571.png"
 ALT="$ \phi=\phi(r,\theta)$">|; 

$key = q/displaystylem_{rmadded}=pileft(a^{,2},sin^2alpha+b^{,2},cos^2alpharight)rho.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2515.png"
 ALT="$\displaystyle m_{\rm added}= \pi\left(a^{ 2} \sin^2\alpha+b^{ 2} \cos^2\alpha\right)\rho.
$">|; 

$key = q/displaystylenablacdot{bfn}=left.frac{1}{r^{,2}},frac{partialr^{,2}}{partialr}right|_{r=R}=frac{2}{R}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1115.png"
 ALT="$\displaystyle \nabla\cdot{\bf n} = \left.\frac{1}{r^{ 2}} \frac{\partial r^{ 2}}{\partial r}\right\vert _{r=R} = \frac{2}{R}.$">|; 

$key = q/displaystylerightarrowbeta_{n,,n'}^{,m,,m'}=-int_{mitOmega}frac{costheta}{sinthec{partial{mitPhi}_{n'}^{,m'}}{partialtheta}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="460" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4944.png"
 ALT="$\displaystyle \rightarrow \beta_{n, n'}^{ m, m'}= -\int_{\mit\Omega}\frac{\c...
...}  \frac{\partial{\mit\Phi}_{n'}^{ m'}}{\partial\theta}\right) d{\mit\Omega},$">|; 

$key = q/u'slashU;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5907.png"
 ALT="$ u'/U$">|; 

$key = q/displaystyleA=frac{b^{,2}}{S}=frac{4}{pi},frac{b}{c_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3512.png"
 ALT="$\displaystyle A = \frac{b^{ 2}}{S}= \frac{4}{\pi} \frac{b}{c_0}.$">|; 

$key = q/lambda_0=2pislashk_0=2pi,V^{,2}slashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4059.png"
 ALT="$ \lambda_0=2\pi/k_0=2\pi V^{ 2}/g$">|; 

$key = q/displaystyle{mitPhi}_n({bfr})rightarrow0[0.5ex];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4360.png"
 ALT="$\displaystyle {\mit\Phi}_n({\bf r})\rightarrow 0\ [0.5ex]$">|; 

$key = q/alpha_1,r,sin(theta-theta_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1752.png"
 ALT="$ \alpha_1 r \sin(\theta-\theta_1)$">|; 

$key = q/m+(1slash2),m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2674.png"
 ALT="$ m+(1/2) m'$">|; 

$key = q/alpha_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3534.png"
 ALT="$ \alpha_c$">|; 

$key = q/displaystylez+{rmi},varpi=f(xi+{rmi},eta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2683.png"
 ALT="$\displaystyle z+ {\rm i} \varpi = f(\xi+{\rm i} \eta),$">|; 

$key = q/partial^{,2}psislashpartialx^{,2}=partial^{,2}psislashpartialy^{,2}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1667.png"
 ALT="$ \partial^{ 2}\psi/\partial x^{ 2}=\partial^{ 2}\psi/\partial y^{ 2}=0$">|; 

$key = q/displaystyle=frac{p_2}{p_1},frac{rho_1}{rho_2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5488.png"
 ALT="$\displaystyle = \frac{p_2}{p_1} \frac{\rho_1}{\rho_2}.$">|; 

$key = q/%latex2htmlidmarker87763(sintheta)^{-1},partial(ref{e129c})slashpartialphi-(sintheta)^{-1},partial[sintheta(ref{e130c})slashpartialtheta];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4668.png"
 ALT="% latex2html id marker 87763
$ (\sin\theta)^{-1} \partial(\ref{e129c})/\partial\phi-(\sin\theta)^{-1} \partial[\sin\theta(\ref{e130c})/\partial\theta]$">|; 

$key = q/H_1>H_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1502.png"
 ALT="$ H_1&gt;H_2$">|; 

$key = q/displaystylepsi(r,theta)=sin^2theta,f(r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3765.png"
 ALT="$\displaystyle \psi(r,\theta) = \sin^2\theta f(r).$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter12slashphasemf.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5045.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter12/phasemf.eps}}
\end{figure}">|; 

$key = q/{rmMa}_infty=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$ {\rm Ma}_\infty =2$">|; 

$key = q/60;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3819.png"
 ALT="$ 60$">|; 

$key = q/25.819;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4621.png"
 ALT="$ 25.819$">|; 

$key = q/displaystyleskew{3}bar{C}_V=left.frac{q}{dT}right|_v.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5302.png"
 ALT="$\displaystyle \skew{3}\bar{C}_V = \left.\frac{q}{dT}\right\vert _v.$">|; 

$key = q/displaystyleaproptofrac{1}{l}int_0^lcosleft(pi,frac{x}{l}right)cos(k,x),dx=frac{sin(pi-k,l)}{pi-k,l},frac{k,l}{pi+k,l}:;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4001.png"
 ALT="$\displaystyle a\propto \frac{1}{l}\int_0^l\cos\left(\pi \frac{x}{l}\right)\cos(k x) dx = \frac{\sin(\pi-k l)}{\pi-k l} \frac{k l}{\pi+k l}:$">|; 

$key = q/|{bfr}-({bfr}cdot{bfn}),{bfn}|=d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6724.png"
 ALT="$ \vert{\bf r} -({\bf r}\cdot{\bf n}) {\bf n}\vert = d$">|; 

$key = q/displaystyle{calG}(x)gtreqless1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1493.png"
 ALT="$\displaystyle {\cal G}(x) \gtreqless 1$">|; 

$key = q/displaystyle=int_Sleft(frac{partialphi}{partialx}-frac{partialpsi}{partialy}right)dS.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2361.png"
 ALT="$\displaystyle =\int_S\left(\frac{\partial\phi}{\partial x} - \frac{\partial\psi}{\partial y}\right)dS.$">|; 

$key = q/displaystyle{mitPsi}=-frac{1}{2},G,rhoointleft[frac{2sum_{i=1,3}x_i^{,2},n_i^{,2a_i^{,2}}{sum_{i=1,3}n_i^{,2}slasha_i^{,2}}right]d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="499" HEIGHT="98" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7252.png"
 ALT="$\displaystyle {\mit\Psi} = -\frac{1}{2} G \rho\oint\left[ \frac{2\sum_{i=1,3}...
...,3}x_i^{ 2}/a_i^{ 2}}{\sum_{i=1,3}n_i^{ 2}/a_i^{ 2}} \right] d{\mit\Omega}.$">|; 

$key = q/displaystylewidehat{omega}^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img895.png"
 ALT="$\displaystyle \widehat{\omega}^{ 2}$">|; 

$key = q/{rmMa}ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="$ {\rm Ma}\ll 1$">|; 

$key = q/muslash(rho,g,a)simeq0.35;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4716.png"
 ALT="$ \mu/(\rho g a)\simeq 0.35$">|; 

$key = q/displaystylec^{,2}=c_0^{,2}-frac{1}{2},(gamma-1)left(u^{,2}+v^{,2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5890.png"
 ALT="$\displaystyle c^{ 2} = c_0^{ 2} -\frac{1}{2} (\gamma-1)\left(u^{ 2}+v^{ 2}\right),$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter02slashfig5.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img959.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter02/fig5.03.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{Chapter10slashlub.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="532" HEIGHT="165" BORDER="0"
 SRC="|."$dir".q|img3689.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{Chapter10/lub.eps}}
\end{figure}">|; 

$key = q/u_1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5526.png"
 ALT="$ u_1=0$">|; 

$key = q/displaystyledelta(x)=int_0^inftyleft[1-frac{v_x(x,y)}{U_0}right]dy,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2987.png"
 ALT="$\displaystyle \delta(x) = \int_0^\infty \left[1-\frac{v_x(x,y)}{U_0}\right]dy,$">|; 

$key = q/V,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4013.png"
 ALT="$ V t$">|; 

$key = q/displaystyle2,a_{ijkl}+a_{jikl}+a_{ikjl}+a_{ilkj};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6962.png"
 ALT="$\displaystyle 2 a_{ijkl} + a_{jikl} +a_{ikjl} +a_{ilkj}$">|; 

$key = q/V_0=s,V=s,a,b,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img664.png"
 ALT="$ V_0=s V = s a b c$">|; 

$key = q/widehat{omega}=0.43257;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img958.png"
 ALT="$ \widehat{\omega}=0.43257$">|; 

$key = q/{bfV}=V_0,(costheta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2423.png"
 ALT="$ {\bf V} = V_0 (\cos\theta_0$">|; 

$key = q/partialTslashpartialt>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6621.png"
 ALT="$ \partial T/\partial t&gt;0$">|; 

$key = q/m'=1.99times10^{30},{rmkg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4595.png"
 ALT="$ m'=1.99\times 10^{30} {\rm kg}$">|; 

$key = q/rhoslashT^{,gamma-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5339.png"
 ALT="$ \rho/T^{ \gamma-1}$">|; 

$key = q/gammarightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5420.png"
 ALT="$ \gamma\rightarrow\infty$">|; 

$key = q/exp(,{rmi},m,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1737.png"
 ALT="$ \exp( {\rm i} m \theta)$">|; 

$key = q/F_x=-D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3055.png"
 ALT="$ F_x=-D$">|; 

$key = q/r_2rightarrowr_1rightarrowr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2628.png"
 ALT="$ r_2\rightarrow r_1\rightarrow r$">|; 

$key = q/displaystyle{mitPhi}_M=oint_Srho,{bfv}cdotd{bfS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="$\displaystyle {\mit\Phi}_M = \oint_S \rho {\bf v}\cdot d{\bf S},$">|; 

$key = q/displaystyledS_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7123.png"
 ALT="$\displaystyle dS_\theta$">|; 

$key = q/displaystylenablatimes(phi,{bfA});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6695.png"
 ALT="$\displaystyle \nabla\times(\phi {\bf A})$">|; 

$key = q/displaystylesimeqfrac{k}{zeta^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2327.png"
 ALT="$\displaystyle \simeq \frac{k}{\zeta^{ 2}},$">|; 

$key = q/displaystyleC_{pL}=frac{2,alpha_0}{sqrt{{rmMa}_1^{,2}-1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5796.png"
 ALT="$\displaystyle C_{pL} = \frac{2 \alpha_0}{\sqrt{{\rm Ma}_1^{ 2}-1}},$">|; 

$key = q/O;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="$ O$">|; 

$key = q/displaystyle{mitGamma}=oint_C(v_x,dx+v_y,dy)=-{rmRe}oint_Cfrac{dF}{dz},dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2374.png"
 ALT="$\displaystyle {\mit\Gamma} = \oint_C (v_x dx+v_y dy) = -{\rm Re}\oint_C \frac{dF}{dz} dz,$">|; 

$key = q/displaystyleV_x,frac{partialV_x}{partialX}+V_y,frac{partialV_y}{partialY};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2899.png"
 ALT="$\displaystyle V_x \frac{\partial V_x}{\partial X} + V_y \frac{\partial V_y}{\partial Y}$">|; 

$key = q/displaystyletimes{bfa},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6401.png"
 ALT="$\displaystyle \times {\bf a},$">|; 

$key = q/{multline}I_{n,,n'}^{,m,,m'}=sum_{p=0,p_{{rm{max}}sum_{p'=0,p'_{{rm{max}}alpha_{Gamma}[(m+m'+2p+2p'+2)slash2]}{{mitGamma}[(n+n'+3)slash2]},{multline};LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="576" HEIGHT="141" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4969.png"
 ALT="\begin{multline}
I_{n, n'}^{ m, m'} = \sum_{p=0,p_{\rm max}}\sum_{p'=0,p'_{\r...
...)/2] {\mit\Gamma}[(m+m'+2p+2p'+2)/2]}{{\mit\Gamma}[(n+n'+3)/2]},
\end{multline}">|; 

$key = q/displaystyledY;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2385.png"
 ALT="$\displaystyle dY$">|; 

$key = q/displaystyle{mitPsi}_n^{,m}(theta,phi)=a_n^{,m},skew{5}hat{P}_n^{,m}(costheta),sin(m,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4942.png"
 ALT="$\displaystyle {\mit\Psi}_n^{ m}(\theta,\phi) = a_n^{ m} \skew{5}\hat{P}_n^{ m}(\cos\theta) \sin(m \phi),$">|; 

$key = q/displaystylenabla{calH}+nabla(v^{,2}slash2)=nabla{calH}_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5869.png"
 ALT="$\displaystyle \nabla {\cal H} +\nabla(v^{ 2}/2) = \nabla {\cal H}_0.$">|; 

$key = q/lambda=lambda_s=-0.1567;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3221.png"
 ALT="$ \lambda=\lambda_s=-0.1567$">|; 

$key = q/displaystyle=left(frac{r}{a}right)^nP_n^{,m}(costheta),cos(m,varphi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4273.png"
 ALT="$\displaystyle = \left(\frac{r}{a}\right)^n P_n^{ m}(\cos\theta) \cos(m \varphi),$">|; 

$key = q/displaystylefrac{partialP}{partialr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3786.png"
 ALT="$\displaystyle \frac{\partial P}{\partial r}$">|; 

$key = q/{calC},S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1406.png"
 ALT="$ {\cal C} S$">|; 

$key = q/displaystyleC_psimeqfrac{2}{U}left(frac{partialphi}{partialx}right)_{y=0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6055.png"
 ALT="$\displaystyle C_p\simeq \frac{2}{U}\left(\frac{\partial\phi}{\partial x}\right)_{y=0}.$">|; 

$key = q/displaystyle=m,frac{partial^{,2}skew{3}bar{phi}}{partialeta^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6023.png"
 ALT="$\displaystyle = m \frac{\partial^{ 2}\skew{3}\bar{\phi}}{\partial \eta^{ 2}}.$">|; 

$key = q/displaystyleleft(frac{h_2}{h_1}right)^2+frac{h_2}{h_1}-frac{2,(u_1+V)^2}{g,h_1}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1515.png"
 ALT="$\displaystyle \left(\frac{h_2}{h_1}\right)^2+\frac{h_2}{h_1}-\frac{2 (u_1+V)^2}{g h_1}=0,$">|; 

$key = q/displaystyled{bfx}cdotd{bfx}=dr^{,2}+r^{,2},dtheta^{,2}+r^{,2},sin^2theta,dphi^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7172.png"
 ALT="$\displaystyle d {\bf x}\cdot d{\bf x} = dr^{ 2} + r^{ 2} d\theta^{ 2} + r^{ 2} \sin^2\theta d\phi^2.$">|; 

$key = q/P(pi-theta)=P(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3093.png"
 ALT="$ P(\pi-\theta)=P(\theta)$">|; 

$key = q/displaystyleint_P^Qleft(f,dx+g,dyright)=int_P^QdF(x,y)=F(Q)-F(P).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6467.png"
 ALT="$\displaystyle \int_P^Q \left(f dx + g dy\right) = \int_P^Q dF(x,y) = F(Q) - F(P).$">|; 

$key = q/e_{13}>0.81267;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img961.png"
 ALT="$ e_{13}&gt;0.81267$">|; 

$key = q/eta=yslashdelta(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3153.png"
 ALT="$ \eta = y/\delta(x)$">|; 

$key = q/(1slash2),rho,g,a^{,2},w,v_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3987.png"
 ALT="$ (1/2) \rho g a^{ 2} w v_g$">|; 

$key = q/displaystyle{mitPhi}_{rmtide}({bfr})={mitPhi}_{rmmoon}({bfr})-{mitPhi}_{rmcentripetal}({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4248.png"
 ALT="$\displaystyle {\mit\Phi}_{\rm tide}({\bf r}) = {\mit\Phi}_{\rm moon}({\bf r}) -{\mit\Phi}_{\rm centripetal}({\bf r})$">|; 

$key = q/{rmRe};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="$ {\rm Re}$">|; 

$key = q/displaystyledelta_{ij},a_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6797.png"
 ALT="$\displaystyle \delta_{ij} a_j$">|; 

$key = q/{bfd};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6670.png"
 ALT="$ {\bf d}$">|; 

$key = q/z=r,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2530.png"
 ALT="$ z=r \cos\theta$">|; 

$key = q/(1slash2),M,a^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7200.png"
 ALT="$ (1/2) M a^{ 2}$">|; 

$key = q/displaystyle(nabla^{,2}{bfA})_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7163.png"
 ALT="$\displaystyle (\nabla^{ 2}{\bf A})_z$">|; 

$key = q/F(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1943.png"
 ALT="$ F(z)$">|; 

$key = q/displaystyley<0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6051.png"
 ALT="$\displaystyle y&lt;0.$">|; 

$key = q/displaystyle{mitPsi}'+{mitPsi}''simeq-lambdaleft(frac{3}{2},x_1^{,2}-frac{1}{2},x_3^{,2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img987.png"
 ALT="$\displaystyle {\mit\Psi}'+{\mit\Psi}'' \simeq -\lambda \left(\frac{3}{2} x_1^{ 2}-\frac{1}{2} x_3^{ 2}\right),$">|; 

$key = q/{figure}{epsfysize{=2.4in{centerline{epsffile{Chapter12slashsemidiurnal2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="547" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img4721.png"
 ALT="\begin{figure}
\epsfysize =2.4in
\centerline{\epsffile{Chapter12/semidiurnal2.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter06slashfree.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="537" HEIGHT="368" BORDER="0"
 SRC="|."$dir".q|img2193.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter06/free.eps}}
\end{figure}">|; 

$key = q/partialv'slashpartialy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5914.png"
 ALT="$ \partial v'/\partial y$">|; 

$key = q/{bfa}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6404.png"
 ALT="$ {\bf a}'$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashcylinder.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1761.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/cylinder.eps}}
\end{figure}">|; 

$key = q/displaystyle{rmarg}(dzeta'')-{rmarg}(dzeta')={rmarg}(dz'')-{rmarg}(dz').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2076.png"
 ALT="$\displaystyle {\rm arg}(d\zeta'')-{\rm arg}(d\zeta') = {\rm arg}(dz'')-{\rm arg}(dz').$">|; 

$key = q/{mitGamma},{mitGamma}'>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2439.png"
 ALT="$ {\mit\Gamma} {\mit\Gamma}'&gt;0$">|; 

$key = q/displaystyleV=frac{a^{,2},g}{3,nu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3868.png"
 ALT="$\displaystyle V = \frac{a^{ 2} g}{3 \nu}.$">|; 

$key = q/{rmMa}_1,sinbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5727.png"
 ALT="$ {\rm Ma}_1 \sin\beta$">|; 

$key = q/deltatheta_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6827.png"
 ALT="$ \delta\theta_i$">|; 

$key = q/v_1=v_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1478.png"
 ALT="$ v_1=v_2$">|; 

$key = q/displaystyle{mitPsi}(theta,phi_pm,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4764.png"
 ALT="$\displaystyle {\mit\Psi}(\theta,\phi_\pm,t)$">|; 

$key = q/MH'=HH'slashtheta=kappa_x^{,2},A_0slashV_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img644.png"
 ALT="$ MH' = HH'/\theta=\kappa_x^{ 2} A_0/V_0$">|; 

$key = q/nablatimes{bfv}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1650.png"
 ALT="$ \nabla\times {\bf v}={\bf0}$">|; 

$key = q/A'E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4012.png"
 ALT="$ A'E$">|; 

$key = q/p_n^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5021.png"
 ALT="$ p_n^{ m}$">|; 

$key = q/omega^{,2}=(8slash9),s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img779.png"
 ALT="$ \omega^{ 2} = (8/9) s$">|; 

$key = q/displaystyleE(gamma,alpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img943.png"
 ALT="$\displaystyle E(\gamma,\alpha)$">|; 

$key = q/r',r''=CslashA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7247.png"
 ALT="$ r' r''=C/A$">|; 

$key = q/displaystyleskew{3}hat{kappa}_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5010.png"
 ALT="$\displaystyle \skew{3}\hat{\kappa}_{n, n'}^{ m, m'}$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter06slashconf2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="546" HEIGHT="252" BORDER="0"
 SRC="|."$dir".q|img2111.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter06/conf2.eps}}
\end{figure}">|; 

$key = q/displaystylenablacdot{bfn}simeq-frac{partial^{,2}zeta}{partialx^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4140.png"
 ALT="$\displaystyle \nabla\cdot{\bf n} \simeq -\frac{\partial^{ 2}\zeta}{\partial x^{ 2}}.$">|; 

$key = q/alpha_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2595.png"
 ALT="$ \alpha_l$">|; 

$key = q/displaystylerho,g,zeta=left.p_1right|_{z=0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3911.png"
 ALT="$\displaystyle \rho g \zeta=\left. p_1\right\vert _{z=0},$">|; 

$key = q/ABCDEFA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2714.png"
 ALT="$ ABCDEFA$">|; 

$key = q/0.7;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$ 0.7$">|; 

$key = q/Vrightarrow(g,d)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3993.png"
 ALT="$ V\rightarrow (g d)^{1/2}$">|; 

$key = q/displaystyleskew{5}bar{zeta}_2(theta,phi,t)={rmRe}sum_jskew{5}bar{zeta}_j(theta,phi,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4538.png"
 ALT="$\displaystyle \skew{5}\bar{\zeta}_2(\theta,\phi,t)={\rm Re}\sum_j \skew{5}\bar{\zeta}_j(\theta,\phi,t),$">|; 

$key = q/displaystyle=-omega^{,2},X+left(frac{m'-m}{m'+m}right)g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1849.png"
 ALT="$\displaystyle = -\omega^{ 2} X + \left(\frac{m'-m}{m'+m}\right)g,$">|; 

$key = q/displaystylefrac{1}{2},rholeft[left(frac{3}{2},G,M,alpha_1-3,lambdaright)x_1^{,2}+left(frac{3}{2},G,M,alpha_3+lambdaright)x_3^{,2}right]=p_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="583" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img993.png"
 ALT="$\displaystyle \frac{1}{2} \rho\left[\left(\frac{3}{2} G M \alpha_1-3 \lamb...
...2^{ 2}+\left(\frac{3}{2} G M \alpha_3+\lambda\right)x_3^{ 2}\right] = p_0,$">|; 

$key = q/F(z)=-(Qslash2pi),ln(z-z_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1982.png"
 ALT="$ F(z)=-(Q/2\pi) \ln(z-z_0)$">|; 

$key = q/displaystyle=oint_CP,(x,dx+y,dy),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2393.png"
 ALT="$\displaystyle =\oint_C P (x dx+y dy),$">|; 

$key = q/u_3=phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7165.png"
 ALT="$ u_3=\phi$">|; 

$key = q/displaystyleu,v,frac{partialv}{partialx}+v^{,2},frac{partialv}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5887.png"
 ALT="$\displaystyle u v \frac{\partial v}{\partial x} + v^{ 2} \frac{\partial v}{\partial y}$">|; 

$key = q/displaystylexgtreqless1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1494.png"
 ALT="$\displaystyle x\gtreqless 1,$">|; 

$key = q/displaystylefrac{d}{dx}left({calE}+frac{p}{rho}+frac{1}{2},u^{,2}right)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5459.png"
 ALT="$\displaystyle \frac{d}{dx}\left({\cal E} + \frac{p}{\rho}+\frac{1}{2} u^{ 2}\right)=0$">|; 

$key = q/|theta_2|>|theta_{rmmax}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6153.png"
 ALT="$ \vert\theta_2\vert&gt;\vert\theta_{\rm max}\vert$">|; 

$key = q/displaystyleP(theta)=p(a,theta)=p_0+frac{1}{2},rholeft(V_0^{,2}-v_t^{,2}right)=p_1+rho,V_0^{,2}left(cos2theta-2,gamma,sinthetaright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="560" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1779.png"
 ALT="$\displaystyle P(\theta)=p(a,\theta) = p_0 + \frac{1}{2} \rho\left(V_0^{ 2}-v_...
...}\right)= p_1 + \rho V_0^{ 2}\left(\cos 2\theta-2 \gamma \sin\theta\right),$">|; 

$key = q/c^{,2}<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4183.png"
 ALT="$ c^{ 2}&lt;0$">|; 

$key = q/displaystyleP=p_0-frac{1}{2},rholeft|frac{dF}{dz}right|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2375.png"
 ALT="$\displaystyle P = p_0 -\frac{1}{2} \rho\left\vert\frac{dF}{dz}\right\vert^{ 2}.$">|; 

$key = q/sigma_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4525.png"
 ALT="$ \sigma_j$">|; 

$key = q/displaystylenabla{calH}=frac{nablap}{rho}+T,nabla{calS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5868.png"
 ALT="$\displaystyle \nabla {\cal H} = \frac{\nabla p}{\rho} + T \nabla{\cal S},$">|; 

$key = q/displaystyle{mitDelta}J=J-J'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2366.png"
 ALT="$\displaystyle {\mit\Delta} J = J-J'.$">|; 

$key = q/displaystylev(x,y)simeqfrac{beta(y)}{sqrt{x_0-x}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3119.png"
 ALT="$\displaystyle v(x,y) \simeq \frac{\beta(y)}{\sqrt{x_0-x}},$">|; 

$key = q/displaystyle=c_x,costheta+c_y,sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6362.png"
 ALT="$\displaystyle = c_x \cos\theta +c_y \sin\theta.$">|; 

$key = q/displaystyle=-frac{2pi,a_1,a_2,a_3}{a_i}int_0^inftyfrac{partial}{partiala_i}!lef=2pi,a_1,a_2,a_3int_0^inftyfrac{du}{(a_i^{,2}+u),{mitDelta}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="509" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7269.png"
 ALT="$\displaystyle =-\frac{2\pi a_1 a_2 a_3}{a_i}\int_0^\infty \frac{\partial}{\p...
...ght)du =2\pi a_1 a_2 a_3\int_0^\infty\frac{du}{(a_i^{ 2}+u) {\mit\Delta}}.$">|; 

$key = q/mu>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="$ \mu&gt;0$">|; 

$key = q/displaystyleY;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2390.png"
 ALT="$\displaystyle Y$">|; 

$key = q/displaystyleleft.frac{partialpsi}{partialtheta}right|_{r=a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3762.png"
 ALT="$\displaystyle \left.\frac{\partial\psi}{\partial\theta}\right\vert _{r=a}$">|; 

$key = q/displaystyled_{ij}=alpha,e_{kk},delta_{ij}+2,beta,e_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="$\displaystyle d_{ij} = \alpha e_{kk} \delta_{ij}+2 \beta e_{ij},$">|; 

$key = q/displaystylep_1,S_1-F-p_2,S_2=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1402.png"
 ALT="$\displaystyle p_1 S_1-F-p_2 S_2 = 0,$">|; 

$key = q/displaystyle=2,l,cosh[ln(cslashl)]=c+l^{,2}slashc,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2128.png"
 ALT="$\displaystyle = 2 l \cosh[\ln(c/l)] = c + l^{ 2}/c,$">|; 

$key = q/displaystyleC_p=frac{2,theta}{sqrt{{rmMa}_1^{,2}-1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5792.png"
 ALT="$\displaystyle C_p =\frac{2 \theta}{\sqrt{{\rm Ma}_1^{ 2}-1}}.$">|; 

$key = q/y(pmaslash2)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7355.png"
 ALT="$ y(\pm a/2)= 0$">|; 

$key = q/displaystylefrac{r_1}{sintheta_2}=frac{r_2}{sintheta_1}=frac{d}{sin(theta_2-theta_1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2622.png"
 ALT="$\displaystyle \frac{r_1}{\sin\theta_2} = \frac{r_2}{\sin\theta_1} =\frac{d}{\sin(\theta_2-\theta_1)}.$">|; 

$key = q/displaystyle{bfh}=-kappa,,nablaT,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6605.png"
 ALT="$\displaystyle {\bf h} = - \kappa   \nabla T,$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter03slashfig6.10.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="371" BORDER="0"
 SRC="|."$dir".q|img1337.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter03/fig6.10.eps}}
\end{figure}">|; 

$key = q/{bfn}cdotnablaphi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3272.png"
 ALT="$ {\bf n}\cdot\nabla\phi=0$">|; 

$key = q/displaystyle({bfa}times{bfb})cdot({bfc}times{bfd});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6677.png"
 ALT="$\displaystyle ({\bf a}\times {\bf b})\cdot({\bf c}\times {\bf d})$">|; 

$key = q/beta_1=-Vslash(2,a^{,3});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2644.png"
 ALT="$ \beta_1=-V/(2 a^{ 3})$">|; 

$key = q/displaystyler,frac{partialxi_{i,n}}{partialr}-xi_{i,n}=n,A_n,r^{,2},frac{partialrtialx_i}+n,B_n,p_n,x_i+(n-2),frac{partialphi_n}{partialx_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4417.png"
 ALT="$\displaystyle r \frac{\partial\xi_{i n}}{\partial r}-\xi_{i n} = n A_n r^{...
...}{\partial x_i} + n B_n p_n x_i +(n-2) \frac{\partial\phi_n}{\partial x_i}.$">|; 

$key = q/-d{mitPsi}slashdr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5176.png"
 ALT="$ -d{\mit\Psi}/dr$">|; 

$key = q/overline{nabla}cdotoverline{bfv}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="$ \overline{\nabla}\cdot\overline{\bf v}=0$">|; 

$key = q/displaystyle2,n,A_n,mu+(3+n),B_n,mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4424.png"
 ALT="$\displaystyle 2 n A_n \mu + (3+n) B_n \mu$">|; 

$key = q/displaystylenablacdot{bfv}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4070.png"
 ALT="$\displaystyle \nabla \cdot{\bf v}_1$">|; 

$key = q/displaystylexi_z(x,-d)=a,sin(k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4121.png"
 ALT="$\displaystyle \xi_z(x,-d) = a \sin(k x),$">|; 

$key = q/h(x,,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6519.png"
 ALT="$ h(x, y)$">|; 

$key = q/{bfv}(r,theta)rightarrowV_0,{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1762.png"
 ALT="$ {\bf v}(r,\theta)\rightarrow V_0 {\bf e}_x$">|; 

$key = q/displaystyle=-frac{1}{rho,r},frac{partialp}{partialtheta}-frac{1}{r}frac{partial}v_theta+frac{2}{r^{,2}},frac{partialv_r}{partialtheta}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img506.png"
 ALT="$\displaystyle = - \frac{1}{\rho r} \frac{\partial p}{\partial \theta} - \frac...
...^{ 2} v_\theta +\frac{2}{r^{ 2}} \frac{\partial v_r}{\partial\theta} \right.$">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="$ T$">|; 

$key = q/displaystyleG(u)=u-zeta,coshu=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1270.png"
 ALT="$\displaystyle G(u) = u - \zeta \cosh u = 0.$">|; 

$key = q/displaystylephi(xi,eta)=-V,aleft{frac{1+coshxi,lnleft[tanh(xislash2)right]}{(1-e)^{,-1}+(1slash2,e)lnleft[(1-e)slash(1+e)right]}right}coseta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="525" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2806.png"
 ALT="$\displaystyle \phi(\xi,\eta) =-V a\left\{\frac{1 +\cosh\xi \ln\left[\tanh(\xi...
...ght]} {(1-e^{ 2})^{ -1}+(1/2 e)\ln\left[(1-e)/(1+e)\right]}\right\}\cos\eta.$">|; 

$key = q/{rmMa}_{,e}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5427.png"
 ALT="$ {\rm Ma}_{ e}=1$">|; 

$key = q/displaystylefrac{rho,D^{,2}}{8pi,a^{,3}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2459.png"
 ALT="$\displaystyle \frac{\rho D^{ 2}}{8\pi a^{ 3}}
$">|; 

$key = q/displaystylefrac{d}{dx}!left(frac{partialF}{partialy_i'}right)-frac{partialF}{partialy_i}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7365.png"
 ALT="$\displaystyle \frac{d}{dx}\!\left(\frac{\partial F}{\partial y_i'}\right)-\frac{\partial F}{\partial y_i} = 0,$">|; 

$key = q/displaystyle=2,l,sinh[ln(rslashl)],sintheta=(r-l^{,2}slashr),sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2123.png"
 ALT="$\displaystyle =2 l \sinh[\ln(r/l)] \sin\theta = (r-l^{ 2}/r) \sin\theta.$">|; 

$key = q/displaystyleK_{rmsphere}=frac{1}{2},m,V^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2680.png"
 ALT="$\displaystyle K_{\rm sphere} = \frac{1}{2} m V^{ 2}.$">|; 

$key = q/displaystyled_{ii}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="$\displaystyle d_{ii} = 0.$">|; 

$key = q/displaystylenabla^{,2}(r^{,m},{calR}_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4334.png"
 ALT="$\displaystyle \nabla^{ 2}(r^{ m} {\cal R}_n)$">|; 

$key = q/displaystyle-frac{partialphi}{partialt}+frac{1}{2},v^{,2}+frac{p}{rho}+{mitPsi}={calC}(t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img547.png"
 ALT="$\displaystyle -\frac{\partial\phi}{\partial t} + \frac{1}{2} v^{ 2}+ \frac{p}{\rho} + {\mit\Psi} ={\cal C}(t).
$">|; 

$key = q/{bfr}={bfr}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1686.png"
 ALT="$ {\bf r}={\bf r}_0$">|; 

$key = q/displaystylea_{y'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6287.png"
 ALT="$\displaystyle a_{y'}$">|; 

$key = q/displaystyleH=F(v),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1428.png"
 ALT="$\displaystyle H =F(v),$">|; 

$key = q/{bfx}=(x_1,,x_2,,x_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6871.png"
 ALT="$ {\bf x}= (x_1, x_2, x_3)$">|; 

$key = q/c_nu=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3552.png"
 ALT="$ c_\nu=0$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter05slashsource.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="367" BORDER="0"
 SRC="|."$dir".q|img1673.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter05/source.eps}}
\end{figure}">|; 

$key = q/displaystyle{bfA}({bfr})=(8,x^{,3}+3,x^{,2},y^{,2},,2,x^{,3},y+6,y,,6).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="325" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6745.png"
 ALT="$\displaystyle {\bf A}({\bf r}) = (8 x^{ 3}+3 x^{ 2} y^{ 2}, 2 x^{ 3} y+6 y, 6).
$">|; 

$key = q/delta(x)llL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2979.png"
 ALT="$ \delta(x)\ll L$">|; 

$key = q/{bfg}=-g,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3747.png"
 ALT="$ {\bf g} = -g {\bf e}_z$">|; 

$key = q/x,dx+y,dy={rmRe}(z,dskew{2}bar{z});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2410.png"
 ALT="$ x dx+ y dy={\rm Re}(z d\skew{2}\bar{z})$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter09slashcyl1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3348.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/cyl1.eps}}
\end{figure}">|; 

$key = q/displaystylepmsqrt{lambda^{,2}-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2310.png"
 ALT="$\displaystyle \pm\sqrt{\lambda^{ 2}-1}$">|; 

$key = q/displaystyle-G(x)=frac{dP}{dx}=frac{{mitDelta}P}{l},frac{1}{d^{,3}(x),langled^{,-3}rangle},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3675.png"
 ALT="$\displaystyle -G(x)= \frac{dP}{dx} = \frac{{\mit\Delta} P}{l} \frac{1}{d^{ 3}(x) \langle d^{ -3}\rangle},$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter06slashimage4.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="277" BORDER="0"
 SRC="|."$dir".q|img2414.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter06/image4.eps}}
\end{figure}">|; 

$key = q/{bfa}times({bfb}times{bfc})=({bfa}cdot{bfc}),{bfb}-({bfa}cdot{bfb}),{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6714.png"
 ALT="$ {\bf a}\times ({\bf b}\times {\bf c}) = ({\bf a}\cdot{\bf c}) {\bf b} - ({\bf a}\cdot{\bf b}) {\bf c}$">|; 

$key = q/displaystylefrac{d{mitGamma}_C}{dt}=-oint_Somega_z,{bfv}cdotd{bfS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1791.png"
 ALT="$\displaystyle \frac{d{\mit\Gamma}_C}{dt} = -\oint_S \omega_z {\bf v}\cdot d{\bf S}.$">|; 

$key = q/psi(xi,eta)=F(xi),sin^2eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2759.png"
 ALT="$ \psi(\xi,\eta)= F(\xi) \sin^2\eta$">|; 

$key = q/omega^{,2}>gslasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1064.png"
 ALT="$ \omega^{ 2}&gt;g/a$">|; 

$key = q/f';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3159.png"
 ALT="$ f'$">|; 

$key = q/sqrt{nu,t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="$ \sqrt{\nu t}$">|; 

$key = q/displaystyle-nablaP=G(x),{bfe}_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3649.png"
 ALT="$\displaystyle -\nabla P = G(x) {\bf e}_x,$">|; 

$key = q/v_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2540.png"
 ALT="$ v_\perp$">|; 

$key = q/displaystylezeta=z^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2113.png"
 ALT="$\displaystyle \zeta = z^{ 2}.$">|; 

$key = q/displaystyle=2,muleft[left(frac{partialv_x}{partialx}right)^2+left(frac{partialvartialv_x}{partialy}+frac{partialv_y}{partialx}right)^2right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="$\displaystyle =2 \mu\left[\left(\frac{\partial v_x}{\partial x}\right)^2+\left...
...\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right)^2\right.$">|; 

$key = q/{bfL}=X,{bfe}_x+Y,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3283.png"
 ALT="$ {\bf L}=X {\bf e}_x + Y {\bf e}_y$">|; 

$key = q/displaystyle=frac{delta_1}{delta_2}=left(frac{3}{10}-frac{{mitLambda}}{120}rightc{{mitLambda}}{945}-frac{{mitLambda}^{,2}}{9072}right)right.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3192.png"
 ALT="$\displaystyle = \frac{\delta_1}{\delta_2} = \left(\frac{3}{10}-\frac{{\mit\Lamb...
...{315}-\frac{{\mit\Lambda}}{945}-\frac{{\mit\Lambda}^{ 2}}{9072}\right)\right.,$">|; 

$key = q/GH;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img620.png"
 ALT="$ GH$">|; 

$key = q/dslashc=zeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1268.png"
 ALT="$ d/c=\zeta$">|; 

$key = q/displaystyle{rmMa}_1^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5489.png"
 ALT="$\displaystyle {\rm Ma}_1^{ 2}$">|; 

$key = q/{bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6225.png"
 ALT="$ {\bf0}$">|; 

$key = q/displaystyleepsilon_{ijk}=left{{array}{lll}+1&mbox{hspace{0.5cm}}&mbox{ifi,j,kisnoddpermutationof1,2,3}[0.5ex]0&&mbox{otherwise}{array}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="98" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6787.png"
 ALT="$\displaystyle \epsilon_{ijk} = \left\{ \begin{array}{lll} +1&amp;\mbox{\hspace{0.5c...
... odd permutation of $1, 2, 3$}\ [0.5ex] 0&amp;&amp;\mbox{otherwise} \end{array}\right.$">|; 

$key = q/pislash2geqphigeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1315.png"
 ALT="$ \pi/2\geq\phi\geq 0$">|; 

$key = q/(4pislash3),a^{,3},rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2751.png"
 ALT="$ (4\pi/3) a^{ 3} \rho$">|; 

$key = q/displaystyle=-left[frac{n,(n-m+1)}{2,n+1}right]P_{n+1}^{,m}(mu)+left[frac{(n+1),(n+m)}{2,n+1}right]P_{n-1}^{,m}(mu).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4692.png"
 ALT="$\displaystyle = -\left[\frac{n (n-m+1)}{2 n+1}\right]P_{n+1}^{ m}(\mu) + \left[\frac{(n+1) (n+m)}{2 n+1}\right] P_{n-1}^{ m}(\mu).$">|; 

$key = q/F_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1469.png"
 ALT="$ F_{21}$">|; 

$key = q/{bfv}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1383.png"
 ALT="$ {\bf v}_1$">|; 

$key = q/displaystylephi(x,y)=frac{U,epsilon}{m},cos(alpha,x),frac{sinh[m,alpha,(b-y)]}{cosh(m,alpha,b)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6220.png"
 ALT="$\displaystyle \phi(x,y) = \frac{U \epsilon}{m} \cos(\alpha x) \frac{\sinh[m \alpha (b-y)]}{\cosh(m \alpha b)}.
$">|; 

$key = q/theta>theta_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3229.png"
 ALT="$ \theta&gt;\theta_s$">|; 

$key = q/displaystyle{mitOmega}(r)={mitOmega}_0,frac{a^{,3}}{r^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3886.png"
 ALT="$\displaystyle {\mit\Omega}(r) = {\mit\Omega}_0 \frac{a^{ 3}}{r^{ 3}},
$">|; 

$key = q/displaystylezeta_j(mu,phi,t)=skew{5}hat{zeta}_j(vartheta),cos(m_j,phi+sigma_j,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4709.png"
 ALT="$\displaystyle \zeta_j(\mu,\phi,t) = \skew{5}\hat{\zeta}_j(\vartheta) \cos(m_j \phi+\sigma_j t).$">|; 

$key = q/ij;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7142.png"
 ALT="$ ij$">|; 

$key = q/xi=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2472.png"
 ALT="$ \xi=\pi$">|; 

$key = q/displaystyleM=-frac{pi}{2},rho,V^{,2},(a^{,2}-b^{,2}),sin(2,alpha).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2506.png"
 ALT="$\displaystyle M = -\frac{\pi}{2} \rho V^{ 2} (a^{ 2}-b^{ 2}) \sin(2 \alpha).
$">|; 

$key = q/displaystyleQ=[g,h^{,3}(0)]^{1slash2}=left(frac{2}{3}right)^{3slash2}left(gleft[H-d(0)right]^{,3}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1534.png"
 ALT="$\displaystyle Q = [g h^{ 3}(0)]^{1/2} = \left(\frac{2}{3}\right)^{3/2}\left(g\left[H-d(0)\right]^{ 3}\right)^{1/2}.$">|; 

$key = q/m=m_ast=-0.0905;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2972.png"
 ALT="$ m=m_\ast = -0.0905$">|; 

$key = q/theta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$ \theta _1$">|; 

$key = q/phi_0=sin^{-1}([1-(1-k^{,2})^{1slash2}]^{1slash2}slashk);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1327.png"
 ALT="$ \phi_0= \sin^{-1}([1-(1-k^{ 2})^{1/2}]^{1/2}/k)$">|; 

$key = q/displaystyle={bf0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3729.png"
 ALT="$\displaystyle = {\bf0},$">|; 

$key = q/displaystyle=q_1int_{x_L}^{x_R}left(C_{pL}-C_{pU}right)dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5845.png"
 ALT="$\displaystyle = q_1\int_{x_L}^{x_R} \left(C_{pL}-C_{pU}\right)dx,$">|; 

$key = q/10.2,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img572.png"
 ALT="$ 10.2 {\rm m}$">|; 

$key = q/displaystyle=3,omega,z^{,-1},a,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img780.png"
 ALT="$\displaystyle = 3 \omega z^{ -1} a,$">|; 

$key = q/37%;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5133.png"
 ALT="$ 37\%$">|; 

$key = q/nablacdot(r^{,n},{bfe}_r)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7203.png"
 ALT="$ \nabla\cdot(r^{ n} {\bf e}_r)=0$">|; 

$key = q/kappa_{calS}=1slash(gamma,p);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5609.png"
 ALT="$ \kappa_{\cal S}=1/(\gamma p)$">|; 

$key = q/displaystylephantom{=}+{rmMa}_infty^{,2},frac{v'}{U}left(frac{partialu'}{partialy}+frac{partialv'}{partialx}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5910.png"
 ALT="$\displaystyle \phantom{=} + {\rm Ma}_\infty^{ 2} \frac{v'}{U}\left(\frac{\partial u'}{\partial y}+\frac{\partial v'}{\partial x}\right).$">|; 

$key = q/displaystylefrac{d}{dxi}left(frac{F}{cosh^2xi}right)=frac{B}{cosh^3xi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2763.png"
 ALT="$\displaystyle \frac{d}{d\xi}\left(\frac{F}{\cosh^2\xi}\right)= \frac{B}{\cosh^3\xi}.$">|; 

$key = q/displaystyleepsilon_{ijk}=epsilon_{jki}=epsilon_{kij}=-epsilon_{kji}=-epsilon_{jik}=-epsilon_{ikj}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6789.png"
 ALT="$\displaystyle \epsilon_{ijk}=\epsilon_{jki}=\epsilon_{kij}=-\epsilon_{kji}=-\epsilon_{jik}=-\epsilon_{ikj}.$">|; 

$key = q/displaystyleE(gamma,alpha)-2,F(gamma,alpha)+left[frac{1+(sinalpha,tanbeta,cosgamma)^2}{cos^2alpha}right]E(gamma,alpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="459" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img940.png"
 ALT="$\displaystyle E(\gamma,\alpha)- 2 F(\gamma,\alpha) +\left[\frac{1+(\sin\alpha \tan\beta \cos\gamma)^2}{\cos^2\alpha}\right] E(\gamma,\alpha)$">|; 

$key = q/displaystyleskew{5}bar{zeta}_{2}(mu,phi,t)=skew{5}tilde{zeta}_{j},P_2^{,m_j}(mu),{rme}^{,{rmi},(m_j,phi+sigma_j,t)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4673.png"
 ALT="$\displaystyle \skew{5}\bar{\zeta}_{2}(\mu,\phi,t) = \skew{5}\tilde{\zeta}_{j} P_2^{ m_j}(\mu) {\rm e}^{ {\rm i} (m_j \phi+\sigma_j t)},$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter15slashdiamond.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="398" HEIGHT="310" BORDER="0"
 SRC="|."$dir".q|img5808.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter15/diamond.eps}}
\end{figure}">|; 

$key = q/pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5724.png"
 ALT="$ \pi/2$">|; 

$key = q/epsilon_{ikj}=-epsilon_{ijk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6886.png"
 ALT="$ \epsilon_{ikj}=-\epsilon_{ijk}$">|; 

$key = q/theta>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img732.png"
 ALT="$ \theta&gt;0$">|; 

$key = q/{bft};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1087.png"
 ALT="$ {\bf t}$">|; 

$key = q/p_r(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4929.png"
 ALT="$ p_r(t)$">|; 

$key = q/displaystyleC_D=frac{D}{q_1,c}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5798.png"
 ALT="$\displaystyle C_D= \frac{D}{q_1 c}.$">|; 

$key = q/displaystyle=(-a_x,sintheta+a_y,costheta),b_z-a_z,(-b_x,sintheta+b_y,costheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6360.png"
 ALT="$\displaystyle = (-a_x  \sin\theta + a_y \cos\theta) b_z - a_z (-b_x  \sin\theta + b_y \cos\theta)$">|; 

$key = q/s<1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img725.png"
 ALT="$ s&lt;1/2$">|; 

$key = q/c=sqrt{d,g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5092.png"
 ALT="$ c=\sqrt{d g}$">|; 

$key = q/displaystyle=-frac{d}{a,sintheta}left[frac{partial}{partialtheta},(sintheta,u)+frac{partialv}{partialphi}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4508.png"
 ALT="$\displaystyle = - \frac{d}{a \sin\theta}\left[\frac{\partial}{\partial\theta} (\sin\theta u)+\frac{\partial v}{\partial\phi}\right],$">|; 

$key = q/displaystylephantom{=}left.+frac{1}{2}left(frac{partialv_r}{partialz}+frac{partialz}+frac{1}{r},frac{partialv_z}{partialtheta}right)^2right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="$\displaystyle \phantom{=}\left.+\frac{1}{2}\left(\frac{\partial v_r}{\partial z...
...{\partial z}+\frac{1}{r} \frac{\partial v_z}{\partial \theta}\right)^2\right].$">|; 

$key = q/displaystyle=(a_y,b_z-a_z,b_y)^2+(a_z,b_x-a_x,b_z)^2+(a_x,b_y-a_y,b_x)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6373.png"
 ALT="$\displaystyle = (a_y  b_z-a_z  b_y)^2 +(a_z  b_x - a_x  b_z)^2 +(a_x  b_y -a_y  b_x)^2$">|; 

$key = q/{figure}{epsfysize{=1.25in{centerline{epsffile{Chapter03slashfig6.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="129" BORDER="0"
 SRC="|."$dir".q|img1088.png"
 ALT="\begin{figure}
\epsfysize =1.25in
\centerline{\epsffile{Chapter03/fig6.01.eps}}
\end{figure}">|; 

$key = q/displaystyle{mitDelta}{calS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5750.png"
 ALT="$\displaystyle {\mit\Delta} {\cal S}$">|; 

$key = q/nablatimes{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6652.png"
 ALT="$ \nabla\times{\bf v}$">|; 

$key = q/T_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5446.png"
 ALT="$ T_2$">|; 

$key = q/displaystylefrac{{mitDelta}rho}{rho_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5510.png"
 ALT="$\displaystyle \frac{{\mit\Delta}\rho}{\rho_1}$">|; 

$key = q/displaystyleX-{rmi},Y=frac{1}{2},{rmi},rhooint_Cleft(frac{dF}{dz}right)^{,2}dz.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2409.png"
 ALT="$\displaystyle X - {\rm i} Y = \frac{1}{2} {\rm i} \rho\oint_C \left(\frac{dF}{dz}\right)^{ 2}dz.$">|; 

$key = q/a_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6262.png"
 ALT="$ a_z$">|; 

$key = q/dF=3,x^{,2},y,dx+x^{,3},dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6470.png"
 ALT="$ dF= 3 x^{ 2} y dx + x^{ 3} dy$">|; 

$key = q/displaystyle{mitPsi}=-frac{1}{4},G,rhooint,(r'^{,2}+r''^{,2}),d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7236.png"
 ALT="$\displaystyle {\mit\Psi} = - \frac{1}{4} G \rho\oint  (r'^{ 2}+r''^{ 2}) d{\mit\Omega}.$">|; 

$key = q/a_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1069.png"
 ALT="$ a_0$">|; 

$key = q/displaystylefrac{dhat{T}}{dhat{z}}=-frac{gamma-1}{gamma}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5145.png"
 ALT="$\displaystyle \frac{d\hat{T}}{d\hat{z}}= - \frac{\gamma-1}{\gamma}.$">|; 

$key = q/displaystylezeta_b(theta,phi)=sum_{n=1,infty}zeta_{b,n}(theta,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4341.png"
 ALT="$\displaystyle \zeta_b(\theta,\phi) = \sum_{n=1,\infty}\zeta_{b n}(\theta,\phi),$">|; 

$key = q/displaystyle{bfA}=A_r,{bfe}_r+A_theta,{bfe}_theta+A_phi,{bfe}_phi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7169.png"
 ALT="$\displaystyle {\bf A} = A_r {\bf e}_r+ A_\theta {\bf e}_\theta + A_\phi {\bf e}_\phi,$">|; 

$key = q/-1slash2pm1slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2104.png"
 ALT="$ -1/2\pm1/2)$">|; 

$key = q/U(x)=-U_0slashx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3233.png"
 ALT="$ U(x)=-U_0/x$">|; 

$key = q/displaystylesimeqfrac{sqrt{15}}{2},widehat{omega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img905.png"
 ALT="$\displaystyle \simeq \frac{\sqrt{15}}{2} \widehat{\omega},$">|; 

$key = q/r_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2619.png"
 ALT="$ r_1$">|; 

$key = q/y=fslash(2,l);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1226.png"
 ALT="$ y=f/(2 l)$">|; 

$key = q/K=I+lambda,J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7331.png"
 ALT="$ K = I + \lambda J$">|; 

$key = q/displaystyle=phi,nablatimes{bfA}+nablaphitimes{bfA},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6696.png"
 ALT="$\displaystyle =\phi  \nabla\times{\bf A} +\nabla\phi\times{\bf A},$">|; 

$key = q/cosalpha_0simeq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5803.png"
 ALT="$ \cos\alpha_0\simeq 1$">|; 

$key = q/displaystylefrac{d}{dtheta}lnleft(frac{f}{h}right)=g(theta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3360.png"
 ALT="$\displaystyle \frac{d}{d\theta}\ln\left(\frac{f}{h}\right)= g(\theta),$">|; 

$key = q/X=Y=dXslashdt=dYslashdt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1852.png"
 ALT="$ X=Y=dX/dt=dY/dt=0$">|; 

$key = q/sqrt{kappa_H,t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="$ \sqrt{\kappa_H t}$">|; 

$key = q/displaystylekappa_H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="$\displaystyle \kappa_H$">|; 

$key = q/y>epsilon,sin(alpha,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5953.png"
 ALT="$ y&gt; \epsilon \sin(\alpha x)$">|; 

$key = q/displaystylefrac{{mitDelta}p}{p_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6160.png"
 ALT="$\displaystyle \frac{{\mit\Delta}p}{p_1}$">|; 

$key = q/displaystyle(nablatimes{bfA})_i={bfe}_icdotnablatimes{bfA}=sum_{j,k=1,3}h_i,fracpsilon_{ijk},frac{h_i}{h},frac{partial(h_k,A_k)}{partialu_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="697" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7076.png"
 ALT="$\displaystyle (\nabla\times {\bf A})_i = {\bf e}_i\cdot\nabla\times {\bf A} = \...
...,k=1,3}\epsilon_{ijk} \frac{h_i}{h} \frac{\partial (h_k A_k)}{\partial u_j}.$">|; 

$key = q/displaystylefrac{partialvarpi}{partialxi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2686.png"
 ALT="$\displaystyle \frac{\partial \varpi}{\partial\xi}$">|; 

$key = q/displaystylefrac{dF}{dzeta}=V,{rme}^{,{rmi},alpha}+{rmi},frac{{mitGamma}}{2pi,zeta}-frac{V,a^{,2},{rme}^{,-{rmi},alpha}}{zeta^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3295.png"
 ALT="$\displaystyle \frac{dF}{d\zeta} = V {\rm e}^{ {\rm i} \alpha} + {\rm i} \fr...
...}}{2\pi \zeta} - \frac{V a^{ 2} {\rm e}^{ -{\rm i} \alpha}}{\zeta^{ 2}}.$">|; 

$key = q/partialskew{3}bar{phi}slashpartialyrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2300.png"
 ALT="$ \partial\skew{3}\bar{\phi}/\partial y\rightarrow 0$">|; 

$key = q/displaystyleA_{-r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4903.png"
 ALT="$\displaystyle A_{-r}$">|; 

$key = q/P-p_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$ P-p_0$">|; 

$key = q/displaystyleepsilon_{n,,n'}^{,m,,m'}=(1+k_n'-h_n'),alpha_n,(a_n^{,m})^{-2},kappa-h_{n'}'),alpha_{n'},(a_{n'}^{,m'})^{-2},kappa_{n',n}^{,m',m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="597" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4988.png"
 ALT="$\displaystyle \epsilon_{n, n'}^{ m, m'} =(1+k_n'-h_n') \alpha_n (a_n^{ m}...
... (1+k_{n'}'-h_{n'}') \alpha_{n'} (a_{n'}^{ m'})^{-2} \kappa_{n',n}^{ m',m}$">|; 

$key = q/(xi,,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6006.png"
 ALT="$ (\xi, \eta)$">|; 

$key = q/skew{3}bar{C}_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5304.png"
 ALT="$ \skew{3}\bar{C}_V$">|; 

$key = q/displaystyleg,h=frac{1}{2},v^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1409.png"
 ALT="$\displaystyle g h = \frac{1}{2} v^{ 2},$">|; 

$key = q/displaystyletheta_{n,,n'}^{,m,,m'}=sum_{n''=1,infty}sum_{m''=0,n''}^{m'',{rmodd}{,m''})^{-2},kappa_{n'',n}^{,m'',m},kappa_{n'',n'}^{,m'',m'};;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="526" HEIGHT="79" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4989.png"
 ALT="$\displaystyle \theta_{n, n'}^{ m, m'}= \sum_{n''=1,\infty}\sum_{m''=0,n''}^{...
... (a_{n''}^{ m''})^{-2} \kappa_{n'',n}^{ m'',m} \kappa_{n'',n'}^{ m'',m'};$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{AppendixAslashfigA.17.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="224" BORDER="0"
 SRC="|."$dir".q|img6491.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{AppendixA/figA.17.eps}}
\end{figure}">|; 

$key = q/r'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4231.png"
 ALT="$ r''$">|; 

$key = q/zetaslasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4488.png"
 ALT="$ \zeta/a$">|; 

$key = q/displaystyleint_{mitOmega}{mitPhi}_r,d{mitOmega}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4812.png"
 ALT="$\displaystyle \int_{\mit\Omega} {\mit\Phi}_r d{\mit\Omega} = 0.$">|; 

$key = q/h_u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7215.png"
 ALT="$ h_u$">|; 

$key = q/displaystylefrac{dzeta}{dz}=K',(zeta-b)^{1slash2},(zeta-c)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2153.png"
 ALT="$\displaystyle \frac{d\zeta}{dz}= K' (\zeta-b)^{1/2} (\zeta-c)^{1/2},$">|; 

$key = q/displaystyledelta_{n,,n'}^{,m,,m'}=delta_{n,n'},delta_{m,m'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4982.png"
 ALT="$\displaystyle \delta_{n, n'}^{ m, m'} = \delta_{n n'} \delta_{m m'}.$">|; 

$key = q/sqrt{gamma,p_0slashrho_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="$ \sqrt{\gamma p_0/\rho_0}$">|; 

$key = q/{bfg}=-g,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1834.png"
 ALT="$ {\bf g} = -g {\bf e}_y$">|; 

$key = q/psi_0(varpi,z)=-(1slash2),V,varpi^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2710.png"
 ALT="$ \psi_0(\varpi,z)= -(1/2) V \varpi^{ 2}$">|; 

$key = q/displaystyle=-frac{g}{2,a},(3,a^{,2}-r^{,2})-3,g,d,frac{skew{3}bar{rho}}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4363.png"
 ALT="$\displaystyle = -\frac{g}{2 a} (3 a^{ 2}-r^{ 2}) - 3 g d \frac{\skew{3}\bar{\rho}}{\rho},$">|; 

$key = q/deltasimLslash{rmRe}^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3088.png"
 ALT="$ \delta\sim L/{\rm Re}^{1/2}$">|; 

$key = q/displaystylephantom{=}left.+left(frac{1}{sinphi},lnleft[frac{sin,(1slash2),(phi'{sin,(1slash2),(phi'-phi)}right]right)_pi^{phi+epsilon}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3498.png"
 ALT="$\displaystyle \phantom{=}\left.+\left(\frac{1}{\sin\phi} \ln\left[\frac{\sin ...
...i'+\phi)} {\sin (1/2) (\phi'-\phi)}\right]\right)_\pi^{\phi+\epsilon}\right\}$">|; 

$key = q/displaystyleU,frac{dU}{dx}=-frac{1}{rho},frac{dP}{dx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2871.png"
 ALT="$\displaystyle U \frac{dU}{dx} = - \frac{1}{\rho} \frac{dP}{dx}$">|; 

$key = q/k_c=(rho,gslashgamma)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4215.png"
 ALT="$ k_c= (\rho g/\gamma)^{1/2}$">|; 

$key = q/displaystyleoint_C{mitOmega}_z,dl={mitGamma}(z),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3449.png"
 ALT="$\displaystyle \oint_C {\mit\Omega}_z dl = {\mit\Gamma}(z),$">|; 

$key = q/displaystyleJ_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2344.png"
 ALT="$\displaystyle J_r$">|; 

$key = q/displaystyleF=h,rho,v^{,2},cosalpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1895.png"
 ALT="$\displaystyle F = h \rho v^{ 2} \cos\alpha.$">|; 

$key = q/delta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img617.png"
 ALT="$ \delta_0$">|; 

$key = q/displaystylefrac{x}{l}=int_{sin^{-1}(zslash2,l)}^{sin^{-1}(hslash2,l)}left(frac{ht)+2,cosuright]_{sin^{-1}(zslash2,l)}^{sin^{-1}(hslash2,l)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="646" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1228.png"
 ALT="$\displaystyle \frac{x}{l} = \int_{\sin^{-1}(z/2 l)}^{\sin^{-1}(h/2 l)}\left(\...
...cos u}{\sin u}\right)+2 \cos u\right]_{\sin^{-1}(z/2 l)}^{\sin^{-1}(h/2 l)},$">|; 

$key = q/displaystyle=left(frac{partial{calH}}{partial{calS}}right)_p,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5582.png"
 ALT="$\displaystyle = \left(\frac{\partial {\cal H}}{\partial {\cal S}}\right)_p,$">|; 

$key = q/F(P);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6463.png"
 ALT="$ F(P)$">|; 

$key = q/0<eta<infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2969.png"
 ALT="$ 0&lt;\eta&lt;\infty$">|; 

$key = q/displaystylenablaP=mu,nablavarphitimesnabla[{calL}(psi)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3745.png"
 ALT="$\displaystyle \nabla P = \mu \nabla\varphi\times \nabla[{\cal L}(\psi)].$">|; 

$key = q/p_c=p(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5200.png"
 ALT="$ p_c= p(0)$">|; 

$key = q/int_0^piy^{,2},dx=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7374.png"
 ALT="$ \int_0^\pi y^{ 2}  dx=2$">|; 

$key = q/displaystyleC_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5946.png"
 ALT="$\displaystyle C_p$">|; 

$key = q/F'(z)rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3256.png"
 ALT="$ F'(z)\rightarrow 0$">|; 

$key = q/displaystyleint_0^pisin(m,phi),cos(m',phi),dphi=left{{array}{lll}2,mslash(m^{,2}e{0.5cm}}&mbox{m+m'odd}[0.5ex]0&&mbox{m+m'even}{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="561" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4956.png"
 ALT="$\displaystyle \int_0^\pi \sin(m \phi) \cos(m' \phi) d\phi = \left\{ \begin{...
...5cm}} &amp;\mbox{$m+m'$ odd}\ [0.5ex] 0&amp;&amp;\mbox{$m+m'$ even} \end{array}\right. .$">|; 

$key = q/displaystylelnfrac{p}{p_0}=-frac{T_0}{H}int_0^zfrac{dz}{T}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5263.png"
 ALT="$\displaystyle \ln\frac{p}{p_0} = -\frac{T_0}{H}\int_0^z \frac{dz}{T}.
$">|; 

$key = q/theta(0)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5276.png"
 ALT="$ \theta(0)=1$">|; 

$key = q/displaystyle{bfn}=frac{nabla(z-f)}{|nabla(z-f)|}=frac{{bfe}_z-f_x,{bfe}_x}{(1+f_x^{,2})^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1156.png"
 ALT="$\displaystyle {\bf n} = \frac{\nabla(z-f)}{\vert\nabla(z-f)\vert}=\frac{{\bf e}_z - f_x {\bf e}_x}{(1+f_x^{ 2})^{1/2}},$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialx^{,2}}-frac{1}{lambda^{,2}},frac{partial^{,2}phi}{partialy^{,2}}simeq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6044.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial x^{ 2}}-\frac{1}{\lambda^{ 2}} \frac{\partial^{ 2}\phi}{\partial y^{ 2}}\simeq 0,$">|; 

$key = q/P_n^{,0}(x)=P_n(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4271.png"
 ALT="$ P_n^{ 0}(x)=P_n(x)$">|; 

$key = q/displaystyle=frac{partialpsi}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img537.png"
 ALT="$\displaystyle =\frac{\partial\psi}{\partial x},$">|; 

$key = q/displaystyle=left(frac{gamma}{gamma-1}right){calR}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5387.png"
 ALT="$\displaystyle =\left(\frac{\gamma}{\gamma-1}\right){\cal R}.$">|; 

$key = q/displaystyleC_{pF}=frac{2,epsilon}{sqrt{{rmMa}_1^{,2}-1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5810.png"
 ALT="$\displaystyle C_{pF} = \frac{2 \epsilon}{\sqrt{{\rm Ma}_1^{ 2}-1}},$">|; 

$key = q/displaystylenablatimesnablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6692.png"
 ALT="$\displaystyle \nabla\times\nabla\phi$">|; 

$key = q/displaystyle{bfv}={bfe}_ztimesnablapsi=nablaztimesnablapsi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1637.png"
 ALT="$\displaystyle {\bf v} = {\bf e}_z\times \nabla\psi= \nabla z\times \nabla \psi.$">|; 

$key = q/displaystylesimeq1.0times10^4,L({rmm}),V_0({rmm,s^{-1}}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="$\displaystyle \simeq 1.0\times 10^4 L({\rm m}) V_0({\rm m s^{-1}}),$">|; 

$key = q/displaystyletheta=frac{1}{(1+xi^{,2}slash3)^{1slash2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5281.png"
 ALT="$\displaystyle \theta = \frac{1}{(1+\xi^{ 2}/3)^{1/2}}
$">|; 

$key = q/displaystylesimeqfrac{1}{gamma},frac{{mitDelta}p}{p_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5511.png"
 ALT="$\displaystyle \simeq \frac{1}{\gamma} \frac{{\mit\Delta}p}{p_1},$">|; 

$key = q/displaystylephantom{=}+frac{A_j}{h_i},frac{h}{h_j^{,3}}left[frac{1}{h_j},frac{pa2}}{partialu_i,partialu_j}!left(frac{h_j^{,2}}{h}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="618" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7101.png"
 ALT="$\displaystyle \phantom{=}+\frac{A_j}{h_i} \frac{h}{h_j^{ 3}}\left[\frac{1}{h_...
...ial^{ 2}}{\partial u_i \partial u_j}\!\left(\frac{h_j^{ 2}}{h}\right)\right]$">|; 

$key = q/displaystyle=-frac{partial{mitPhi}_r'}{partialtheta}-frac{1}{sintheta},frac{partial{mitPsi}_r''}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4847.png"
 ALT="$\displaystyle = - \frac{\partial{\mit\Phi}_r'}{\partial\theta} -\frac{1}{\sin\theta} \frac{\partial{\mit\Psi}_r''}{\partial\phi},$">|; 

$key = q/{bfr}=(x,,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5084.png"
 ALT="$ {\bf r}=(x, y)$">|; 

$key = q/displaystyle=nablatimes{bfv}=-nablatimesnablaphi={bf0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1586.png"
 ALT="$\displaystyle =\nabla\times {\bf v} = -\nabla\times\nabla\phi= {\bf0}.$">|; 

$key = q/omega=alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2306.png"
 ALT="$ \omega=\alpha$">|; 

$key = q/displaystylepsi(r,theta)=-frac{A}{1-m},r^{,1-m},cosleft[(1-m),(theta-pislash2)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1879.png"
 ALT="$\displaystyle \psi(r,\theta) = -\frac{A}{1-m} r^{ 1-m} \cos\left[(1-m) (\theta-\pi/2)\right]$">|; 

$key = q/displaystylep_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5244.png"
 ALT="$\displaystyle p_c$">|; 

$key = q/displaystyle=frac{4,a^{,3},s,(1-s)-(1slash3),a^{,3},tan^2theta}{4,a^{,2},s}=(1-s),a-frac{a}{12,s},tan^2theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="484" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img750.png"
 ALT="$\displaystyle = \frac{4 a^{ 3} s (1-s)-(1/3) a^{ 3} \tan^2\theta}{4 a^{ 2} s}=(1-s) a - \frac{a}{12 s} \tan^2\theta,$">|; 

$key = q/int_0^1y,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7375.png"
 ALT="$ \int_0^1y dx$">|; 

$key = q/displaystyleu,frac{partialu}{partialx}+v,frac{partialu}{partialy}simeq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3125.png"
 ALT="$\displaystyle u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \simeq 0.$">|; 

$key = q/displaystyle=nu({rmMa}_1)+frac{pi}{2}-left(frac{gamma+1}{gamma-1}right)^{1slash2}z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6127.png"
 ALT="$\displaystyle = \nu({\rm Ma}_1)+\frac{\pi}{2}-\left(\frac{\gamma+1}{\gamma-1}\right)^{1/2}z,$">|; 

$key = q/displaystyleint_P^Qleft(y,dx+x^{,3},dyright)=int_{0}^1left[y,dy+(y+1)^3,dyright]=frac{17}{4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6459.png"
 ALT="$\displaystyle \int_P^Q \left( y dx + x^{ 3} dy\right)= \int_{0}^1\left[y dy + (y+1)^3 dy\right] = \frac{17}{4}.$">|; 

$key = q/displaystyle={rmi}^{,n+n'},beta_{-n,,n'}^{,-m,,m'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5003.png"
 ALT="$\displaystyle = {\rm i}^{ n+n'} \beta_{-n, n'}^{ -m, m'},$">|; 

$key = q/displaystyleh_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7119.png"
 ALT="$\displaystyle h_z$">|; 

$key = q/nablacdot{bfE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7211.png"
 ALT="$ \nabla\cdot{\bf E}$">|; 

$key = q/displaystyleU=langleint_{-d}^zetarho,g,z,dzrangle+frac{1}{2},rho,g,d^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3978.png"
 ALT="$\displaystyle U = \langle\int_{-d}^\zeta \rho g z dz\rangle + \frac{1}{2} \rho g d^{ 2},$">|; 

$key = q/b^{,2}=Qslash(2pi,V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2840.png"
 ALT="$ b^{ 2} = Q/(2\pi V)$">|; 

$key = q/displaystyle=Z_1(x+c,t,0),{rme}^{+s,yslashl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5097.png"
 ALT="$\displaystyle = Z_1(x+c t,0) {\rm e}^{+s y/l},$">|; 

$key = q/displaystyle{mitOmega}llfrac{sqrt{muslashrho}}{a}:;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4385.png"
 ALT="$\displaystyle {\mit\Omega}\ll \frac{\sqrt{\mu/\rho}}{a}:$">|; 

$key = q/displaystyleq=m,(k,dT+p,dv).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5301.png"
 ALT="$\displaystyle q = m (k dT + p dv).$">|; 

$key = q/h=z(x=0)=z(skew{5}tilde{phi}=pislash2)simeq2,lslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1217.png"
 ALT="$ h = z(x=0)=z(\skew{5}\tilde{\phi}=\pi/2)\simeq 2 l/k$">|; 

$key = q/deltallc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3352.png"
 ALT="$ \delta\ll c$">|; 

$key = q/|u|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3564.png"
 ALT="$ \vert u\vert$">|; 

$key = q/displaystyleint_P^Qleft(f,dx+g,dyright)=F(Q)-F(P);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6462.png"
 ALT="$\displaystyle \int_P^Q \left(f dx + g dy\right) = F(Q) - F(P)$">|; 

$key = q/{mitPsi}_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4680.png"
 ALT="$ {\mit\Psi}_n$">|; 

$key = q/displaystyleoint{bfA}cdotd{bfr}=left(frac{partialA_y}{partialx}-frac{partialA_x}{partialy}right)dS_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6641.png"
 ALT="$\displaystyle \oint {\bf A} \cdot d{\bf r} = \left(\frac{\partial A_y}{\partial x}- \frac{\partial A_x}{\partial y}\right) dS_z.$">|; 

$key = q/-alpha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5793.png"
 ALT="$ -\alpha_0$">|; 

$key = q/displaystylef=pmfrac{2,l}{k},(1-k^{,2},sin^2phi)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1169.png"
 ALT="$\displaystyle f = \pm\frac{2 l}{k} (1-k^{ 2} \sin^2\phi)^{1/2}.$">|; 

$key = q/{bfgrad},h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6524.png"
 ALT="$ {\bf grad} h$">|; 

$key = q/displaystylenablacdot{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7069.png"
 ALT="$\displaystyle \nabla\cdot{\bf A}$">|; 

$key = q/k=sinbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2247.png"
 ALT="$ k=\sin\beta$">|; 

$key = q/Pequiv(x_1,,x_2,,x_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7229.png"
 ALT="$ P\equiv (x_1, x_2, x_3)$">|; 

$key = q/RR';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2538.png"
 ALT="$ RR'$">|; 

$key = q/displaystylepsi(xi_0,eta)=-frac{1}{2},V,b^{,2},sin^2eta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2807.png"
 ALT="$\displaystyle \psi(\xi_0,\eta)= -\frac{1}{2} V b^{ 2} \sin^2\eta,$">|; 

$key = q/displaystyle=overline{p}_0-p_0+(rho-overline{rho}),g,a,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3860.png"
 ALT="$\displaystyle = \overline{p}_0-p_0 +(\rho-\overline{\rho}) g a \cos\theta$">|; 

$key = q/left.v_zright|_{z=-d}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4089.png"
 ALT="$ \left.v_z\right\vert _{z=-d}=0$">|; 

$key = q/eta=q(xi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6027.png"
 ALT="$ \eta=q(\xi)$">|; 

$key = q/y=x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6760.png"
 ALT="$ y= x_2$">|; 

$key = q/{rmMa}_2=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$ {\rm Ma}_2=1$">|; 

$key = q/n=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2489.png"
 ALT="$ n=3$">|; 

$key = q/beta_{n,,-n'}^{,m,,-m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5019.png"
 ALT="$ \beta_{n, -n'}^{ m, -m'}$">|; 

$key = q/displaystylev_x(x,y)=U_0,f'!left[1.22,frac{|y|}{delta(x)}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2991.png"
 ALT="$\displaystyle v_x(x,y) = U_0 f'\!\left[1.22 \frac{\vert y\vert}{\delta(x)}\right],$">|; 

$key = q/stackrel{displaystylerightarrow}{DB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="29" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6702.png"
 ALT="$ \stackrel{\displaystyle \rightarrow}{DB}$">|; 

$key = q/displaystylew=c,{rmMa}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5767.png"
 ALT="$\displaystyle w = c {\rm Ma}.$">|; 

$key = q/displaystyle(nabla^{,2}{bfA})_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7197.png"
 ALT="$\displaystyle (\nabla^{ 2}{\bf A})_\phi$">|; 

$key = q/S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$ S$">|; 

$key = q/{bfv}_1=-nablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4077.png"
 ALT="$ {\bf v}_1=-\nabla\phi$">|; 

$key = q/displaystylefrac{v'}{U}simeq-frac{partialfslashpartialx}{partialfslashpartialy}=frac{dy}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5937.png"
 ALT="$\displaystyle \frac{v'}{U} \simeq -\frac{\partial f/\partial x}{\partial f/\partial y} = \frac{dy}{dx},$">|; 

$key = q/h=d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3963.png"
 ALT="$ h=d$">|; 

$key = q/{mitPhi}_n({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4371.png"
 ALT="$ {\mit\Phi}_n({\bf r})$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter13slashfig4.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5224.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter13/fig4.03.eps}}
\end{figure}">|; 

$key = q/beta=2,{mitOmega},sinthetaslasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5106.png"
 ALT="$ \beta=2 {\mit \Omega} \sin\theta/a$">|; 

$key = q/(x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2054.png"
 ALT="$ (x$">|; 

$key = q/displaystyleomega=frac{partial^{,2}psi}{partialy^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2911.png"
 ALT="$\displaystyle \omega = \frac{\partial^{ 2}\psi}{\partial y^{ 2}},$">|; 

$key = q/displaystyleK=K_{rmfluid}+K_{rmcylinder}=frac{1}{2},(m+m'),V_x^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1826.png"
 ALT="$\displaystyle K = K_{\rm fluid}+K_{\rm cylinder} = \frac{1}{2} (m+m') V_x^{ 2}.$">|; 

$key = q/displaystyleQ=int_0^dv_x,dy=frac{G(x),d^{,3}(x)}{12,mu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3666.png"
 ALT="$\displaystyle Q = \int_0^d v_x dy = \frac{G(x) d^{ 3}(x)}{12 \mu}.$">|; 

$key = q/displaystyle=left[frac{x^{,2}slashl^{,2}+y^{,2}slashl^{,2}+4+left([x^{,2}slashl^]^2-16,x^{,2}slashl^{,2}right)^{1slash2}}{8}right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2141.png"
 ALT="$\displaystyle =\left[\frac{x^{ 2}/l^{ 2}+y^{ 2}/l^{ 2}+4+\left([x^{ 2}/l^{ 2}+y^{ 2}/l^{ 2}+4]^2-16 x^{ 2}/l^{ 2}\right)^{1/2}}{8}\right]^{1/2}.$">|; 

$key = q/dskew{2}bar{z}=dx-{rmi},dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2399.png"
 ALT="$ d\skew{2}\bar{z} = dx -{\rm i} dy$">|; 

$key = q/p_2slashp_1>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5504.png"
 ALT="$ p_2/p_1&gt;1$">|; 

$key = q/displaystyletheta_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6202.png"
 ALT="$\displaystyle \theta_2'$">|; 

$key = q/displaystylefrac{partialF}{partialy_k'}={rmconst}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7367.png"
 ALT="$\displaystyle \frac{\partial F}{\partial y_k'} = {\rm const}.$">|; 

$key = q/displaystylephi(r,theta,t)rightarrow-frac{mitGamma}{2pi,r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1828.png"
 ALT="$\displaystyle \phi(r,\theta,t)\rightarrow -\frac{\mit\Gamma}{2\pi r}$">|; 

$key = q/displaystylez-z_0=int_b^r,frac{df}{f_z}=pmint_b^rfrac{df}{(f^{,2}slashb^{,2}-1)^{1slash2}}=pmb,cosh^{-1}(rslashb),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1257.png"
 ALT="$\displaystyle z-z_0 = \int_b^r \frac{df}{f_z} = \pm\int_b^r\frac{df}{(f^{ 2}/b^{ 2}-1)^{1/2}}=\pm b \cosh^{-1}(r/b),$">|; 

$key = q/x_0-x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3112.png"
 ALT="$ x_0-x$">|; 

$key = q/skew{5}hat{zeta}_j(vartheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4708.png"
 ALT="$ \skew{5}\hat{\zeta}_j(\vartheta)$">|; 

$key = q/displaystylenablacdotleft(kappa,nablaTright)=c,frac{partialT}{partialt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6609.png"
 ALT="$\displaystyle \nabla\cdot\left(\kappa  \nabla T\right) = c \frac{\partial T}{\partial t}.$">|; 

$key = q/5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$ 5$">|; 

$key = q/Z_1(x,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5100.png"
 ALT="$ Z_1(x,0)$">|; 

$key = q/delta(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3375.png"
 ALT="$ \delta(y)$">|; 

$key = q/displaystyle=1.79times10^{-1},{rmm}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4581.png"
 ALT="$\displaystyle = 1.79\times 10^{-1} {\rm m}.$">|; 

$key = q/alll;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1144.png"
 ALT="$ a\ll l$">|; 

$key = q/displaystylenabla(phi,psi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6681.png"
 ALT="$\displaystyle \nabla(\phi \psi)$">|; 

$key = q/displaystylep=p_0'-rholeft[{mitPsi}-frac{1}{2},omega^{,2},(x_1^{,2}+x_2^{,2})right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img855.png"
 ALT="$\displaystyle p = p_0'-\rho\left[{\mit\Psi}-\frac{1}{2} \omega^{ 2} (x_1^{ 2}+x_2^{ 2})\right],$">|; 

$key = q/displaystylefrac{partialF}{partialy'}={rmconst}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7319.png"
 ALT="$\displaystyle \frac{\partial F}{\partial y'} = {\rm const}.$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}-frac{partialphi}{partialt}=frac{p_0}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1808.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} -\frac{\partial\phi}{\partial t} = \frac{p_0}{\rho},$">|; 

$key = q/displaystyle{rmMa}_2lesseqgtr1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5481.png"
 ALT="$\displaystyle {\rm Ma}_2\lesseqgtr 1$">|; 

$key = q/v_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4169.png"
 ALT="$ v_c$">|; 

$key = q/displaystylesigma_{yz}=sigma_{zy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="$\displaystyle \sigma_{yz}=\sigma_{zy}$">|; 

$key = q/displaystyle{rmRe}=frac{U_0,L}{nu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2892.png"
 ALT="$\displaystyle {\rm Re} = \frac{U_0 L}{\nu}$">|; 

$key = q/displaystyle-nablaleft(frac{p}{rho}+{mitPhi}+chiright)simeq{bf0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4387.png"
 ALT="$\displaystyle -\nabla\left(\frac{p}{\rho}+{\mit\Phi}+\chi\right)\simeq {\bf0},$">|; 

$key = q/displaystyle=nabla^{,2}A_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7164.png"
 ALT="$\displaystyle =\nabla^{ 2} A_z.$">|; 

$key = q/r(xi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2730.png"
 ALT="$ r(\xi)$">|; 

$key = q/displaystyle=-nablaP+mu,nabla^{,2}{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3572.png"
 ALT="$\displaystyle = - \nabla P + \mu \nabla^{ 2}{\bf v},$">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1052.png"
 ALT="$ k$">|; 

$key = q/{bfu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7023.png"
 ALT="$ {\bf u}$">|; 

$key = q/{bfF}=-(rho,Q^{,2}slash4pi,a),{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2421.png"
 ALT="$ {\bf F} = -(\rho Q^{ 2}/4\pi a) {\bf e}_y$">|; 

$key = q/{bfa}equiv(a_x,,a_y,,a_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6263.png"
 ALT="$ {\bf a} \equiv (a_x, a_y, a_z)$">|; 

$key = q/h_1=h_2=h_3=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7090.png"
 ALT="$ h_1=h_2=h_3=1$">|; 

$key = q/partialu'slashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5913.png"
 ALT="$ \partial u'/\partial x$">|; 

$key = q/displaystylefrac{V_0}{V}=s,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img599.png"
 ALT="$\displaystyle \frac{V_0}{V} = s,$">|; 

$key = q/displaystyleepsilon_{mis},a_{sjkl}+epsilon_{mjs},a_{iskl}+epsilon_{mks},a_{ijsl}+epsilon_{mls},a_{ijks}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6935.png"
 ALT="$\displaystyle \epsilon_{mis} a_{sjkl} + \epsilon_{mjs} a_{iskl}+\epsilon_{mks} a_{ijsl} +\epsilon_{mls} a_{ijks}=0.$">|; 

$key = q/displaystylesim{calO}left(frac{v_x^{,2}}{l}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3655.png"
 ALT="$\displaystyle \sim {\cal O}\left(\frac{v_x^{ 2}}{l}\right),$">|; 

$key = q/{bfn};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$ {\bf n}$">|; 

$key = q/{bfv}=D{bfv}slashDt={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img558.png"
 ALT="$ {\bf v} = D{\bf v}/Dt= {\bf0}$">|; 

$key = q/{bfc}cdot({bfa}times{bfb})=({bfb}times{bfa})cdot{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6712.png"
 ALT="$ {\bf c}\cdot ({\bf a}\times {\bf b}) = ({\bf b}\times {\bf a})\cdot{\bf c}$">|; 

$key = q/displaystylefrac{3sqrt{2},rho,Q^{,2}}{8pi,a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2442.png"
 ALT="$\displaystyle \frac{3\sqrt{2} \rho Q^{ 2}}{8\pi a}
$">|; 

$key = q/displaystyle=frac{t}{36,s^{,2}}left[t^{,2}-12,s,(1-s)+2right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img757.png"
 ALT="$\displaystyle = \frac{t}{36 s^{ 2}}\left[t^{ 2}-12 s (1-s)+2\right],$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partialg(theta,phi_pm)}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4800.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial g(\theta,\phi_\pm)}{\partial \phi}$">|; 

$key = q/zeta(l)<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3999.png"
 ALT="$ \zeta(l)&lt;0$">|; 

$key = q/displaystylef'(x)=frac{U}{lambda}left(frac{dy}{dx}right)_U.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6053.png"
 ALT="$\displaystyle f'(x)= \frac{U}{\lambda}\left(\frac{dy}{dx}\right)_U.$">|; 

$key = q/displaystyle(U+u'),frac{partialf}{partialx}+v',frac{partialf}{partialy}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5936.png"
 ALT="$\displaystyle (U+u') \frac{\partial f}{\partial x} + v' \frac{\partial f}{\partial y} =0.$">|; 

$key = q/displaystyle{rmRe}(x)=frac{U(x),x}{nu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2949.png"
 ALT="$\displaystyle {\rm Re}(x) = \frac{U(x) x}{\nu}$">|; 

$key = q/displaystyle{bfv}=-nablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1587.png"
 ALT="$\displaystyle {\bf v} = -\nabla\phi$">|; 

$key = q/gammageq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1770.png"
 ALT="$ \gamma\geq 0$">|; 

$key = q/displaystylepsi=-frac{1}{2},V,varpi^{,2}+frac{Q}{4pi},(costheta_1-costheta_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2835.png"
 ALT="$\displaystyle \psi = - \frac{1}{2} V \varpi^{ 2} + \frac{Q}{4\pi} (\cos\theta_1-\cos\theta_2),$">|; 

$key = q/displaystyle(g,pi)^{1slash2},(n^{,2},l^{,-2}+m^{,2},b^{,-2})^{1slash4},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4203.png"
 ALT="$\displaystyle (g \pi)^{1/2} (n^{ 2} l^{ -2}+m^{ 2} b^{ -2})^{1/4},
$">|; 

$key = q/displaystyle=V,dp+T,dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5332.png"
 ALT="$\displaystyle = V dp + T dS,$">|; 

$key = q/displaystyle{mitPsi}=-frac{1}{2},G,rhoointleft[frac{2sum_{i,j=1,3}x_i,x_j,n_i,n_a_i^{,2}}{sum_{i=1,3}n_i^{,2}slasha_i^{,2}}right]d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="576" HEIGHT="98" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7250.png"
 ALT="$\displaystyle {\mit\Psi} = -\frac{1}{2} G \rho\oint\left[ \frac{2\sum_{i,j=1,...
...,3}x_i^{ 2}/a_i^{ 2}}{\sum_{i=1,3}n_i^{ 2}/a_i^{ 2}} \right] d{\mit\Omega}.$">|; 

$key = q/epsilon_{mjt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6910.png"
 ALT="$ \epsilon_{mjt}$">|; 

$key = q/displaystyleC_{p_0}simeqfrac{2}{U},frac{partialskew{3}bar{phi}}{partialxi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6034.png"
 ALT="$\displaystyle C_{p_0} \simeq \frac{2}{U} \frac{\partial\skew{3}\bar{\phi}}{\partial\xi}.$">|; 

$key = q/displaystyle{bfS}=sum_idelta{bfS}_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6347.png"
 ALT="$\displaystyle {\bf S} = \sum_i \delta {\bf S}_i.$">|; 

$key = q/displaystylebeta-theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6119.png"
 ALT="$\displaystyle \beta-\theta$">|; 

$key = q/displaystylerho,frac{Dv_i}{Dt}=F_i-frac{partialp}{partialx_i}+muleft(frac{partiarac{1}{3},frac{partial^{,2}v_j}{partialx_i,partialx_j}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="$\displaystyle \rho  \frac{D v_i}{Dt}= F_i - \frac{\partial p}{\partial x_i} +\...
..._j}+ \frac{1}{3} \frac{\partial^{ 2} v_j}{\partial x_i \partial x_j}\right).$">|; 

$key = q/{mitDelta}=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6112.png"
 ALT="$ {\mit\Delta}=\pi$">|; 

$key = q/displaystyle=-frac{(m+1),U_0^{,2},x^{,2,m-1}}{2,nu},f''',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2935.png"
 ALT="$\displaystyle = -\frac{(m+1) U_0^{ 2} x^{ 2 m-1}}{2 \nu} f''',$">|; 

$key = q/displaystylefrac{int_{A_0}int_0^{delta'}y,dx,dy,dz}{V_0}=frac{int_{A_0}y,[delta(x,y)-theta,y],dx,dy}{V_0}=-theta,frac{kappa_x^{,2},A_0}{V_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="477" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img639.png"
 ALT="$\displaystyle \frac{\int_{A_0}\int_0^{\delta'}y dx dy dz}{V_0}=\frac{\int_{A...
...[\delta(x,y)-\theta y] dx dy}{V_0}=-\theta \frac{\kappa_x^{ 2} A_0}{V_0},$">|; 

$key = q/displaystylefrac{p}{rho}={rmconstant}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5116.png"
 ALT="$\displaystyle \frac{p}{\rho} = {\rm constant}.$">|; 

$key = q/displaystyle{bfq};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img520.png"
 ALT="$\displaystyle {\bf q}$">|; 

$key = q/displaystylec_{ijkl}=K,delta_{ij},delta_{kl}+mu,[delta_{ik},delta_{jl}+delta_{il},delta_{jk}-(2slash3),delta_{ij},delta_{kl}].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7022.png"
 ALT="$\displaystyle c_{ijkl} = K \delta_{ij} \delta_{kl} + \mu [\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk} - (2/3) \delta_{ij} \delta_{kl}].
$">|; 

$key = q/dS_1=h_2,h_3,du_2,du_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7051.png"
 ALT="$ dS_1 = h_2 h_3 du_2 du_3$">|; 

$key = q/dS_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$ dS_i$">|; 

$key = q/displaystyle=lnleft{left[frac{1-gamma+2,gamma,{rmMa}_1^{,2},sin^2beta}{gamma+1}r{rmMa}_1^{,2},sin^2beta}right]^{-frac{gamma}{1-gamma}}right},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="559" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5691.png"
 ALT="$\displaystyle =\ln\left\{\left[ \frac{1-\gamma+2 \gamma {\rm Ma}_1^{ 2} \si...
...a-1) {\rm Ma}_1^{ 2} \sin^2\beta}\right]^{-\frac{\gamma}{1-\gamma}}\right\},$">|; 

$key = q/displaystyletau_1=2pi,r_1^{,2},sigma_{rtheta}(r_1)=-4pi,muleft(frac{{mitOmega}_1-{mitOmega}_2}{r_1^{,-2}-r_2^{,-2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3642.png"
 ALT="$\displaystyle \tau_1 = 2\pi r_1^{ 2} \sigma_{r\theta}(r_1) = -4\pi \mu\left(\frac{{\mit\Omega}_1-{\mit\Omega}_2}{r_1^{ -2}-r_2^{ -2}}\right).$">|; 

$key = q/+8.95times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4631.png"
 ALT="$ +8.95\times 10^{-2}$">|; 

$key = q/nablatimes(phi,{bfa})=phi,nablatimes{bfa}+nablaphitimes{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6988.png"
 ALT="$ \nabla\times (\phi {\bf a})= \phi \nabla\times {\bf a} + \nabla\phi\times {\bf a}$">|; 

$key = q/ngeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4238.png"
 ALT="$ n\geq 0$">|; 

$key = q/displaystyle{bfF}=frac{rho}{4pi}int_Sint_{S'}frac{[mbox{boldmathOmega}_S({bfr}')mesmbox{boldmathOmega}_S({bfr})}{|{bfr}-{bfr}'|^{,3}},dS,dS'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3457.png"
 ALT="$\displaystyle {\bf F} = \frac{\rho}{4\pi}\int_S\int_{S'}\frac{[\mbox{\boldmath$...
...mbox{\boldmath$\Omega$}_S({\bf r})}{\vert{\bf r}-{\bf r}'\vert^{ 3}} dS dS'.$">|; 

$key = q/displaystylea_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img926.png"
 ALT="$\displaystyle a_2$">|; 

$key = q/r'gga;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4227.png"
 ALT="$ r'\gg a$">|; 

$key = q/displaystyle=frac{U,epsilon}{m},cos(alpha,x),{rme}^{-m,alpha,y},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5975.png"
 ALT="$\displaystyle =\frac{U \epsilon}{m} \cos(\alpha x) {\rm e}^{-m \alpha y},$">|; 

$key = q/2,{rmh},39,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1076.png"
 ALT="$ 2 {\rm h} 39 {\rm m}$">|; 

$key = q/displaystyleleft(frac{p_0}{p_{rmatm}}right)_{rmcrit}=left(frac{gamma+1}{2}right)^{gammaslash(gamma-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5428.png"
 ALT="$\displaystyle \left(\frac{p_0}{p_{\rm atm}}\right)_{\rm crit} = \left(\frac{\gamma+1}{2}\right)^{\gamma/(\gamma-1)}.$">|; 

$key = q/nablacdot{bfn}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1110.png"
 ALT="$ \nabla\cdot{\bf n}=0$">|; 

$key = q/{rmRe}^{,-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3274.png"
 ALT="$ {\rm Re}^{ -1/2}$">|; 

$key = q/p_2';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5782.png"
 ALT="$ p_2'$">|; 

$key = q/displaystyle{mitOmega}={rmi},theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2221.png"
 ALT="$\displaystyle {\mit\Omega}={\rm i} \theta,$">|; 

$key = q/displaystylephantom{=}-int_{mitOmega}frac{1}{sintheta}left(frac{partial{mitPsi}'ltheta},frac{partial{mitPhi}_r}{partialphi}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4900.png"
 ALT="$\displaystyle \phantom{=}-\int_{\mit\Omega}\frac{1}{\sin\theta}\left(\frac{\par...
...\partial\theta} \frac{\partial{\mit\Phi}_r}{\partial\phi}\right)d{\mit\Omega}.$">|; 

$key = q/partialslashpartialxsim{calO}(1slashl);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3651.png"
 ALT="$ \partial/\partial x\sim {\cal O}(1/l)$">|; 

$key = q/b=c-l^{,2}slashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2505.png"
 ALT="$ b=c-l^{ 2}/c$">|; 

$key = q/J=J_r+{rmi},J_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2343.png"
 ALT="$ J=J_r+{\rm i} J_i$">|; 

$key = q/displaystylefrac{costheta}{sintheta},frac{partial{mitPsi}_r}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4862.png"
 ALT="$\displaystyle \frac{\cos\theta}{\sin\theta} \frac{\partial{\mit\Psi}_r}{\partial\phi}$">|; 

$key = q/displaystyle=frac{1}{m},frac{partial^{,2}skew{3}bar{phi}}{partialxi^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6021.png"
 ALT="$\displaystyle = \frac{1}{m} \frac{\partial^{ 2}\skew{3}\bar{\phi}}{\partial \xi^{ 2}},$">|; 

$key = q/B_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5968.png"
 ALT="$ B_1$">|; 

$key = q/displaystylefrac{partialzeta}{partialt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4507.png"
 ALT="$\displaystyle \frac{\partial \zeta}{\partial t}$">|; 

$key = q/displaystyleT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="$\displaystyle T$">|; 

$key = q/Q={mitPsi}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4768.png"
 ALT="$ Q={\mit\Psi}'$">|; 

$key = q/{bfU}(P);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3425.png"
 ALT="$ {\bf U}(P)$">|; 

$key = q/displaystyle=-frac{rho,D_1,D_2}{pi,x^{,3}},cos(alpha_1+alpha_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2457.png"
 ALT="$\displaystyle = - \frac{\rho D_1 D_2}{\pi x^{ 3}} \cos(\alpha_1+\alpha_2),$">|; 

$key = q/displaystyle=frac{T,v,alpha^{,2}}{kappa_T},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5603.png"
 ALT="$\displaystyle = \frac{T v \alpha^{ 2}}{\kappa_T},$">|; 

$key = q/zrightarrow-z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1190.png"
 ALT="$ z\rightarrow -z$">|; 

$key = q/xslashd;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2521.png"
 ALT="$ x/d$">|; 

$key = q/{multline}left(1-[2,n,A_n+(2+n),B_n],muright)left(p_n,x_iright)_{r=a}-muleft[(2,1}right)frac{skew{3}bar{rho}}{rho},zeta_nright](x_i)_{r=a},{multline};LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="689" HEIGHT="115" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4419.png"
 ALT="\begin{multline}
\left(1-[2 n A_n+(2+n) B_n] \mu\right)\left(p_n x_i\right)...
...ight)
\frac{\skew{3}\bar{\rho}}{\rho} \zeta_n\right](x_i)_{r=a},
\end{multline}">|; 

$key = q/displaystylea_{ijkl}=a_{jilk}=a_{klij}=a_{lkji};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6964.png"
 ALT="$\displaystyle a_{ijkl} = a_{jilk} = a_{klij}=a_{lkji}$">|; 

$key = q/0leqxleqp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7382.png"
 ALT="$ 0\leq x\leq p$">|; 

$key = q/phi=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1174.png"
 ALT="$ \phi=\pi/2$">|; 

$key = q/{mitGamma}_r(r)=2pi,r,v_theta(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1718.png"
 ALT="$ {\mit\Gamma}_r(r) = 2\pi r v_\theta(r)$">|; 

$key = q/displaystylenablapsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7128.png"
 ALT="$\displaystyle \nabla \psi$">|; 

$key = q/displaystyle=-nabla[nablacdot(psi,nablavarphi)]+nabla^{,2}(psi,nablavarphi)+nabla{mitOmega}_varphitimesnablavarphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2561.png"
 ALT="$\displaystyle = -\nabla[\nabla\cdot(\psi \nabla\varphi)] + \nabla^{ 2}(\psi \nabla\varphi)+\nabla{\mit\Omega}_\varphi\times \nabla\varphi$">|; 

$key = q/displaystyleH_2=h_2+frac{1}{2},frac{v_2^{,2}}{g},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1500.png"
 ALT="$\displaystyle H_2= h_2+\frac{1}{2} \frac{v_2^{ 2}}{g},$">|; 

$key = q/displaystylealpha=frac{d_1-d_2}{l}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3694.png"
 ALT="$\displaystyle \alpha = \frac{d_1-d_2}{l}.$">|; 

$key = q/{bfe}_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7171.png"
 ALT="$ {\bf e}_\phi$">|; 

$key = q/displaystyleP=p+rho,{mitPsi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3573.png"
 ALT="$\displaystyle P = p + \rho {\mit\Psi},$">|; 

$key = q/psi(r)=(1slash2),omega,a^{,2},ln(rslasha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1912.png"
 ALT="$ \psi(r)=(1/2) \omega a^{ 2} \ln(r/a)$">|; 

$key = q/F=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1404.png"
 ALT="$ F=0$">|; 

$key = q/aslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2495.png"
 ALT="$ a/2)$">|; 

$key = q/rho,{mitGamma}^{,2}slash(4pi,a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2441.png"
 ALT="$ \rho {\mit\Gamma}^{ 2}/(4\pi a)$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialx^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6020.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial x^{ 2}}$">|; 

$key = q/partialslashpartialysim{calO}(1slashd);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3652.png"
 ALT="$ \partial/\partial y\sim {\cal O}(1/d)$">|; 

$key = q/displaystyle=left[1+left(frac{gamma-1}{gamma+1}right)left({rmMa}^{,2}-1right)right]^{-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5402.png"
 ALT="$\displaystyle =\left[1+\left(\frac{\gamma-1}{\gamma+1}\right)\left({\rm Ma}^{ 2}-1\right)\right]^{-1},$">|; 

$key = q/displaystylesqrt{frac{p_2}{p_1}}=frac{c_4}{c_1}frac{sqrt{2,gamma_1,(gamma_1+1)}}{(gamma_4-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5653.png"
 ALT="$\displaystyle \sqrt{\frac{p_2}{p_1}} = \frac{c_4}{c_1}\frac{\sqrt{2 \gamma_1 (\gamma_1+1)}}{(\gamma_4-1)},
$">|; 

$key = q/displaystylesimeq-left(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right)|theta|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6161.png"
 ALT="$\displaystyle \simeq -\left(\frac{\gamma {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right)\vert\theta\vert,$">|; 

$key = q/sinalpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2446.png"
 ALT="$ \sin\alpha)$">|; 

$key = q/displaystylefrac{dy_U}{dx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5841.png"
 ALT="$\displaystyle \frac{dy_U}{dx}$">|; 

$key = q/displaystyle-left(frac{delta{mitPhi}}{g}right)_{r=a}=sum_{n=1,infty}left[(1+k_2)skew{5}bar{zeta}_2,delta_{n,2}+(1+k_n'),alpha_n,zeta_nright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="444" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4439.png"
 ALT="$\displaystyle - \left(\frac{\delta {\mit\Phi}}{g}\right)_{r=a} =\sum_{n=1,\inft...
..._2) \skew{5}\bar{\zeta}_2 \delta_{n 2} + (1+k_n') \alpha_n \zeta_n\right],$">|; 

$key = q/displaystyle=frac{1-gamma+2,gamma,M_1^{,2}}{gamma+1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5484.png"
 ALT="$\displaystyle = \frac{1-\gamma+2 \gamma M_1^{ 2}}{\gamma+1},$">|; 

$key = q/displaystyleQ=int_0^hv_x,dy=frac{G,h^{,3}}{3,mu}=frac{g,sinalpha,h^{,3}}{3,nu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3598.png"
 ALT="$\displaystyle Q = \int_0^h v_x dy = \frac{G h^{ 3}}{3 \mu} = \frac{g \sin\alpha h^{ 3}}{3 \nu},$">|; 

$key = q/displaystylefrac{{mitDelta}T}{T_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5512.png"
 ALT="$\displaystyle \frac{{\mit\Delta}T}{T_1}$">|; 

$key = q/displaystyle(AB)^2=(OA)^2+(OB)^2-2,(OA),(OB),costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6314.png"
 ALT="$\displaystyle (AB)^2 = (OA)^2 + (OB)^2 - 2  (OA) (OB) \cos\theta,$">|; 

$key = q/displaystyled{calH}=frac{dp}{rho};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5338.png"
 ALT="$\displaystyle d{\cal H} = \frac{dp}{\rho}$">|; 

$key = q/displaystyle=int_0^piint_{-1}^{1}left[(1-mu^{,2}),frac{partial{mitPhi}_n^{,m}}{prac{partial{mitPsi}_{n'}^{,m'}}{partialphi}right]mu,dmu,dphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4953.png"
 ALT="$\displaystyle = \int_0^\pi\int_{-1}^{1}\left[(1-\mu^{ 2})  \frac{\partial{\mi...
...  \frac{\partial{\mit\Psi}_{n'}^{ m'}}{\partial\phi}\right] \mu d\mu d\phi,$">|; 

$key = q/f_x=-1slashtantheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1195.png"
 ALT="$ f_x=-1/\tan\theta$">|; 

$key = q/displaystylelambda={rmi},expleft(-frac{pi}{2},frac{bar{F}}{skew{3}bar{psi}_1}right)={rmi},sinhleft(frac{pi}{2,alpha},{mitOmega}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2242.png"
 ALT="$\displaystyle \lambda = {\rm i} \exp\left(-\frac{\pi}{2} \frac{\bar{F}}{\skew...
...\psi}_1}\right)={\rm i} \sinh\left(\frac{\pi}{2 \alpha} {\mit\Omega}\right),$">|; 

$key = q/displaystylefrac{Drho}{Dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$\displaystyle \frac{D\rho}{Dt}$">|; 

$key = q/dxslashdt=c_2-V_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5553.png"
 ALT="$ dx/dt= c_2-V_p$">|; 

$key = q/displaystylea=c,{rme}^{,xi_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2731.png"
 ALT="$\displaystyle a = c {\rm e}^{ \xi_0}$">|; 

$key = q/1slashn^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5757.png"
 ALT="$ 1/n^{ 2}$">|; 

$key = q/{bfb}=(1slash!sqrt{2},,-1slash!sqrt{2},,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6356.png"
 ALT="$ {\bf b} = (1/\!\sqrt{2},  -1/\!\sqrt{2}, 0)$">|; 

$key = q/displaystyle=left(frac{p_2}{p_1}right)left[frac{gamma+1+(gamma-1),(p_2slashp_1)}{gamma-1+(gamma+1),(p_2slashp_1)}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5492.png"
 ALT="$\displaystyle =\left(\frac{p_2}{p_1}\right)\left[\frac{\gamma+1+(\gamma-1) (p_2/p_1)}{\gamma-1+(\gamma+1) (p_2/p_1)}\right].$">|; 

$key = q/displaystylec<c_0=sqrt{frac{2}{3}},b.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img679.png"
 ALT="$\displaystyle c &lt; c_0 = \sqrt{\frac{2}{3}} b.$">|; 

$key = q/displaystyle=-frac{G,(m+m')}{r'^{,3}},{bfr}''cdot{bfr}=-frac{G,m'}{r'^{,3}},{bfr}'cdot{bfr}=-frac{G,m'}{r'^{,2}},r,cosgamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="440" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4234.png"
 ALT="$\displaystyle =-\frac{G (m+m')}{r'^{ 3}} {\bf r}''\cdot{\bf r}= - \frac{G m'}{r'^{ 3}} {\bf r}'\cdot{\bf r}=-\frac{G m'}{r'^{ 2}} r \cos\gamma$">|; 

$key = q/nablacdot{bfn};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1112.png"
 ALT="$ \nabla\cdot{\bf n}$">|; 

$key = q/r_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1346.png"
 ALT="$ r_+$">|; 

$key = q/j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$ j$">|; 

$key = q/displaystyle{bfW}=-W,{bfe}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img594.png"
 ALT="$\displaystyle {\bf W} =- W {\bf e}_z,$">|; 

$key = q/displaystylep=p_0+rho,V^{,2},sin^2(thetaslash2)left[frac{3}{2},sin^2(thetaslash2)-1right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2821.png"
 ALT="$\displaystyle p = p_0 + \rho V^{ 2} \sin^2(\theta/2)\left[\frac{3}{2} \sin^2(\theta/2)-1\right],
$">|; 

$key = q/displaystyledelta=left[frac{1+[(gamma-1)slash2],{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right]theta_1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6176.png"
 ALT="$\displaystyle \delta = \left[\frac{1+[(\gamma-1)/2] {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right]\theta_1.
$">|; 

$key = q/displaystylepsi(r,theta)=R(r),{mitTheta}(theta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1730.png"
 ALT="$\displaystyle \psi(r,\theta) = R(r) {\mit\Theta}(\theta).$">|; 

$key = q/displaystyleleft(frac{h_2}{h_1}right)^2+frac{h_2}{h_1}-2,{rmFr}_1^{,2}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1482.png"
 ALT="$\displaystyle \left(\frac{h_2}{h_1}\right)^2+\frac{h_2}{h_1}-2 {\rm Fr}_1^{ 2} = 0,$">|; 

$key = q/0^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1134.png"
 ALT="$ 0^\circ$">|; 

$key = q/displaystylef(rrightarrowinfty);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3770.png"
 ALT="$\displaystyle f(r\rightarrow\infty)$">|; 

$key = q/p(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5157.png"
 ALT="$ p(z)$">|; 

$key = q/v(x,0)=-v_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3259.png"
 ALT="$ v(x,0)=-v_s$">|; 

$key = q/{bfW};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img585.png"
 ALT="$ {\bf W}$">|; 

$key = q/displaystylefrac{d}{dxi}left(frac{1}{sinhxi},frac{dF}{dxi}right)-frac{2}{sinhxi},F=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2800.png"
 ALT="$\displaystyle \frac{d}{d\xi}\left(\frac{1}{\sinh\xi} \frac{dF}{d\xi}\right) - \frac{2}{\sinh\xi} F=0.$">|; 

$key = q/displaystylembox{boldmath{{omega{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img808.png"
 ALT="$\displaystyle \mbox{\boldmath$\omega$}$">|; 

$key = q/y=y(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7369.png"
 ALT="$ y = y(x)$">|; 

$key = q/{bfC};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6734.png"
 ALT="$ {\bf C}$">|; 

$key = q/displaystyleG(x)=-frac{dp}{dx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3697.png"
 ALT="$\displaystyle G(x) = - \frac{dp}{dx}$">|; 

$key = q/displaystyle=int_0^inftyleft[(U-u),frac{dU}{dx}+u,frac{partial(U-u)}{partialx}-(U-u),frac{partialv}{partialy}right]dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="428" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3082.png"
 ALT="$\displaystyle =\int_0^\infty\left[(U-u) \frac{dU}{dx} + u \frac{\partial(U-u)}{\partial x} -(U-u) \frac{\partial v}{\partial y}\right]dy$">|; 

$key = q/h_+=Qslashv_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1440.png"
 ALT="$ h_+= Q/v_+$">|; 

$key = q/a_x,,a_y,,a_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6291.png"
 ALT="$ a_x,  a_y,  a_z$">|; 

$key = q/displaystylev_varpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2546.png"
 ALT="$\displaystyle v_\varpi$">|; 

$key = q/displaystyleoint_S{bfA}cdotd{bfS}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6590.png"
 ALT="$\displaystyle \oint_S {\bf A}\cdot d{\bf S} =0$">|; 

$key = q/displaystylefrac{5}{4},M,g+frac{3}{16},M,omega^{,2},a,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1066.png"
 ALT="$\displaystyle \frac{5}{4} M g+ \frac{3}{16} M \omega^{ 2} a,
$">|; 

$key = q/displaystyle(nablaphi)_i=frac{1}{h_i},frac{partialphi}{partialu_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7067.png"
 ALT="$\displaystyle (\nabla\phi)_i = \frac{1}{h_i} \frac{\partial \phi}{\partial u_i}.$">|; 

$key = q/n=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5279.png"
 ALT="$ n=0$">|; 

$key = q/B=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4790.png"
 ALT="$ B=0$">|; 

$key = q/alpha=1slashsqrt{f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3117.png"
 ALT="$ \alpha=1/\sqrt{f}$">|; 

$key = q/dbar{F}slashdzrightarrowV,(sinalpha+{rmi},cosalpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2292.png"
 ALT="$ d\bar{F}/dz\rightarrow V (\sin\alpha+{\rm i} \cos\alpha)$">|; 

$key = q/alpha=pislash12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$ \alpha =\pi /12$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashbs.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img2953.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/bs.eps}}
\end{figure}">|; 

$key = q/displaystylek_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4440.png"
 ALT="$\displaystyle k_n$">|; 

$key = q/displaystyle=frac{partialpsi}{partialr},{bfe}_r+frac{1}{r}frac{partialpsi}{partia+frac{1}{r,sintheta},frac{partialpsi}{partialphi},{bfe}_phi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7178.png"
 ALT="$\displaystyle = \frac{\partial \psi}{\partial r} {\bf e}_r + \frac{1}{r}\frac{...
...a + \frac{1}{r \sin\theta} \frac{\partial \psi}{\partial \phi} {\bf e}_\phi,$">|; 

$key = q/displaystyleX=X_0,sinbeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4045.png"
 ALT="$\displaystyle X = X_0 \sin\beta,$">|; 

$key = q/y=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$ y=0$">|; 

$key = q/nu=1.0times10^{-6},{rmm^2slashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3817.png"
 ALT="$ \nu=1.0\times 10^{-6} {\rm m^2/s}$">|; 

$key = q/displaystylev_eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2695.png"
 ALT="$\displaystyle v_\eta$">|; 

$key = q/{mitPsi}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img979.png"
 ALT="$ {\mit\Psi}'$">|; 

$key = q/displaystylekappa_{n,,n'}^{,m,,m'}=left(frac{2,m}{m^{,2}-m'^{,2}}right)lambda_{n'}^{,m'},c_n^{,m},c_{n'}^{,m'},I_{n,,n'}^{,m,,m'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4984.png"
 ALT="$\displaystyle \kappa_{n, n'}^{ m, m'} = \left(\frac{2 m}{m^{ 2}-m'^{ 2}}\right)\lambda_{n'}^{ m'} c_n^{ m} c_{n'}^{ m'} I_{n, n'}^{ m, m'},$">|; 

$key = q/n+n'-m-m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4968.png"
 ALT="$ n+n'-m-m'$">|; 

$key = q/G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img596.png"
 ALT="$ G$">|; 

$key = q/mu({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="$ \mu({\bf r},t)$">|; 

$key = q/(d_1-d_2)slashd_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3709.png"
 ALT="$ (d_1-d_2)/d_1$">|; 

$key = q/dot{W};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="$ \dot{W}$">|; 

$key = q/{calE}({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="$ {\cal E}({\bf r},t)$">|; 

$key = q/{mitOmega}-2,omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4532.png"
 ALT="$ {\mit\Omega}-2 \omega_1$">|; 

$key = q/R<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6113.png"
 ALT="$ R&lt;0$">|; 

$key = q/F(z)=z^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1958.png"
 ALT="$ F(z) = z^{ 2}$">|; 

$key = q/f_x=dfslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1157.png"
 ALT="$ f_x = df/dx$">|; 

$key = q/-C_{p0}slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6075.png"
 ALT="$ -C_{p0}/2$">|; 

$key = q/displaystylew(z)=frac{{mitGamma}_0}{2,b}left{{array}{lll}1&mbox{hspace{1cm}}&|z|ash(z^{,2}-b^{,2}slash4)^{1slash2}&&|z|>bslash2{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3504.png"
 ALT="$\displaystyle w(z) = \frac{{\mit\Gamma}_0}{2 b}\left\{ \begin{array}{lll} 1&amp;\m...
... 1-\vert z\vert/(z^{ 2}-b^{ 2}/4)^{1/2}&amp;&amp;\vert z\vert&gt;b/2 \end{array}\right..$">|; 

$key = q/displaystyle=frac{2}{sqrt{{rmMa}_1^{,2}-1}},frac{dy_U}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5838.png"
 ALT="$\displaystyle = \frac{2}{\sqrt{{\rm Ma}_1^{ 2}-1}} \frac{dy_U}{dx},$">|; 

$key = q/displaystyle=int_0^piint_{-1}^{1}left(frac{partial{mitPsi}_n^{,m}}{partialmu},frfrac{partial{mitPsi}_{n'}^{,m'}}{partialmu}right)mu,dmu,dphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4955.png"
 ALT="$\displaystyle = \int_0^\pi\int_{-1}^{1}\left( \frac{\partial{\mit\Psi}_n^{ m}}...
...}  \frac{\partial{\mit\Psi}_{n'}^{ m'}}{\partial\mu}\right) \mu d\mu d\phi,$">|; 

$key = q/displaystyle(l^{,2}-a^{,2})^2=2,a,b^{,2},l,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2839.png"
 ALT="$\displaystyle (l^{ 2}-a^{ 2})^2 =2 a b^{ 2} l,
$">|; 

$key = q/displaystyleintint_Sx^{,2},y,dx,dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6497.png"
 ALT="$\displaystyle \int\int_S x^{ 2}  y dx dy$">|; 

$key = q/{mitDelta}p=p_1-p_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1108.png"
 ALT="$ {\mit\Delta} p = p_1-p_2$">|; 

$key = q/displaystyleP_2^{,1}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4267.png"
 ALT="$\displaystyle P_2^{ 1}(x)$">|; 

$key = q/d{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="$ d{\bf S}$">|; 

$key = q/displaystyleF_x(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1812.png"
 ALT="$\displaystyle F_x (t)$">|; 

$key = q/{bfr}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1684.png"
 ALT="$ {\bf r}={\bf0}$">|; 

$key = q/AA'=v_1,deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1366.png"
 ALT="$ AA'=v_1 \delta t$">|; 

$key = q/displaystyle=int_{-b}^b(p-p_0)_{x=0},dy=int_{-b}^bleft(frac{1}{2},rho,V^{,2}-frac{1}{2},rho,v^{,2}right)dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="398" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2276.png"
 ALT="$\displaystyle =\int_{-b}^b(p-p_0)_{x=0} dy=\int_{-b}^b\left(\frac{1}{2} \rho V^{ 2}-\frac{1}{2} \rho v^{ 2}\right)dy$">|; 

$key = q/displaystylefrac{partialeta}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6013.png"
 ALT="$\displaystyle \frac{\partial \eta}{\partial y}$">|; 

$key = q/displaystyle=left.left(frac{3}{2,n-2}right)rightslashleft[1+frac{(2,n+1),(2,n^{,2}+4,n+3)}{(6+n^{,2})},frac{mu}{rho,g,a}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4441.png"
 ALT="$\displaystyle = \left.\left(\frac{3}{2 n-2}\right)\right/\left[1+\frac{(2 n+1) (2 n^{ 2}+4 n+3)}{(6+n^{ 2})} \frac{\mu}{\rho g a}\right],$">|; 

$key = q/displaystylep(x,y)simeqp_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3020.png"
 ALT="$\displaystyle p(x,y)\simeq p_0$">|; 

$key = q/{bfS}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1538.png"
 ALT="$ {\bf S}_2$">|; 

$key = q/displaystyleR=r,frac{dr}{dp}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img710.png"
 ALT="$\displaystyle R = r \frac{dr}{dp}.$">|; 

$key = q/displaystylefrac{R}{a_0'}>2.455left(frac{rho'}{rho}right)^{1slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1037.png"
 ALT="$\displaystyle \frac{R}{a_0'} &gt; 2.455\left(\frac{\rho'}{\rho}\right)^{1/3},$">|; 

$key = q/V=(4slash3),pi,a_1,a_2,a_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7272.png"
 ALT="$ V=(4/3) \pi a_1 a_2 a_3$">|; 

$key = q/displaystylefrac{1}{gamma^{,2},{rmMa}_p^{,2}}=frac{1}{beta^{,2}}+frac{gamma+1}{2,gamma,beta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5539.png"
 ALT="$\displaystyle \frac{1}{\gamma^{ 2} {\rm Ma}_p^{ 2}} = \frac{1}{\beta^{ 2}}+\frac{\gamma+1}{2 \gamma \beta}.$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}+g,r,costheta-frac{partialphi}{partialt}=frac{p_0}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2664.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} + g r \cos\theta -\frac{\partial\phi}{\partial t} = \frac{p_0}{\rho},$">|; 

$key = q/{bfb}=3,{bfe}_x+2,{bfe}_y+{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6709.png"
 ALT="$ {\bf b} = 3 {\bf e}_x+ 2 {\bf e}_y+{\bf e}_z$">|; 

$key = q/+pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4050.png"
 ALT="$ +\pi/2$">|; 

$key = q/displaystyleI_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7279.png"
 ALT="$\displaystyle I_1$">|; 

$key = q/displaystylephi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6109.png"
 ALT="$\displaystyle \phi$">|; 

$key = q/displaystylephi(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1953.png"
 ALT="$\displaystyle \phi(x, y)$">|; 

$key = q/displaystyleoint_S{bfA}cdotd{bfS}=int_Vnablacdot{bfA},,dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6584.png"
 ALT="$\displaystyle \oint_S {\bf A}\cdot d{\bf S} = \int_V \nabla\cdot{\bf A}  dV,$">|; 

$key = q/displaystyle=frac{int_{-a}^aint_{a-h(x)}^ax,dx,dy}{int_{-a}^aint_{a-h(x)}^adx,dy(2slash3),a^{,3},tantheta}{4,a^{,2},s}=frac{a}{6,s},tantheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img747.png"
 ALT="$\displaystyle = \frac{\int_{-a}^a\int_{a-h(x)}^a x dx dy}{\int_{-a}^a\int_{a-...
... = \frac{(2/3) a^{ 3} \tan\theta}{4 a^{ 2} s}=\frac{a}{6 s} \tan\theta,$">|; 

$key = q/displaystyleskew{3}bar{C}_p=left.frac{q}{dT}right|_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5311.png"
 ALT="$\displaystyle \skew{3}\bar{C}_p = \left.\frac{q}{dT}\right\vert _p$">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$ z$">|; 

$key = q/_xcdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$ _x \cdot d{\bf S}$">|; 

$key = q/displaystyle{bfv}=-nablaphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1584.png"
 ALT="$\displaystyle {\bf v} = -\nabla\phi.$">|; 

$key = q/N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1338.png"
 ALT="$ N$">|; 

$key = q/phi({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6870.png"
 ALT="$ \phi({\bf x})$">|; 

$key = q/displaystylelambda_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img696.png"
 ALT="$\displaystyle \lambda_2$">|; 

$key = q/zeta_c=1slashsinhu_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1280.png"
 ALT="$ \zeta_c=1/\sinh u_c$">|; 

$key = q/displaystyle=frac{pi}{8},rho,sum_{nu=1,3,5,cdots}nu,{mitGamma}_nu^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3550.png"
 ALT="$\displaystyle =\frac{\pi}{8} \rho \sum_{\nu=1,3,5,\cdots} \nu {\mit\Gamma}_\nu^{ 2},$">|; 

$key = q/displaystyle[rho,({bfv}cdotnabla),{bfv}]_r=rholeft(v_r,frac{partial,v_r}{partialta}-frac{v_theta^{,2}}{r}right)simfrac{rho,V^{,2},a}{r^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="435" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3829.png"
 ALT="$\displaystyle [\rho ({\bf v}\cdot\nabla) {\bf v}]_r = \rho\left(v_r \frac{\p...
...\theta}- \frac{v_\theta^{ 2}}{r}\right) \sim \frac{\rho V^{ 2} a}{r^{ 2}},$">|; 

$key = q/s<1slash2-1slashsqrt{12}=0.2113;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img765.png"
 ALT="$ s&lt;1/2-1/\sqrt{12}= 0.2113$">|; 

$key = q/{mitDelta}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="$ {\mit\Delta}=0$">|; 

$key = q/displaystyle=oint_{C'}F(z),dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2365.png"
 ALT="$\displaystyle =\oint_{C'} F(z) dz,$">|; 

$key = q/displaystyle[({bfa}cdotnabla){bfb}]_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="$\displaystyle [({\bf a}\cdot\nabla){\bf b}]_i$">|; 

$key = q/displaystyle=left[frac{1+[(gamma-1)slash2],{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right]theta_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6188.png"
 ALT="$\displaystyle = \left[\frac{1+[(\gamma-1)/2] {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right]\theta_2.$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter10slashstokes1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3794.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter10/stokes1.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{partialpsi}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2928.png"
 ALT="$\displaystyle \frac{\partial\psi}{\partial y}$">|; 

$key = q/{mitOmega}-2,omega_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4558.png"
 ALT="$ {\mit\Omega}-2 \omega_E$">|; 

$key = q/M_0,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img825.png"
 ALT="$ M_0 g$">|; 

$key = q/({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4374.png"
 ALT="$ ({\bf r})$">|; 

$key = q/X(beta)=V,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4040.png"
 ALT="$ X(\beta)=V t$">|; 

$key = q/displaystyle=sindelta,cos(omega_1,t)+{calO}(e,sindelta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4300.png"
 ALT="$\displaystyle =\sin\delta \cos(\omega_1 t)+{\cal O}(e \sin\delta),$">|; 

$key = q/{bfa}cdot{bfa}times{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6367.png"
 ALT="$ {\bf a}\cdot {\bf a}\times{\bf b}$">|; 

$key = q/displaystyleA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img838.png"
 ALT="$\displaystyle A$">|; 

$key = q/zrightarrowpislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6152.png"
 ALT="$ z\rightarrow\pi/2$">|; 

$key = q/displaystyleleft[frac{-4,A_n+(n-2),B_n}{2,(n-1)}right]aleft.p_nright|_{r=a}=zeta_{b,n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4421.png"
 ALT="$\displaystyle \left[\frac{-4 A_n+(n-2) B_n}{2 (n-1)}\right]a\left.p_n\right\vert _{r=a} = \zeta_{b n},$">|; 

$key = q/yleq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5093.png"
 ALT="$ y\leq 0$">|; 

$key = q/v'slashUll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5901.png"
 ALT="$ v'/U\ll 1$">|; 

$key = q/displaystyle=a_{sisk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6958.png"
 ALT="$\displaystyle =a_{sisk},$">|; 

$key = q/U(x)=U_0,x^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2921.png"
 ALT="$ U(x)=U_0 x^{ m}$">|; 

$key = q/nablatimes(nablatimes{bfa})=nabla(nablacdot{bfa})-nabla^{,2}{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6991.png"
 ALT="$ \nabla\times(\nabla\times {\bf a}) = \nabla(\nabla\cdot{\bf a}) - \nabla^{ 2} {\bf a}$">|; 

$key = q/displaystyle=frac{2,m,m'}{m^{,2}-m'^{,2}}int_{-1}^1left(P_n^{,m},frac{dP_{n'}^{,m'}}{dmu}+frac{dP_n^{,m}}{dmu},P_{n'}^{,m'}right)mu,dmu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="398" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4963.png"
 ALT="$\displaystyle =\frac{2 m m'}{m^{ 2}-m'^{ 2}}\int_{-1}^1 \left(P_n^{ m}  \...
...dP_{n'}^{ m'}}{d\mu} + \frac{dP_n^{ m}}{d\mu} P_{n'}^{ m'}\right)\mu d\mu,$">|; 

$key = q/displaystyle{bfe}_parallel=-cosalpha,{bfe}_x+sinalpha,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3321.png"
 ALT="$\displaystyle {\bf e}_\parallel = -\cos\alpha {\bf e}_x + \sin\alpha {\bf e}_y$">|; 

$key = q/dxslashdt=c_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5552.png"
 ALT="$ dx/dt=c_1$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter05slashstream.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="249" BORDER="0"
 SRC="|."$dir".q|img1614.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter05/stream.eps}}
\end{figure}">|; 

$key = q/displaystyle=left[2,g,lleft(frac{1-s}{1+s}right)right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4177.png"
 ALT="$\displaystyle = \left[2 g l\left(\frac{1-s}{1+s}\right)\right]^{1/2}.$">|; 

$key = q/displaystylec'=c+u=c_0+left(frac{gamma+1}{2}right)u,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5371.png"
 ALT="$\displaystyle c' = c + u = c_0+\left(\frac{\gamma+1}{2}\right)u,$">|; 

$key = q/sigma>rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4208.png"
 ALT="$ \sigma&gt;\rho$">|; 

$key = q/displaystylep_1(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1470.png"
 ALT="$\displaystyle p_1(z)$">|; 

$key = q/displaystylev_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2548.png"
 ALT="$\displaystyle v_z$">|; 

$key = q/m'=(4pislash3),a^{,3},rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2671.png"
 ALT="$ m' = (4\pi/3) a^{ 3} \rho$">|; 

$key = q/alpha<alpha_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3535.png"
 ALT="$ \alpha&lt;\alpha_c$">|; 

$key = q/AB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img708.png"
 ALT="$ AB$">|; 

$key = q/kappa_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5018.png"
 ALT="$ \kappa_{n, n'}^{ m, m'}$">|; 

$key = q/deltas;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1630.png"
 ALT="$ \delta s$">|; 

$key = q/displaystylem_{rmadded}=frac{4}{3},pi,rho,a,b^{,2},G(e),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2812.png"
 ALT="$\displaystyle m_{\rm added} = \frac{4}{3} \pi \rho a b^{ 2} G(e),$">|; 

$key = q/uslashu_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$ u/u_1$">|; 

$key = q/displaystylesimeq2sqrt{frac{g}{h}},u.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3568.png"
 ALT="$\displaystyle \simeq 2\sqrt{\frac{g}{h}} u.$">|; 

$key = q/v_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3569.png"
 ALT="$ v_0$">|; 

$key = q/u_2=V_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5528.png"
 ALT="$ u_2=V_p$">|; 

$key = q/(x',,y',,z');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6276.png"
 ALT="$ (x',  y',  z')$">|; 

$key = q/displaystylerho,frac{D{bfv}}{Dt}equivrho!left[frac{partial{bfv}}{partialt}+({bfvuleft[nabla^{,2}{bfv}+frac{1}{3},nabla(nablacdot{bfv})right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="526" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="$\displaystyle \rho \frac{D{\bf v}}{Dt}\equiv \rho\!\left[\frac{\partial {\bf v...
...\mu\left[\nabla^{ 2} {\bf v} + \frac{1}{3} \nabla(\nabla\cdot{\bf v})\right],$">|; 

$key = q/displaystyle=left(frac{partial{calH}}{partialp}right)_{calS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5583.png"
 ALT="$\displaystyle = \left(\frac{\partial {\cal H}}{\partial p}\right)_{\cal S},$">|; 

$key = q/displaystylepsi(varpi,z)=-frac{1}{2},V,varpi^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2600.png"
 ALT="$\displaystyle \psi(\varpi,z) = - \frac{1}{2} V \varpi^{ 2}.$">|; 

$key = q/P={bff}cdot{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6332.png"
 ALT="$ P={\bf f}\cdot {\bf v}$">|; 

$key = q/h=k,gslashW;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3560.png"
 ALT="$ h=k g/W$">|; 

$key = q/lambda_rrightarrowlambda_n^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4935.png"
 ALT="$ \lambda_r\rightarrow\lambda_n^{ m}$">|; 

$key = q/2,{rmRe}^{1slash2},(xslashL)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3069.png"
 ALT="$ 2 {\rm Re}^{1/2} (x/L)^{1/2}$">|; 

$key = q/displaystyleC_{pR}=-frac{2,epsilon}{sqrt{{rmMa}_1^{,2}-1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5811.png"
 ALT="$\displaystyle C_{pR} =- \frac{2 \epsilon}{\sqrt{{\rm Ma}_1^{ 2}-1}}.$">|; 

$key = q/displaystyle=frac{partial}{partialx}!left(frac{partialT}{partialx}right)+frac{pa)+frac{partial}{partialz}!left(frac{partialT}{partialz}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6613.png"
 ALT="$\displaystyle = \frac{\partial}{\partial x}\!\left(\frac{\partial T}{\partial x...
...right)+ \frac{\partial}{\partial z}\!\left(\frac{\partial T}{\partial z}\right)$">|; 

$key = q/displaystyle=-left(frac{partial{calF}}{partialT}right)_v,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5587.png"
 ALT="$\displaystyle = -\left(\frac{\partial {\cal F}}{\partial T}\right)_v,$">|; 

$key = q/23.934;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4627.png"
 ALT="$ 23.934$">|; 

$key = q/deltatheta_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6398.png"
 ALT="$ \delta\theta_y$">|; 

$key = q/displaystyle=z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6280.png"
 ALT="$\displaystyle = z.$">|; 

$key = q/displaystyle{bfg}_{,rmmoon}({bfr})=-nabla{mitPhi}_{rmmoon},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4244.png"
 ALT="$\displaystyle {\bf g}_{ \rm moon}({\bf r}) = - \nabla{\mit\Phi}_{\rm moon},$">|; 

$key = q/displaystyle{bfc}={bfa}+{bfb}={bfb}+{bfa}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6235.png"
 ALT="$\displaystyle {\bf c} = {\bf a} + {\bf b} = {\bf b} + {\bf a}.$">|; 

$key = q/D_2,{rme}^{,{rmi},alpha_2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2455.png"
 ALT="$ D_2 {\rm e}^{ {\rm i} \alpha_2}$">|; 

$key = q/betagg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5544.png"
 ALT="$ \beta\gg 1$">|; 

$key = q/displaystylenablatimes(nablatimes{bfA});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6693.png"
 ALT="$\displaystyle \nabla\times(\nabla\times{\bf A})$">|; 

$key = q/displaystyleV'>(1+s^{,-1})^{1slash2},c_0=6.6,{rmmslashs}=12.8,{rmkts}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4201.png"
 ALT="$\displaystyle V' &gt; (1+s^{ -1})^{1/2} c_0 = 6.6 {\rm m/s} = 12.8 {\rm kts}.$">|; 

$key = q/displaystyleF'''+2,F,F''+2,F'^{,2}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3254.png"
 ALT="$\displaystyle F''' +2 F F'' + 2 F'^{ 2} = 0,
$">|; 

$key = q/alpha=4,sigmaslashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5184.png"
 ALT="$ \alpha=4 \sigma/c$">|; 

$key = q/displaystylenablacdot{bfA}=frac{partialA_x}{partialx}+frac{partialA_y}{partialy}+frac{partialA_z}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6601.png"
 ALT="$\displaystyle \nabla\cdot{\bf A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} +\frac{\partial A_z}{\partial z},$">|; 

$key = q/displaystylev_theta(a,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3752.png"
 ALT="$\displaystyle v_\theta(a,\theta)$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter13slashfig4.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5197.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter13/fig4.01.eps}}
\end{figure}">|; 

$key = q/ineqj;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7251.png"
 ALT="$ i\neq j$">|; 

$key = q/displaystyle=2left(frac{37}{315}-frac{{mitLambda}}{945}-frac{{mitLambda}^{,2}}{9}right){mitLambda}^{,2}+frac{2}{9072},{mitLambda}^{,3}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="580" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3196.png"
 ALT="$\displaystyle = 2\left(\frac{37}{315}-\frac{{\mit\Lambda}}{945}-\frac{{\mit\Lam...
...{1}{120}\right){\mit\Lambda}^{ 2}+ \frac{2}{9072} {\mit\Lambda}^{ 3}\right].$">|; 

$key = q/displaystyle=frac{sinbeta}{cosbeta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4043.png"
 ALT="$\displaystyle =\frac{\sin\beta}{\cos\beta},$">|; 

$key = q/displaystylefrac{partialu}{partialt}+u,frac{partialu}{partialx}+frac{1}{rho},frac{partialp}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5346.png"
 ALT="$\displaystyle \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + \frac{1}{\rho} \frac{\partial p}{\partial x}$">|; 

$key = q/P=(infty,0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6479.png"
 ALT="$ P=(\infty, 0, 0)$">|; 

$key = q/displaystyle={mitOmega}_x,delta(y),{bfe}_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3374.png"
 ALT="$\displaystyle = {\mit\Omega}_x \delta(y) {\bf e}_x,$">|; 

$key = q/beta<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1323.png"
 ALT="$ \beta&lt;0$">|; 

$key = q/displaystylexi_r(a,theta,phi)=zeta_b(theta,phi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4404.png"
 ALT="$\displaystyle \xi_r(a,\theta,\phi) = \zeta_b(\theta,\phi).$">|; 

$key = q/{rmMa}=qslashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5928.png"
 ALT="$ {\rm Ma}=q/c$">|; 

$key = q/displaystylefrac{1}{4},pi,a^{,4},rho,omega^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1063.png"
 ALT="$\displaystyle \frac{1}{4} \pi a^{ 4} \rho \omega^{ 2}.
$">|; 

$key = q/displaystylebeta_{r,,s}=-int_{mitOmega}frac{costheta}{sintheta}left(frac{partiallphi},frac{partial{mitPhi}_s}{partialtheta}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4869.png"
 ALT="$\displaystyle \beta_{r, s}= -\int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\le...
...\partial\phi} \frac{\partial{\mit\Phi}_s}{\partial\theta}\right)d{\mit\Omega}.$">|; 

$key = q/displaystyley(x)=-h,[lambda'+cosh(-xslashh+c)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7353.png"
 ALT="$\displaystyle y(x) =-h [\lambda' + \cosh(-x/h + c)].$">|; 

$key = q/displaystyle=left(frac{r}{a}right)^nP_n^{,0}(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4275.png"
 ALT="$\displaystyle = \left(\frac{r}{a}\right)^n P_n^{ 0}(\cos\theta),$">|; 

$key = q/u';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5898.png"
 ALT="$ u'$">|; 

$key = q/displaystylefrac{BQ'}{BP'}=tantheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img772.png"
 ALT="$\displaystyle \frac{BQ'}{BP'} = \tan\theta.$">|; 

$key = q/cdotd{bfS}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1547.png"
 ALT="$ \cdot d{\bf S}=0$">|; 

$key = q/{rmFr}={rmFr}_c=1slashsqrt{pi}=0.56;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4005.png"
 ALT="$ {\rm Fr}={\rm Fr}_c= 1/\sqrt{\pi}= 0.56$">|; 

$key = q/displaystylep_0(r)simeqfrac{rho}{2,a}left(g-frac{2}{3},{mitOmega}^{,2},aright)(a^{,2}-r^{,2})+skew{3}bar{rho},g,d+p_{rmatm},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="415" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4396.png"
 ALT="$\displaystyle p_0(r) \simeq \frac{\rho}{2 a}\left(g-\frac{2}{3} {\mit\Omega}^{ 2} a\right)(a^{ 2}-r^{ 2}) + \skew{3}\bar{\rho} g d+ p_{\rm atm},$">|; 

$key = q/x=y+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6457.png"
 ALT="$ x=y+1$">|; 

$key = q/bar{F}^{,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2258.png"
 ALT="$ \bar{F}^{ 1/2}$">|; 

$key = q/e_M=0.05488;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4587.png"
 ALT="$ e_M= 0.05488$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter10slashdrop1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3853.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter10/drop1.eps}}
\end{figure}">|; 

$key = q/displaystyleoint_C{bft}times{bfg},dr=int_Sleft[(nabla{bfg})cdot{bfn}-(nablacdot{bfg}),{bfn}right]dS.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1101.png"
 ALT="$\displaystyle \oint_C {\bf t}\times {\bf g} dr = \int_S\left[(\nabla {\bf g})\cdot{\bf n} - (\nabla\cdot{\bf g}) {\bf n}\right]dS.$">|; 

$key = q/-rho_0,V_0,z_H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img703.png"
 ALT="$ -\rho_0 V_0 z_H$">|; 

$key = q/displaystyle=1-frac{1}{2},(gamma-1)left(frac{v}{c_0}right)^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5617.png"
 ALT="$\displaystyle = 1-\frac{1}{2} (\gamma-1)\left(\frac{v}{c_0}\right)^2,$">|; 

$key = q/v_theta(a_-,theta)=v_theta(a_+,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3849.png"
 ALT="$ v_\theta(a_-,\theta)=v_\theta(a_+,\theta)$">|; 

$key = q/-aslash2leqxleq+aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7340.png"
 ALT="$ -a/2\leq x \leq +a/2$">|; 

$key = q/displaystylesigma_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7019.png"
 ALT="$\displaystyle \sigma_{ij}$">|; 

$key = q/|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$ \vert$">|; 

$key = q/ointnablatimes{bfA}cdotd{bfS}=intnablacdot(nablatimes{bfA}),dV=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6666.png"
 ALT="$ \oint\nabla\times{\bf A}\cdot d{\bf S} = \int \nabla\cdot(\nabla\times{\bf A})  dV
=0$">|; 

$key = q/[({bfA}cdotnabla),{bfB}]_i={bfA}cdotnablaB_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7089.png"
 ALT="$ [({\bf A}\cdot\nabla) {\bf B}]_i={\bf A}\cdot\nabla B_i$">|; 

$key = q/displaystylefrac{Dv_x}{Dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="$\displaystyle \frac{Dv_x}{Dt}$">|; 

$key = q/displaystyleH-d(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1528.png"
 ALT="$\displaystyle H-d(0)$">|; 

$key = q/partial^{,2}Tslashpartialx^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6617.png"
 ALT="$ \partial^{ 2} T/\partial x^{ 2}$">|; 

$key = q/y(0)=y(pi)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7373.png"
 ALT="$ y(0)=y(\pi)=0$">|; 

$key = q/displaystyle=-frac{3}{5},frac{(cosbeta,cosgamma)^{1slash3}}{singamma},F(gamma,alcos^2beta}{(cosbeta,cosgamma)^{2slash3}},widehat{omega}^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="448" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img949.png"
 ALT="$\displaystyle =-\frac{3}{5} \frac{(\cos\beta \cos\gamma)^{1/3}}{\sin\gamma} ...
...} \frac{1+\cos^2\beta}{(\cos\beta \cos\gamma)^{2/3}} \widehat{\omega}^{ 2},$">|; 

$key = q/int_P^Q{bfgrad},Tcdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6550.png"
 ALT="$ \int_P^Q {\bf grad } T\cdot d{\bf r}$">|; 

$key = q/displaystyle={rmRe}left[frac{c,sintheta-{rmi},delta,costheta}{(c^{,2},sin^2thetalta^{,2},cos^2theta)^{1slash2}},frac{dF}{dz}right]_{|zeta|=a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3334.png"
 ALT="$\displaystyle = {\rm Re}\left[\frac{c \sin\theta-{\rm i} \delta \cos\theta}{...
...ta+\delta^{ 2} \cos^2\theta)^{1/2}} \frac{dF}{dz}\right]_{\vert\zeta\vert=a}$">|; 

$key = q/zeta_t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4348.png"
 ALT="$ \zeta_t$">|; 

$key = q/displaystyle=c,sinhxi,coseta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2753.png"
 ALT="$\displaystyle = c \sinh\xi \cos\eta,$">|; 

$key = q/e_{13}>0.95289;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img963.png"
 ALT="$ e_{13}&gt;0.95289$">|; 

$key = q/40;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3824.png"
 ALT="$ 40$">|; 

$key = q/Tproptopslashrho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5115.png"
 ALT="$ T\propto p/\rho$">|; 

$key = q/v_z(a)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3614.png"
 ALT="$ v_z(a)=0$">|; 

$key = q/displaystyleskew{3}hat{beta}_{n,,-n'}^{,m,,-m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5000.png"
 ALT="$\displaystyle \skew{3}\hat{\beta}_{n, -n'}^{ m, -m'}$">|; 

$key = q/nablapsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1740.png"
 ALT="$ \nabla\psi$">|; 

$key = q/alpha=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2252.png"
 ALT="$ \alpha=\pi$">|; 

$key = q/deltasrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1633.png"
 ALT="$ \delta s\rightarrow 0$">|; 

$key = q/omega_r=omega_theta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3734.png"
 ALT="$ \omega_r=\omega_\theta=0$">|; 

$key = q/displaystylerightarrowfrac{1}{2},V,r^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3771.png"
 ALT="$\displaystyle \rightarrow \frac{1}{2} V r^{ 2}.$">|; 

$key = q/displaystyleP(theta)=p_1+rho,U_0^{,2},cos(2,theta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3092.png"
 ALT="$\displaystyle P(\theta) = p_1 +\rho U_0^{ 2} \cos (2 \theta),$">|; 

$key = q/displaystylezeta_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4503.png"
 ALT="$\displaystyle \zeta_b$">|; 

$key = q/(1-k,Vslashbar{F})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2262.png"
 ALT="$ (1-k V/\bar{F})^{1/2}$">|; 

$key = q/y=h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3038.png"
 ALT="$ y=h$">|; 

$key = q/k=(d_1-d_2)slash(d_1+d_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3717.png"
 ALT="$ k = (d_1-d_2)/(d_1+d_2)$">|; 

$key = q/c=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7357.png"
 ALT="$ c=0$">|; 

$key = q/P_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1619.png"
 ALT="$ P_2$">|; 

$key = q/displaystyle{mitPhi}_{rmcentripetal}({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4233.png"
 ALT="$\displaystyle {\mit\Phi}_{\rm centripetal}({\bf r})$">|; 

$key = q/displaystylefrac{T_2}{T_1}=frac{1+(1slash2),(gamma-1),{rmMa}_1^{,2}}{1+(1slash2),(gamma-1),{rmMa}_2^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5474.png"
 ALT="$\displaystyle \frac{T_2}{T_1}=\frac{1+(1/2) (\gamma-1) {\rm Ma}_1^{ 2}}{1+(1/2) (\gamma-1) {\rm Ma}_2^{ 2}}.$">|; 

$key = q/v_x(infty);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2968.png"
 ALT="$ v_x(\infty)$">|; 

$key = q/S_{sa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4548.png"
 ALT="$ S_{sa}$">|; 

$key = q/displaystylenabla^{,2}{bfv}=frac{nablaP}{mu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3579.png"
 ALT="$\displaystyle \nabla^{ 2}{\bf v} = \frac{\nabla P}{\mu},$">|; 

$key = q/aslash(k,d);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3961.png"
 ALT="$ a/(k d)$">|; 

$key = q/displaystylev_y(x,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2874.png"
 ALT="$\displaystyle v_y(x,0)$">|; 

$key = q/displaystylenablaP=mu,nabla^{,2}{bfv}=-mu,nablatimes(nablatimes{bfv})=-mu,nablatimes;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3728.png"
 ALT="$\displaystyle \nabla P = \mu \nabla^{ 2}{\bf v} = -\mu \nabla\times(\nabla\times{\bf v}) = -\mu \nabla\times$">|; 

$key = q/theta_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$ \theta _\pm $">|; 

$key = q/displaystyle=frac{pi}{8},rho,{mitGamma}_0^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3510.png"
 ALT="$\displaystyle = \frac{\pi}{8} \rho {\mit\Gamma}_0^{ 2}.$">|; 

$key = q/displaystylefrac{partialP}{partialtheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3788.png"
 ALT="$\displaystyle \frac{\partial P}{\partial \theta}$">|; 

$key = q/displaystyle[({bfv}cdotnabla),{bfv}]_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3654.png"
 ALT="$\displaystyle [({\bf v}\cdot\nabla) {\bf v}]_x$">|; 

$key = q/m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1823.png"
 ALT="$ m'$">|; 

$key = q/stackrel{displaystylerightarrow}{SR};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="30" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6234.png"
 ALT="$ \stackrel{\displaystyle \rightarrow}{SR}$">|; 

$key = q/displaystylefrac{F_{21}}{rho,g}=frac{1}{2},h_2^{,2}-frac{1}{2},h_1^{,2}+frac{Q}{g},(v_2-v_1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1474.png"
 ALT="$\displaystyle \frac{F_{21}}{\rho g} = \frac{1}{2} h_2^{ 2} -\frac{1}{2} h_1^{ 2}+\frac{Q}{g} (v_2-v_1).$">|; 

$key = q/pi,a^{,2},l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7199.png"
 ALT="$ \pi a^{ 2} l$">|; 

$key = q/displaystyleP_r=lambda_rint_{mitOmega}(1+k_2-h_2),skew{5}bar{zeta}_2,{mitPhi}_r,d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4917.png"
 ALT="$\displaystyle P_r = \lambda_r\int_{\mit\Omega}(1+k_2-h_2) \skew{5}\bar{\zeta}_2 {\mit\Phi}_r d{\mit\Omega},$">|; 

$key = q/displaystyle=-frac{partialphi}{partialx}=frac{U,epsilon,alpha}{m},sin(alpha,x),{rme}^{-m,alpha,y},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5976.png"
 ALT="$\displaystyle = -\frac{\partial\phi}{\partial x} = \frac{U \epsilon \alpha}{m} \sin(\alpha x) {\rm e}^{-m \alpha y},$">|; 

$key = q/displaystyle+f_jleft[frac{(n-1),(n+1),(n-m_j)}{2,n-1}right]{mitPsi}_{n-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4695.png"
 ALT="$\displaystyle +f_j\left[\frac{(n-1) (n+1) (n-m_j)}{2 n-1}\right]{\mit\Psi}_{n-1}$">|; 

$key = q/displaystyle=frac{n,(n+1)}{beta}left(1-[1+k_n'-h_n'],alpha_nright)+frac{m_j,f_j}{n,(n+1)}-f_j^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="424" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4703.png"
 ALT="$\displaystyle = \frac{n (n+1)}{\beta}\left(1-[1+k_n'-h_n'] \alpha_n\right) + \frac{m_j f_j}{n (n+1)}-f_j^{ 2}$">|; 

$key = q/beta_{-r,s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4887.png"
 ALT="$ \beta_{-r,s}$">|; 

$key = q/{bfa}={bfe}_x-2,{bfe}_y+{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6710.png"
 ALT="$ {\bf a} = {\bf e}_x - 2 {\bf e}_y+{\bf e}_z$">|; 

$key = q/displaystylebeta_{r,,-s}=int_{mitOmega}costhetaleft(frac{partial{mitPhi}_r}{partialphi},frac{partial{mitPsi}_s}{partialphi}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="412" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4880.png"
 ALT="$\displaystyle \beta_{r, -s}= \int_{\mit\Omega}\cos\theta\left(\frac{\partial{\...
...}{\partial\phi} \frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}.$">|; 

$key = q/displaystyle{calE}={calC}_V,T=frac{c_V}{M},T.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5308.png"
 ALT="$\displaystyle {\cal E} ={\cal C}_V T = \frac{c_V}{ M} T.$">|; 

$key = q/d=0.65,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1266.png"
 ALT="$ d=0.65 c$">|; 

$key = q/|zeta_b|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4349.png"
 ALT="$ \vert\zeta_b\vert$">|; 

$key = q/OPequiv|{bfr}|=r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6268.png"
 ALT="$ OP\equiv \vert{\bf r}\vert=r$">|; 

$key = q/displaystyle{calH}+frac{1}{2},v^{,2}={calH}_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5861.png"
 ALT="$\displaystyle {\cal H} + \frac{1}{2} v^{ 2} = {\cal H}_0,$">|; 

$key = q/displaystylephantom{=}+frac{mu}{rho}left(nabla^{,2}v_theta+frac{2}{r^{,2}},frac{2}}+frac{1}{3,r},frac{partial{mitDelta}}{partialtheta}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="$\displaystyle \phantom{=}+\frac{\mu}{\rho}\left(\nabla^{ 2} v_\theta +\frac{2}...
...{r^{ 2}}+ \frac{1}{3 r} \frac{\partial{\mit\Delta}}{\partial \theta}\right),$">|; 

$key = q/displaystylell1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3660.png"
 ALT="$\displaystyle \ll 1,$">|; 

$key = q/u=r^{,2}slasha^{,2}=(bar{x}^{,2}+bar{y}^{,2})slasha^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img752.png"
 ALT="$ u=r^{ 2}/a^{ 2}=(\bar{x}^{ 2}+ \bar{y}^{ 2})/a^{ 2}$">|; 

$key = q/displaystyle{bfv}=v_x(y),{bfe}_x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3577.png"
 ALT="$\displaystyle {\bf v} = v_x(y) {\bf e}_x.$">|; 

$key = q/s_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6982.png"
 ALT="$ s_{ij}$">|; 

$key = q/W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img593.png"
 ALT="$ W$">|; 

$key = q/displaystyle{calC}_p,T+frac{1}{2},v^{,2}={calC}_p,T_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5378.png"
 ALT="$\displaystyle {\cal C}_p T+ \frac{1}{2} v^{ 2} = {\cal C}_p T_0,$">|; 

$key = q/=omega,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img811.png"
 ALT="$ = \omega {\bf e}_z$">|; 

$key = q/displaystylefrac{D}{rho,V^{,2}}=pilim_{xrightarrowinfty}left(frac{y^{,2}}{4,x}right)_{rmcavity~boundary},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2339.png"
 ALT="$\displaystyle \frac{D}{\rho V^{ 2}} =\pi\lim_{x\rightarrow\infty}\left(\frac{y^{ 2}}{4 x}\right)_{\rm cavity&nbsp;boundary},$">|; 

$key = q/T_2slashT_1>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5506.png"
 ALT="$ T_2/T_1&gt;1$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi}=C,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1594.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} + {\mit\Psi} = C,$">|; 

$key = q/displaystylephi(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1971.png"
 ALT="$\displaystyle \phi (r,\theta)$">|; 

$key = q/3.65times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4617.png"
 ALT="$ 3.65\times 10^{-2}$">|; 

$key = q/z=-d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3906.png"
 ALT="$ z=-d$">|; 

$key = q/displaystyleoverline{alpha}=alpha_0+overline{alpha_C},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5853.png"
 ALT="$\displaystyle \overline{\alpha} = \alpha_0 + \overline{\alpha_C},$">|; 

$key = q/AO;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2263.png"
 ALT="$ AO$">|; 

$key = q/displaystylec_f(x)=frac{(m+1)^{1slash2},sqrt{2},f''(0)}{[{rmRe}(x)]^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2948.png"
 ALT="$\displaystyle c_f(x) = \frac{(m+1)^{1/2} \sqrt{2} f''(0)}{[{\rm Re}(x)]^{1/2}},$">|; 

$key = q/displaystyle=lambda_sint_{mitOmega}{mitPhi}_r'',{mitPhi}_s,d{mitOmega}=-int_{mitOmega}{mitPhi}_r'',D,{mitPhi}_s,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4871.png"
 ALT="$\displaystyle =\lambda_s\int_{\mit\Omega} {\mit\Phi}_r'' {\mit\Phi}_s d{\mit\Omega} = -\int_{\mit\Omega} {\mit\Phi}_r'' D {\mit\Phi}_s d{\mit\Omega}$">|; 

$key = q/displaystylephantom{=}-frac{f_j,(n-1)^2,(n+1),(n-m_j),(n+m_j)}{n,(2,n-1),(2,n+1),[m_j-n,(n-1),f_j]};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4704.png"
 ALT="$\displaystyle \phantom{=}- \frac{f_j (n-1)^2 (n+1) (n-m_j) (n+m_j)}{n (2 n-1) (2 n+1) [m_j-n (n-1) f_j]}$">|; 

$key = q/Lsim10^{-3},{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="$ L\sim 10^{-3} {\rm m}$">|; 

$key = q/displaystyle{calL}^{,2}(psi)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3759.png"
 ALT="$\displaystyle {\cal L}^{ 2}(\psi)=0.$">|; 

$key = q/displaystyle=|{bfa}|^{,2},|{bfb}|^{,2}-({bfa}cdot{bfb})^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6375.png"
 ALT="$\displaystyle = \vert{\bf a}\vert^{ 2}  \vert{\bf b}\vert^{ 2} - ({\bf a}\cdot {\bf b})^2$">|; 

$key = q/h_-=Qslashv_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1439.png"
 ALT="$ h_-= Q/v_-$">|; 

$key = q/displaystyleu,frac{d{calE}}{dx}=-frac{p}{rho},frac{du}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5455.png"
 ALT="$\displaystyle u \frac{d{\cal E}}{dx}=-\frac{p}{\rho} \frac{du}{dx},$">|; 

$key = q/displaystyle=frac{T,v,alpha^{,2}}{c_p}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5605.png"
 ALT="$\displaystyle = \frac{T v \alpha^{ 2}}{c_p}.$">|; 

$key = q/{bfr}cdot{bfn}=b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6722.png"
 ALT="$ {\bf r}\cdot{\bf n} = b$">|; 

$key = q/displaystyle[{bfa}times({bfb}times{bfc})]_i=epsilon_{ijk},a_j,(epsilon_{klm},b_l,c_m),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6817.png"
 ALT="$\displaystyle [{\bf a}\times ({\bf b}\times {\bf c})]_i = \epsilon_{ijk} a_j (\epsilon_{klm} b_l c_m),$">|; 

$key = q/{mitLambda}=-12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$ {\mit \Lambda }=-12$">|; 

$key = q/displaystyle{rmMa}_1^{,2},sin^2beta-1=left(frac{gamma+1}{2}right){rmMa}_1^{,2},frac{sinbeta,sintheta}{cos(beta-theta)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5714.png"
 ALT="$\displaystyle {\rm Ma}_1^{ 2} \sin^2\beta-1 =\left(\frac{\gamma+1}{2}\right){\rm Ma}_1^{ 2} \frac{\sin\beta \sin\theta}{\cos(\beta-\theta)}.$">|; 

$key = q/xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2052.png"
 ALT="$ \xi$">|; 

$key = q/displaystylepsi(r,theta)=sin^2thetaleft(frac{overline{A}}{r}+overline{B},r+overline{C},r^{,2}+overline{D},r^{,4}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3834.png"
 ALT="$\displaystyle \psi(r,\theta) = \sin^2\theta\left(\frac{\overline{A}}{r}+\overline{B} r+\overline{C} r^{ 2} + \overline{D} r^{ 4}\right),$">|; 

$key = q/x=x_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3209.png"
 ALT="$ x=x_s$">|; 

$key = q/displaystylefrac{rho,v_z,a^{,2}}{mu,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3687.png"
 ALT="$\displaystyle \frac{\rho v_z a^{ 2}}{\mu l}$">|; 

$key = q/displaystyledelta_1=(pislash2-1)left(frac{8}{4-pi}right)^{1slash2}left(frac{nu,x}{U_0}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3269.png"
 ALT="$\displaystyle \delta_1 = (\pi/2-1)\left(\frac{8}{4-\pi}\right)^{1/2}\left(\frac{\nu x}{U_0}\right)^{1/2},
$">|; 

$key = q/displaystyleJ=int_a^bG(y,y',x),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7330.png"
 ALT="$\displaystyle J = \int_a^b G(y,y',x) dx$">|; 

$key = q/xi^{,2}+eta^{,2}=c^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2124.png"
 ALT="$ \xi^{ 2}+\eta^{ 2}=c^{ 2}$">|; 

$key = q/displaystyle=-p_0+rho,g,r,costheta+mu,costhetaleft(frac{12,A}{r^{,4}}+frac{6,B}{r^{,2}}+12,D,rright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="435" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3840.png"
 ALT="$\displaystyle = - p_0 +\rho g r \cos\theta+\mu \cos\theta\left(\frac{12 A}{r^{ 4}}+\frac{6 B}{r^{ 2}}+12 D r\right)$">|; 

$key = q/displaystyleomega^{,2}=g,k+frac{gamma,k^{,3}}{rho}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4148.png"
 ALT="$\displaystyle \omega^{ 2} = g k + \frac{\gamma k^{ 3}}{\rho}.$">|; 

$key = q/z=sinhxi,coseta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2792.png"
 ALT="$ z=\sinh\xi \cos\eta$">|; 

$key = q/displaystylef(theta)=sin(alpha+theta)+sinalpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3359.png"
 ALT="$\displaystyle f(\theta) = \sin(\alpha+\theta)+\sin\alpha.$">|; 

$key = q/displaystylefrac{{mitDelta}p}{p_1}simeqleft(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right)theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6116.png"
 ALT="$\displaystyle \frac{{\mit\Delta}p}{p_1}\simeq \left(\frac{\gamma {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right)\theta.
$">|; 

$key = q/sigma_{11}=-(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$ \sigma_{11}= -($">|; 

$key = q/{mitGamma}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4970.png"
 ALT="$ {\mit\Gamma}(x)$">|; 

$key = q/displaystylefrac{partialx_i}{partialx_j'}={calR}_{ji}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6877.png"
 ALT="$\displaystyle \frac{\partial x_i}{\partial x_j'} = {\cal R}_{ji}.$">|; 

$key = q/M_0<M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img832.png"
 ALT="$ M_0&lt;M$">|; 

$key = q/{bfa}=(0,,1,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6352.png"
 ALT="$ {\bf a} =
(0, 1, 0)$">|; 

$key = q/dV=dx,dy,dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6504.png"
 ALT="$ dV = dx  dy  dz$">|; 

$key = q/|zeta|ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2318.png"
 ALT="$ \vert\zeta\vert\ll 1$">|; 

$key = q/displaystyle=oint_C{bfA}cdotd{bfr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2348.png"
 ALT="$\displaystyle =\oint_C {\bf A}\cdot d{\bf r},$">|; 

$key = q/displaystylephi(x,z,t)=Aleft[{rme}^{,k,z}+{rme}^{-k,(z+2,d)}right]cos(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3921.png"
 ALT="$\displaystyle \phi(x,z,t) = A\left[{\rm e}^{ k z} + {\rm e}^{-k (z+2 d)}\right]\cos(\omega t-k x),$">|; 

$key = q/displaystyle=sin^2beta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6088.png"
 ALT="$\displaystyle = \sin^2\beta,$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter08slashw.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="371" BORDER="0"
 SRC="|."$dir".q|img2996.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter08/w.eps}}
\end{figure}">|; 

$key = q/displaystylep(x,0_-,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4098.png"
 ALT="$\displaystyle p(x,0_-,t)$">|; 

$key = q/k=0.6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$ k=0.6$">|; 

$key = q/alpha=2,dbetaslashdy=A,du_0slashdy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3132.png"
 ALT="$ \alpha=2 d\beta/dy=A du_0/dy$">|; 

$key = q/psi_1(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2010.png"
 ALT="$ \psi_1(x,y)$">|; 

$key = q/B'OB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2295.png"
 ALT="$ B'OB$">|; 

$key = q/displaystyle=frac{costheta}{sintheta},frac{partial{mitPsi}_r}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4851.png"
 ALT="$\displaystyle = \frac{\cos\theta}{\sin\theta} \frac{\partial{\mit\Psi}_r}{\partial\phi},$">|; 

$key = q/{bfe}_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7113.png"
 ALT="$ {\bf e}_\theta$">|; 

$key = q/displaystyle=int_0^phi(1-k^{,2},sin^2phi)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1181.png"
 ALT="$\displaystyle = \int_0^\phi (1-k^{ 2} \sin^2\phi)^{1/2},$">|; 

$key = q/displaystyle=(k+s)^{1slash2},s^{1slash2}-klnleft[left(1+frac{s}{k}right)^{1slash2}+left(frac{s}{k}right)^{1slash2}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2272.png"
 ALT="$\displaystyle =(k+s)^{1/2} s^{1/2} -k\ln\left[\left(1+\frac{s}{k}\right)^{1/2}+\left(\frac{s}{k}\right)^{1/2}\right],$">|; 

$key = q/displaystyleF(xi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2764.png"
 ALT="$\displaystyle F(\xi)$">|; 

$key = q/x=d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1202.png"
 ALT="$ x=d$">|; 

$key = q/displaystyle=-frac{1}{rho},frac{partialp}{partialz}-frac{partial{mitPsi}}{partiaa^{,2}v_z+frac{1}{3},frac{partial{mitDelta}}{partialz}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="$\displaystyle = - \frac{1}{\rho} \frac{\partial p}{\partial z} - \frac{\partia...
...\nabla^{ 2} v_z + \frac{1}{3} \frac{\partial{\mit\Delta}}{\partial z}\right),$">|; 

$key = q/{bfv}={bfe}_ztimesnablapsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1928.png"
 ALT="$ {\bf v} = {\bf e}_z\times \nabla\psi$">|; 

$key = q/displaystyle=frac{sintheta}{r},frac{df}{dr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3767.png"
 ALT="$\displaystyle = \frac{\sin\theta}{r} \frac{df}{dr},$">|; 

$key = q/displaystyle={bfV}+{bfv}_{mitSigma}(P_-)+{bfv}_S(P_-).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3423.png"
 ALT="$\displaystyle = {\bf V} + {\bf v}_{\mit\Sigma}(P_-)+ {\bf v}_S(P_-).$">|; 

$key = q/displaystyle=-p(a,theta)=-p_0+rho,g,a,costheta-frac{3,mu,V}{2,a},costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3803.png"
 ALT="$\displaystyle = -p(a,\theta) = -p_0+ \rho g a \cos\theta-\frac{3 \mu V}{2 a} \cos\theta,$">|; 

$key = q/displaystyle=a,{rme}^{,k,z},sin(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3940.png"
 ALT="$\displaystyle = a {\rm e}^{ k z} \sin(\omega t-k x),$">|; 

$key = q/2,omega_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4547.png"
 ALT="$ 2 \omega_E$">|; 

$key = q/displaystylelambda={rmi},sinh{mitOmega}=frac{{rmi}}{2}left(V,frac{dz}{dbar{F}}-frac{1}{V},frac{dbar{F}}{dz}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2286.png"
 ALT="$\displaystyle \lambda={\rm i} \sinh{\mit\Omega} = \frac{{\rm i}}{2}\left(V \frac{dz}{d\bar{F}}-\frac{1}{V} \frac{d\bar{F}}{dz}\right).$">|; 

$key = q/displaystylefrac{omega(x,y)}{U_0slashdelta}=-frac{Q}{U_0,delta},frac{2}{sqrt{pi}},frac{y}{delta},exp(-y^{,2}slashdelta^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3033.png"
 ALT="$\displaystyle \frac{\omega(x,y)}{U_0/\delta}= -\frac{Q}{U_0 \delta} \frac{2}{\sqrt{\pi}} \frac{y}{\delta} \exp(-y^{ 2}/\delta^{ 2})$">|; 

$key = q/hat{t}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3341.png"
 ALT="$ \hat{t}=0$">|; 

$key = q/p=p_0-(1slash2),rho,v^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3431.png"
 ALT="$ p=p_0-(1/2) \rho v^{ 2}$">|; 

$key = q/displaystylehat{p}=hat{rho}^{,gamma}=hat{T}^{,gammaslash(gamma-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5141.png"
 ALT="$\displaystyle \hat{p}=\hat{\rho}^{ \gamma}=\hat{T}^{ \gamma/(\gamma-1)}.$">|; 

$key = q/X_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4046.png"
 ALT="$ X_0$">|; 

$key = q/displaystyle{mitdiv},{bfA}=nablacdot{bfA}=frac{partialA_x}{partialx}+frac{partialA_y}{partialy}+frac{partialA_z}{partialz}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6581.png"
 ALT="$\displaystyle {\mit div} {\bf A} = \nabla\cdot {\bf A} = \frac{\partial A_x}{\partial x} +\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}.$">|; 

$key = q/displaystyle=2,muleft[left(frac{partialv_r}{partialr}right)^2+left(frac{1}{r},frfrac{partialv_theta}{partialr}-frac{v_theta}{r}right)^2right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="586" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="$\displaystyle =2 \mu\left[\left(\frac{\partial v_r}{\partial r}\right)^2+\left...
...\theta}+\frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right)^2\right.$">|; 

$key = q/0<beta<2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2964.png"
 ALT="$ 0&lt;\beta&lt;2$">|; 

$key = q/displaystyle=({bfc}cdot{bfa}),{bfb}-({bfc}cdot{bfb}),{bfa},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6676.png"
 ALT="$\displaystyle = ({\bf c}\cdot{\bf a}) {\bf b} - ({\bf c}\cdot{\bf b}) {\bf a},$">|; 

$key = q/x-lambda,y={rmconst.};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5988.png"
 ALT="$ x-\lambda y={\rm const.}$">|; 

$key = q/displaystylefrac{1}{f_x}=frac{dx}{df}=pmfrac{1}{tan(2,phi)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1171.png"
 ALT="$\displaystyle \frac{1}{f_x}= \frac{dx}{df} = \pm \frac{1}{\tan(2 \phi)}.$">|; 

$key = q/zeta_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4347.png"
 ALT="$ \zeta_b$">|; 

$key = q/displaystylesimeq1.0times10^6,L({rmm}),V_0({rmm,s^{-1}}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="$\displaystyle \simeq 1.0\times 10^6 L({\rm m}) V_0({\rm m s^{-1}}),$">|; 

$key = q/displaystyletau_i=int_Vepsilon_{ijk},x_j,F_k,dV+int_Vepsilon_{ijk},sigma_{kj},dVnt_Vepsilon_{ijk},x_j,frac{partialsigma_{kl}}{partialx_l},dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="466" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$\displaystyle \tau_i = \int_V \epsilon_{ijk} x_j F_k dV + \int_V \epsilon_{i...
... dV+ \int_V\epsilon_{ijk} x_j \frac{\partial \sigma_{kl}}{\partial x_l} dV,$">|; 

$key = q/M_ast=1.989times10^{30},{rmkg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5235.png"
 ALT="$ M_\ast=1.989\times 10^{30} {\rm kg}$">|; 

$key = q/displaystylefrac{pi}{2},frac{V}{skew{3}bar{psi}_1},(z-{rmi},d)={rmi},(lambda-1)--lambda^{,2})^{1slash2}+tanh^{-1}[(1-lambda^{,2})^{1slash2}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="503" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2216.png"
 ALT="$\displaystyle \frac{\pi}{2} \frac{V}{\skew{3}\bar{\psi}_1} (z-{\rm i} d)= {\rm i} (\lambda-1)-(1-\lambda^{ 2})^{1/2}+\tanh^{-1}[(1-\lambda^{ 2})^{1/2}],$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter10slashplane.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="307" BORDER="0"
 SRC="|."$dir".q|img3588.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter10/plane.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{partial^{,2}v_x}{partialy^{,2}}simeq-frac{G}{mu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3663.png"
 ALT="$\displaystyle \frac{\partial^{ 2} v_x}{\partial y^{ 2}}\simeq -\frac{G}{\mu}.$">|; 

$key = q/V';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4085.png"
 ALT="$ V'$">|; 

$key = q/displaystyle=int_S(nablatimes{bfB})_z,dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2359.png"
 ALT="$\displaystyle =\int_S (\nabla\times {\bf B})_z dS,$">|; 

$key = q/{rmMa}_infty=Uslashc_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5894.png"
 ALT="$ {\rm Ma}_\infty= U/c_\infty$">|; 

$key = q/displaystyle=-frac{1}{rho,r},frac{partialp}{partialtheta}-frac{1}{r}frac{partial{mitPsi}}{partialtheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="$\displaystyle = - \frac{1}{\rho r} \frac{\partial p}{\partial \theta} - \frac{1}{r}\frac{\partial{\mit\Psi}}{\partial \theta}$">|; 

$key = q/displaystyle=frac{2,(1-e_{13}^{,2})^{1slash2}}{e_{13}^{,3}}left[int_0^{e_{13}}fr2})int_0^{e_{13}}frac{z^{,2},dz}{(1-z^{,2})^{3slash2}}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="527" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img880.png"
 ALT="$\displaystyle = \frac{2 (1-e_{13}^{ 2})^{1/2}}{e_{13}^{ 3}}\left[\int_0^{e_{...
...}}-(1-e_{13}^{ 2})\int_0^{e_{13}}\frac{z^{ 2} dz}{(1-z^{ 2})^{3/2}}\right].$">|; 

$key = q/[A,B]={bfe}_zcdotnablaAtimesnablaB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1929.png"
 ALT="$ [A,B] = {\bf e}_z\cdot\nabla A \times\nabla B$">|; 

$key = q/F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1397.png"
 ALT="$ F$">|; 

$key = q/displaystylefrac{(alpha+beta),f}{(1+f_z^{,2})^{1slash2}}=f^{,2}+alpha,beta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1293.png"
 ALT="$\displaystyle \frac{(\alpha+\beta) f}{(1+f_z^{ 2})^{1/2}}= f^{ 2} + \alpha \beta,$">|; 

$key = q/partialuslashpartialx+partialvslashpartialy=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3108.png"
 ALT="$ \partial u/\partial x+\partial v/\partial y=0$">|; 

$key = q/displaystylepsi(xi,eta)=F(xi),sin^2eta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2735.png"
 ALT="$\displaystyle \psi(\xi,\eta)=F(\xi) \sin^2\eta.$">|; 

$key = q/displaystylev_theta(r)=frac{{mitGamma}}{2pi,r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1706.png"
 ALT="$\displaystyle v_\theta(r) = \frac{{\mit\Gamma}}{2\pi r}.$">|; 

$key = q/nu=muslashrho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3599.png"
 ALT="$ \nu=\mu/\rho$">|; 

$key = q/displaystyle=kleft(theta-sinthetaright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7379.png"
 ALT="$\displaystyle =k\left(\theta-\sin\theta\right),$">|; 

$key = q/z_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1259.png"
 ALT="$ z_0$">|; 

$key = q/skew{3}bar{rho},{mitOmega}^{,2},a,d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4400.png"
 ALT="$ \skew{3}\bar{\rho} {\mit\Omega}^{ 2} a d$">|; 

$key = q/p_4slashp_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5644.png"
 ALT="$ p_4/p_1$">|; 

$key = q/displaystyleoint{bfA}cdotd{bfr}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6559.png"
 ALT="$\displaystyle \oint {\bf A}\cdot d{\bf r} = 0,$">|; 

$key = q/p_{-r}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4930.png"
 ALT="$ p_{-r}(t)$">|; 

$key = q/displaystyle{bfa}cdot{bfb}times{bfc}={bfa}times{bfb}cdot{bfc}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6411.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b}\times{\bf c} = {\bf a} \times{\bf b} \cdot{\bf c}.$">|; 

$key = q/lambda'=lambdaslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7348.png"
 ALT="$ \lambda' = \lambda/k$">|; 

$key = q/displaystyle{bfF}=frac{rho}{8pi}int_Sint_{S'}frac{[mbox{boldmathOmega}_S({bfr}')esmbox{boldmathOmega}_S({bfr}')}{|{bfr}-{bfr}'|^{,3}},dS,dS'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="644" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3459.png"
 ALT="$\displaystyle {\bf F} = \frac{\rho}{8\pi}\int_S\int_{S'}\frac{[\mbox{\boldmath$...
...box{\boldmath$\Omega$}_S({\bf r}')}{\vert{\bf r}-{\bf r}'\vert^{ 3}} dS dS'.$">|; 

$key = q/displaystylerho,g,h=frac{2,gamma,costheta}{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1141.png"
 ALT="$\displaystyle \rho g h = \frac{2 \gamma \cos\theta}{a},$">|; 

$key = q/d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1285.png"
 ALT="$ d$">|; 

$key = q/l=z_2-z_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3684.png"
 ALT="$ l=z_2-z_1$">|; 

$key = q/displaystyle{bfb}cdotoint_C{bft}times{bfg},dr={bfb}cdot!int_S[(nabla{bfg})cdot{bfn}-(nablacdot{bfg}),{bfn}],dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1099.png"
 ALT="$\displaystyle {\bf b}\cdot\oint_C {\bf t}\times {\bf g} dr = {\bf b}\cdot\!\int_S[(\nabla {\bf g})\cdot{\bf n} - (\nabla\cdot{\bf g}) {\bf n}] dS,$">|; 

$key = q/y<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2469.png"
 ALT="$ y&lt;0$">|; 

$key = q/varpi=r,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2529.png"
 ALT="$ \varpi= r \sin\theta$">|; 

$key = q/a_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6770.png"
 ALT="$ a_i$">|; 

$key = q/displaystyleh=2,l,sin(pislash4-thetaslash2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1224.png"
 ALT="$\displaystyle h=2 l \sin(\pi/4-\theta/2).$">|; 

$key = q/u'slashUll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5900.png"
 ALT="$ u'/U\ll 1$">|; 

$key = q/displaystylesimeqfrac{2}{U},frac{partialphi}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5947.png"
 ALT="$\displaystyle \simeq \frac{2}{U} \frac{\partial\phi}{\partial x},$">|; 

$key = q/displaystyleM=int_Vrho,dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="$\displaystyle M = \int_V \rho dV,$">|; 

$key = q/displaystyleF=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1403.png"
 ALT="$\displaystyle F= 0.$">|; 

$key = q/phi=phi_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4724.png"
 ALT="$ \phi=\phi_+$">|; 

$key = q/displaystyleB=frac{V,a^{,2},e^{,2}}{e,(1-e^{,2})^{,-1}+(1slash2)lnleft[(1-e)slash(1+e)right]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2803.png"
 ALT="$\displaystyle B = \frac{V a^{ 2} e^{ 2}}{e (1-e^{ 2})^{ -1}+(1/2)\ln\left[(1-e)/(1+e)\right]}.$">|; 

$key = q/kappa({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="$ \kappa({\bf r},t)$">|; 

$key = q/displaystyleomega(x,y)=-{rmsgn}(y),1.22,frac{U_0}{delta(x)},f''!left[1.22,frac{|y|}{delta(x)}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2992.png"
 ALT="$\displaystyle \omega(x,y) = - {\rm sgn}(y) 1.22 \frac{U_0}{\delta(x)} f''\!\left[1.22 \frac{\vert y\vert}{\delta(x)}\right],$">|; 

$key = q/(dFslashdz)^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2413.png"
 ALT="$ (dF/dz)^{ 2}$">|; 

$key = q/displaystyleP_2(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4242.png"
 ALT="$\displaystyle P_2(x)$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{AppendixAslashfigA.18.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="227" BORDER="0"
 SRC="|."$dir".q|img6496.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{AppendixA/figA.18.eps}}
\end{figure}">|; 

$key = q/-p_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1245.png"
 ALT="$ -p_0$">|; 

$key = q/skew{3}bar{psi}_1=b,V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2228.png"
 ALT="$ \skew{3}\bar{\psi}_1=b V$">|; 

$key = q/delta(x)=2,(nu,xslashU_0)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3065.png"
 ALT="$ \delta(x)= 2 (\nu x/U_0)^{1/2}$">|; 

$key = q/d{bfS}=(0,,0,,dS_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6640.png"
 ALT="$ d{\bf S} = (0, 0, dS_z)$">|; 

$key = q/displaystylefrac{1}{2},BP',,BQ'=4,a^{,2},s.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img774.png"
 ALT="$\displaystyle \frac{1}{2} BP'  BQ'= 4 a^{ 2} s.$">|; 

$key = q/displaystyle=frac{partialpsi}{partialr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1647.png"
 ALT="$\displaystyle = \frac{\partial\psi}{\partial r}.$">|; 

$key = q/displaystyleb=lim_{srightarrowinfty}y(s)=d-frac{2}{pi},frac{skew{3}bar{psi}_1}{V}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2225.png"
 ALT="$\displaystyle b= \lim_{s\rightarrow\infty} y(s)= d-\frac{2}{\pi} \frac{\skew{3}\bar{\psi}_1}{V}.$">|; 

$key = q/displaystylesimeq-left(frac{1+[(gamma-1)slash2],{rmMa}_1^{,2}}{gamma,{rmMa}_1^{,2}}right)frac{{mitDelta}p}{p_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6164.png"
 ALT="$\displaystyle \simeq -\left(\frac{1+[(\gamma-1)/2] {\rm Ma}_1^{ 2}}{\gamma {\rm Ma}_1^{ 2}}\right)\frac{{\mit\Delta}p}{p_1},$">|; 

$key = q/0leqlambdaleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2518.png"
 ALT="$ 0\leq\lambda\leq 1$">|; 

$key = q/displaystyle<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3146.png"
 ALT="$\displaystyle &lt;0,$">|; 

$key = q/displaystylefrac{partial^{,2}psi}{partialy^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2930.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\psi}{\partial y^{ 2}}$">|; 

$key = q/q_1=(1slash2),rho_1,w_1^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5788.png"
 ALT="$ q_1=(1/2) \rho_1 w_1^{ 2}$">|; 

$key = q/z=z_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1243.png"
 ALT="$ z=z_2$">|; 

$key = q/D=-2pi,V,a^{,3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2648.png"
 ALT="$ D = -2\pi V a^{ 3}$">|; 

$key = q/MG;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img626.png"
 ALT="$ MG$">|; 

$key = q/beta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2634.png"
 ALT="$ \beta_0$">|; 

$key = q/omega<2slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img796.png"
 ALT="$ \omega&lt;2/3$">|; 

$key = q/displaystyleC_D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5801.png"
 ALT="$\displaystyle C_D$">|; 

$key = q/displaystylepsi_Q({bfr})=-frac{Q}{2pi},theta=-frac{Q}{2pi},tan^{-1}(yslashx);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1683.png"
 ALT="$\displaystyle \psi_Q({\bf r})= -\frac{Q}{2\pi} \theta =- \frac{Q}{2\pi} \tan^{-1}(y/x)$">|; 

$key = q/displaystyle{rmFr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="$\displaystyle {\rm Fr}$">|; 

$key = q/displaystyleleft.frac{partialphi}{partialz}right|_{z=0}=-g^{,-1}left.frac{partial^{,2}phi}{partialt^{,2}}right|_{z=0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3913.png"
 ALT="$\displaystyle \left.\frac{\partial \phi}{\partial z}\right\vert _{z=0} = -g^{ -1}\left.\frac{\partial^{ 2}\phi}{\partial t^{ 2}}\right\vert _{z=0}.$">|; 

$key = q/A'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4021.png"
 ALT="$ A''$">|; 

$key = q/displaystylelambda(theta)=0.47,frac{costheta}{sin^6theta}int_0^thetasin^5theta',dtheta'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3220.png"
 ALT="$\displaystyle \lambda(\theta) = 0.47 \frac{\cos\theta}{\sin^6\theta}\int_0^\theta \sin^5 \theta' d\theta'.$">|; 

$key = q/beta>2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2963.png"
 ALT="$ \beta&gt;2$">|; 

$key = q/overline{nabla}=L,nabla;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="$ \overline{\nabla} = L \nabla$">|; 

$key = q/f''(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$ f''(0)$">|; 

$key = q/displaystyle{bfa}cdot{bfb}times{bfc}=-{bfb}cdot{bfa}times{bfc}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6413.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b} \times{\bf c} = - {\bf b} \cdot {\bf a} \times{\bf c}.$">|; 

$key = q/c=(a+b)slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2135.png"
 ALT="$ c=(a+b)/2$">|; 

$key = q/displaystylegamma_{r,,s}=sum_{n=1,infty}sum_{m=0,n}(1+k_n'-h_n'),alpha_n,{mitGamma}_{n,,r}^{,m},{mitGamma}_{n,,s}^{,m,ast},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4918.png"
 ALT="$\displaystyle \gamma_{r, s} = \sum_{n=1,\infty}\sum_{m=0,n} (1+k_n'-h_n') \alpha_n {\mit\Gamma}_{n, r}^{ m} {\mit\Gamma}_{n, s}^{ m \ast},$">|; 

$key = q/d^{,2}uslashdz^{,2}<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img794.png"
 ALT="$ d^{ 2}u/dz^{ 2}&lt;0$">|; 

$key = q/displaystyleleft(frac{dP}{dx}right)_{x=x_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3147.png"
 ALT="$\displaystyle \left(\frac{dP}{dx}\right)_{x=x_0}$">|; 

$key = q/AD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4052.png"
 ALT="$ AD$">|; 

$key = q/Q>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2832.png"
 ALT="$ Q&gt;0$">|; 

$key = q/displaystyletimes{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4449.png"
 ALT="$\displaystyle \times {\bf v}$">|; 

$key = q/x>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2114.png"
 ALT="$ x&gt;0$">|; 

$key = q/displaystyle=oint_C{bfB}cdotd{bfr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2349.png"
 ALT="$\displaystyle =\oint_C {\bf B}\cdot d{\bf r},$">|; 

$key = q/1slash2<{calC}<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1423.png"
 ALT="$ 1/2&lt;{\cal C}&lt; 1$">|; 

$key = q/{rmMa}_infty<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5916.png"
 ALT="$ {\rm Ma}_\infty &lt; 1$">|; 

$key = q/displaystylebeta_w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6103.png"
 ALT="$\displaystyle \beta_w$">|; 

$key = q/displaystyle{mitGamma}_r(r)=left{{array}{lll}{mitGamma},(rslasha)^2&mbox{hspace{1cm}}&rleqa[0.5ex]{mitGamma}&&r>a{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1719.png"
 ALT="$\displaystyle {\mit\Gamma}_r(r)= \left\{\begin{array}{lll} {\mit\Gamma} (r/a)^2&amp;\mbox{\hspace{1cm}}&amp;r\leq a\ [0.5ex] {\mit\Gamma}&amp;&amp;r&gt;a \end{array}\right..$">|; 

$key = q/displaystyledh={bfgrad},hcdotd{bfr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6537.png"
 ALT="$\displaystyle dh = {\bf grad} h \cdot d{\bf r}.$">|; 

$key = q/displaystylezeta_E=8.14times10^{-2},{rmm}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4598.png"
 ALT="$\displaystyle \zeta_E = 8.14\times 10^{-2} {\rm m}.$">|; 

$key = q/u=infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img938.png"
 ALT="$ u=\infty$">|; 

$key = q/displaystylefrac{b}{d}=frac{pi}{pi+2}=0.61.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2229.png"
 ALT="$\displaystyle \frac{b}{d}=\frac{\pi}{\pi+2}= 0.61.$">|; 

$key = q/kappa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="$ \kappa$">|; 

$key = q/displaystylefrac{da_x}{dt}=frac{d}{dt}!left(phi,b_xright)=frac{dphi}{dt},b_x+phi,frac{db_x}{dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6437.png"
 ALT="$\displaystyle \frac{d a_x}{dt} = \frac{d}{dt}\!\left(\phi  b_x\right) = \frac{d\phi}{dt}  b_x + \phi   \frac{d b_x}{dt},$">|; 

$key = q/180^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$ 180^\circ $">|; 

$key = q/displaystyle=frac{1}{r},frac{partialpsi}{partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1756.png"
 ALT="$\displaystyle =\frac{1}{r} \frac{\partial\psi}{\partial\theta},$">|; 

$key = q/displaystylefrac{z}{a}=-{rmi},zetapmsqrt{1-zeta^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1996.png"
 ALT="$\displaystyle \frac{z}{a} = -{\rm i} \zeta \pm \sqrt{1-\zeta^{ 2}},$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter02slashfig5.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img901.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter02/fig5.01.eps}}
\end{figure}">|; 

$key = q/|x|<c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2473.png"
 ALT="$ \vert x\vert&lt;c$">|; 

$key = q/displaystyleD,P=-frac{1}{sintheta}left[frac{partial}{partialtheta},(sintheta,X)+frac{partialY}{partialphi}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4735.png"
 ALT="$\displaystyle D P = -\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta} (\sin\theta X) +\frac{\partial Y}{\partial\phi}\right],$">|; 

$key = q/nu_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5775.png"
 ALT="$ \nu_{\rm max}$">|; 

$key = q/a_{ii};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6778.png"
 ALT="$ a_{ii}$">|; 

$key = q/displaystyle{calH}_{1,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5471.png"
 ALT="$\displaystyle {\cal H}_{1,2}$">|; 

$key = q/Oy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6254.png"
 ALT="$ Oy$">|; 

$key = q/displaystylephantom{simeqfrac{G,m',a^{2}}{R^{,3}}}left.+{calO}(e^{,2})+{calO}(e,sindelta)+{calO}(sin^2delta)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4309.png"
 ALT="$\displaystyle \phantom{\simeq \frac{G m' a^{2}}{R^{ 3}}}\left. +{\cal O}(e^{ 2})+{\cal O}(e \sin\delta) + {\cal O}(\sin^2\delta)\right].$">|; 

$key = q/lambda>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img675.png"
 ALT="$ \lambda&gt;0$">|; 

$key = q/Q';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2433.png"
 ALT="$ Q'$">|; 

$key = q/pslashrho-v^{,2}slash2+{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1560.png"
 ALT="$ p/\rho -v^{ 2}/2+{\mit\Psi}$">|; 

$key = q/displaystyle=sinthetaleft(-frac{A}{r^{,3}}+frac{B}{r}+2,C+4,D,r^{,2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3836.png"
 ALT="$\displaystyle = \sin\theta\left(-\frac{A}{r^{ 3}}+\frac{B}{r}+ 2 C + 4 D r^{ 2}\right),$">|; 

$key = q/h>2,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1044.png"
 ALT="$ h&gt;2 a$">|; 

$key = q/9.2,{rmkm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4650.png"
 ALT="$ 9.2 {\rm km}$">|; 

$key = q/B=(V-c),zeta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4092.png"
 ALT="$ B=(V-c) \zeta_0$">|; 

$key = q/displaystylephantom{=}left.-frac{2cottheta,v_theta}{r^{,2}}-frac{2}{r^{,2},sinthalvarphi}+frac{1}{3},frac{partial{mitDelta}}{partialr}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img504.png"
 ALT="$\displaystyle \phantom{=}\left.-\frac{2\cot\theta v_\theta}{r^{ 2}}-\frac{2}{...
...{\partial\varphi}+ \frac{1}{3} \frac{\partial{\mit\Delta}}{\partial r}\right),$">|; 

$key = q/p_1=p_0-(1slash2),rho,V_0^{,2},(1+gamma^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1780.png"
 ALT="$ p_1=p_0-(1/2) \rho V_0^{ 2} (1+\gamma^{ 2})$">|; 

$key = q/V_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5524.png"
 ALT="$ V_p$">|; 

$key = q/displaystyleint!int!intz,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6512.png"
 ALT="$\displaystyle \int\!\int\!\int z dV$">|; 

$key = q/displaystyle=int_0^inftyfrac{du}{(a_i^{,2}+u),{mitDelta}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7270.png"
 ALT="$\displaystyle =\int_0^\infty \frac{du}{(a_i^{ 2}+u) {\mit\Delta}}.$">|; 

$key = q/displaystylerho_1,u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5462.png"
 ALT="$\displaystyle \rho_1 u_1$">|; 

$key = q/displaystyleD=(p_2-p_3),t=(p_2-p_3),epsilon,c=frac{4,epsilon^{,2}}{sqrt{{rmMa}_1^{,2}-1}},q_1,c,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5813.png"
 ALT="$\displaystyle D = (p_2-p_3) t = (p_2-p_3) \epsilon c= \frac{4 \epsilon^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}} q_1 c,$">|; 

$key = q/{mitPsi}_rrightarrow{mitPsi}_n^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4940.png"
 ALT="$ {\mit\Psi}_r\rightarrow {\mit\Psi}_n^{ m}$">|; 

$key = q/displaystyleC_{pU}=-frac{2,alpha_0}{sqrt{{rmMa}_1^{,2}-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5795.png"
 ALT="$\displaystyle C_{pU} = -\frac{2 \alpha_0}{\sqrt{{\rm Ma}_1^{ 2}-1}}$">|; 

$key = q/12.421;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4633.png"
 ALT="$ 12.421$">|; 

$key = q/displaystylegamma=frac{c_p}{c_V}=frac{c_V+R}{c_V};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="$\displaystyle \gamma = \frac{c_p}{c_V} = \frac{c_V+R}{c_V}$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{AppendixAslashfigA.10.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="255" BORDER="0"
 SRC="|."$dir".q|img6387.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{AppendixA/figA.10.eps}}
\end{figure}">|; 

$key = q/displaystylewidehat{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img899.png"
 ALT="$\displaystyle \widehat{E}$">|; 

$key = q/displaystyleS,rho,g,h={calC},S,rho,v^{,2}=2,{calC},S,rho,g,h,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1418.png"
 ALT="$\displaystyle S \rho g h = {\cal C} S \rho v^{ 2}= 2 {\cal C} S \rho g h,$">|; 

$key = q/displaystylefrac{2}{S}left(frac{A_1,A_2}{A_1+A_2}right)left(frac{h}{2,g}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1603.png"
 ALT="$\displaystyle \frac{2}{S}\left(\frac{A_1 A_2}{A_1+A_2}\right)\left(\frac{h}{2 g}\right)^{1/2}.
$">|; 

$key = q/displaystyle{bfa}cdot{bfb}=a_1,b_1+a_2,b_2+a_3,b_3.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6772.png"
 ALT="$\displaystyle {\bf a}\cdot{\bf b} = a_1 b_1+a_2 b_2+a_3 b_3.$">|; 

$key = q/{bfe}_z=nablazslash|nablaz|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7111.png"
 ALT="$ {\bf e}_z=\nabla z/\vert\nabla z\vert$">|; 

$key = q/{mitOmega}_varphi=v_varphi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2570.png"
 ALT="$ {\mit\Omega}_\varphi=v_\varphi=0$">|; 

$key = q/a_1geqa_2geqa_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img925.png"
 ALT="$ a_1\geq a_2\geq a_3$">|; 

$key = q/AP';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img734.png"
 ALT="$ AP'$">|; 

$key = q/displaystylep(x,0_+,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4102.png"
 ALT="$\displaystyle p(x,0_+,t)$">|; 

$key = q/rho,Q,Q'slash(2pi,r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2434.png"
 ALT="$ \rho Q Q'/(2\pi r)$">|; 

$key = q/0leqthetaleqpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2099.png"
 ALT="$ 0\leq \theta\leq \pi$">|; 

$key = q/displaystyle{rmMa}_1^{,2},sin^2beta-1simeqleft(frac{gamma+1}{2}right)left(frac{{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right)theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5726.png"
 ALT="$\displaystyle {\rm Ma}_1^{ 2} \sin^2\beta-1\simeq \left(\frac{\gamma+1}{2}\right)\left(\frac{{\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right)\theta.$">|; 

$key = q/displaystyleK=frac{1}{2},rholeft[int_{C_2}phi,dpsi-int_{C_1}phi,dpsi-int_{C_3}[phi],dpsiright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2508.png"
 ALT="$\displaystyle K= \frac{1}{2} \rho\left[\int_{C_2}\phi d\psi-\int_{C_1}\phi d\psi-\int_{C_3}[\phi] d\psi\right],
$">|; 

$key = q/e_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4605.png"
 ALT="$ e_E$">|; 

$key = q/displaystyleleft(frac{dU}{dx}right)_{x=x_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3145.png"
 ALT="$\displaystyle \left(\frac{dU}{dx}\right)_{x=x_0}$">|; 

$key = q/displaystylec_{ijk}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6852.png"
 ALT="$\displaystyle c_{ijk}'$">|; 

$key = q/displaystyleF_{rmc}=frac{3}{2}left(frac{Q^{,2}}{g}right)^{1slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1431.png"
 ALT="$\displaystyle F_{\rm c} = \frac{3}{2}\left(\frac{Q^{ 2}}{g}\right)^{1/3},$">|; 

$key = q/beta_{r,-s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4888.png"
 ALT="$ \beta_{r,-s}$">|; 

$key = q/displaystyle=nabla(v^{,2}slash2)+frac{nablap}{rho}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5867.png"
 ALT="$\displaystyle =\nabla(v^{ 2}/2) + \frac{\nabla p}{\rho}.$">|; 

$key = q/displaystyle{mitPhi}_{rmtide}(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4279.png"
 ALT="$\displaystyle {\mit\Phi}_{\rm tide}(r,\theta,\varphi)$">|; 

$key = q/{calR}_n^{,(pmm)}(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4324.png"
 ALT="$ {\cal R}_n^{ (\pm m)}(r,\theta,\varphi)$">|; 

$key = q/p_1=rho,(partialphislashpartialt);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4145.png"
 ALT="$ p_1=\rho (\partial \phi/\partial t)$">|; 

$key = q/displaystyleM=frac{rho,D_1,D_2}{2pi,x^{,2}},sin(alpha_1+alpha_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2456.png"
 ALT="$\displaystyle M = \frac{\rho D_1 D_2}{2\pi x^{ 2}} \sin(\alpha_1+\alpha_2),
$">|; 

$key = q/displaystylefrac{v_g}{v_p}simeqfrac{3}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4163.png"
 ALT="$\displaystyle \frac{v_g}{v_p}\simeq \frac{3}{2}.$">|; 

$key = q/displaystyle{calE}=gammaint_{z_1}^{z_2}2pi,f,(1+f_z^{,2})^{1slash2},dz-p_0int_{z_1}^{z_2}pi,f^{,2},dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1246.png"
 ALT="$\displaystyle {\cal E} = \gamma\int_{z_1}^{z_2} 2\pi f (1+f_z^{ 2})^{1/2} dz-p_0\int_{z_1}^{z_2}\pi f^{ 2} dz,$">|; 

$key = q/yrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3246.png"
 ALT="$ y\rightarrow\infty$">|; 

$key = q/displaystyleD,{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4761.png"
 ALT="$\displaystyle D {\mit\Psi}$">|; 

$key = q/displaystylea_{issl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6959.png"
 ALT="$\displaystyle a_{issl}$">|; 

$key = q/0leqxi_0leqpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2464.png"
 ALT="$ 0\leq \xi_0\leq \pi$">|; 

$key = q/rgeqa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3887.png"
 ALT="$ r\geq a$">|; 

$key = q/displaystyle=a_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6290.png"
 ALT="$\displaystyle = a_z.$">|; 

$key = q/zeta_0+dzeta'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2067.png"
 ALT="$ \zeta_0+d\zeta''$">|; 

$key = q/displaystyle{bfa}'simeq{bfa}+deltatheta_z,{bfe}_ztimes{bfa},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6394.png"
 ALT="$\displaystyle {\bf a}' \simeq {\bf a} +\delta\theta_z  {\bf e}_z\times {\bf a},$">|; 

$key = q/displaystylefrac{partialu}{partialt}+u,frac{partialu}{partialx}+frac{c^{,2}}{rho},frac{partialrho}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5350.png"
 ALT="$\displaystyle \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + \frac{c^{ 2}}{\rho} \frac{\partial \rho}{\partial x}$">|; 

$key = q/displaystyley'=frac{c}{sqrt{1-c^{,2}}}={rmconst}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7328.png"
 ALT="$\displaystyle y' = \frac{c}{\sqrt{1-c^{ 2}}} = {\rm const}.$">|; 

$key = q/pma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2020.png"
 ALT="$ \pm a)$">|; 

$key = q/BQ_3P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2534.png"
 ALT="$ B Q_3 P$">|; 

$key = q/displaystyle{mitDelta}p=frac{2,gamma}{R}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1116.png"
 ALT="$\displaystyle {\mit\Delta} p = \frac{2 \gamma}{R}.$">|; 

$key = q/displaystyleomega_z=frac{1}{r},frac{partial}{partialr}!left(r,frac{partialpsi}{pht)+frac{1}{r^{,2}},frac{partial^{,2}psi}{partialtheta^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1649.png"
 ALT="$\displaystyle \omega_z =\frac{1}{r} \frac{\partial}{\partial r}\!\left(r \fra...
...}\right) + \frac{1}{r^{ 2}} \frac{\partial^{ 2}\psi}{\partial \theta^{ 2}}.$">|; 

$key = q/displaystylegamma_{23},deltar-gamma_{13},deltar-gamma_{12},deltar,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1128.png"
 ALT="$\displaystyle \gamma_{23} \delta r - \gamma_{13} \delta r - \gamma_{12} \delta r \cos\theta.$">|; 

$key = q/displaystyle=frac{h}{2},(1-sinalpha).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1893.png"
 ALT="$\displaystyle = \frac{h}{2} (1-\sin\alpha).$">|; 

$key = q/v_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6762.png"
 ALT="$ v_i$">|; 

$key = q/displaystylealpha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img858.png"
 ALT="$\displaystyle \alpha_0$">|; 

$key = q/p_0simeq10^5,{rmN,m}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img567.png"
 ALT="$ p_0\simeq 10^5 {\rm N m}^{-2}$">|; 

$key = q/displaystyle=Y+frac{1}{sintheta},frac{partialP}{partialphi}-frac{partialQ}{partialtheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4744.png"
 ALT="$\displaystyle = Y + \frac{1}{\sin\theta} \frac{\partial P}{\partial\phi} - \frac{\partial Q}{\partial\theta}.$">|; 

$key = q/{calR}_n(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4320.png"
 ALT="$ {\cal R}_n(r,\theta,\varphi)$">|; 

$key = q/displaystyleint_0^py,dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7383.png"
 ALT="$\displaystyle \int_0^p y dx.
$">|; 

$key = q/displaystyle=0+int_0^1y^{,2},dy=frac{1}{3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6451.png"
 ALT="$\displaystyle = 0 + \int_0^1 y^{ 2} dy = \frac{1}{3}.$">|; 

$key = q/displaystylezeta_0+dzeta'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2062.png"
 ALT="$\displaystyle \zeta_0+d\zeta''$">|; 

$key = q/displaystylec(phi)=sum_{nu=1,3,5,cdots}c_nu,sin(nu,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3544.png"
 ALT="$\displaystyle c(\phi) = \sum_{\nu=1,3,5,\cdots}c_\nu \sin(\nu \phi),
$">|; 

$key = q/|{bfF}|,|{bfr}|,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6384.png"
 ALT="$ \vert{\bf F}\vert \vert{\bf r}\vert \sin\theta$">|; 

$key = q/displaystylesimeq3.2times10^{-1},V_0({rmm,s^{-1}})]slash[L({rmm})]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="$\displaystyle \simeq 3.2\times 10^{-1} V_0({\rm m s^{-1}})]/[L({\rm m})]^{1/2}.$">|; 

$key = q/displaystylev_g=frac{domega}{dk}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3931.png"
 ALT="$\displaystyle v_g = \frac{d\omega}{dk}.$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter13slashfig4.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img5239.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter13/fig4.04.eps}}
\end{figure}">|; 

$key = q/z=zeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2133.png"
 ALT="$ z=\zeta$">|; 

$key = q/q_i({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="$ q_i({\bf r},t)$">|; 

$key = q/displaystylefrac{dF}{dz}=2,z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1959.png"
 ALT="$\displaystyle \frac{dF}{dz} = 2  z$">|; 

$key = q/-deltatheta_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6393.png"
 ALT="$ -\delta\theta_z$">|; 

$key = q/displaystyleskew{3}ddot{eta}+2,{mitOmega},costheta,skew{5}dot{xi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4657.png"
 ALT="$\displaystyle \skew{3}\ddot{\eta} + 2 {\mit\Omega} \cos\theta \skew{5}\dot{\xi}$">|; 

$key = q/psi(x,y,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1655.png"
 ALT="$ \psi(x,y,t)$">|; 

$key = q/widehat{L}<0.30375;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img962.png"
 ALT="$ \widehat{L}&lt;0.30375$">|; 

$key = q/M_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4551.png"
 ALT="$ M_m$">|; 

$key = q/{calT}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1363.png"
 ALT="$ {\cal T}_1$">|; 

$key = q/r''=-|PR|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7234.png"
 ALT="$ r''=-\vert PR\vert$">|; 

$key = q/{mitGamma}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$ {\mit \Gamma }=0$">|; 

$key = q/k,drightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4186.png"
 ALT="$ k d\rightarrow \infty$">|; 

$key = q/r'>r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4246.png"
 ALT="$ r'&gt;r$">|; 

$key = q/0.25;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$ 0.25$">|; 

$key = q/displaystylefrac{C_L}{C_{L0}}=1-frac{1}{2,{calA}_e},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6083.png"
 ALT="$\displaystyle \frac{C_L}{C_{L0}} =1-\frac{1}{2 {\cal A}_e},$">|; 

$key = q/ointnablatimes{bfA}cdotd{bfS}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6665.png"
 ALT="$ \oint\nabla\times {\bf A}\cdot d{\bf S} = 0$">|; 

$key = q/displaystylelambda=frac{1-cosalpha,cosomega}{cosalpha-cosomega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2304.png"
 ALT="$\displaystyle \lambda = \frac{1-\cos\alpha \cos\omega}{\cos\alpha-\cos\omega},$">|; 

$key = q/displaystyle=frac{1}{r}frac{partial}{partialr}!left(r,frac{partial}{partialr}rigl^{,2}}{partialtheta^{,2}}+frac{partial^{,2}}{partialz^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="$\displaystyle =\frac{1}{r}\frac{\partial}{\partial r}\!\left(r \frac{\partial}...
...\partial^{ 2}}{\partial \theta^{ 2}}+\frac{\partial^{ 2}}{\partial z^{ 2}},$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partial{mitPhi}_r'(theta,phi_pm)}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4843.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial {\mit\Phi}_r'(\theta,\phi_\pm)}{\partial\phi}$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{AppendixAslashfigA.13.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="226" BORDER="0"
 SRC="|."$dir".q|img6444.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{AppendixA/figA.13.eps}}
\end{figure}">|; 

$key = q/{mitGamma}_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3347.png"
 ALT="$ {\mit\Gamma}_\infty$">|; 

$key = q/displaystylepsi(r,theta)=sin^2thetaleft(frac{A}{r}+B,r+C,r^{,2}+D,r^{,4}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3833.png"
 ALT="$\displaystyle \psi(r,\theta) = \sin^2\theta\left(\frac{A}{r}+B r+C r^{ 2} + D r^{ 4}\right),$">|; 

$key = q/displaystylephantom{-frac{G,m',a^{2}}{r'^{,3}}}+frac{1}{3},P_2^{,1},(costheta'),sinvarphi',{calR}_2^{,(+1)}(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="366" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4282.png"
 ALT="$\displaystyle \phantom{ - \frac{G m' a^{2}}{r'^{ 3}}}+\frac{1}{3} P_2^{ 1} (\cos\theta') \sin\varphi' {\cal R}_2^{ (+1)}(r,\theta,\varphi)$">|; 

$key = q/displaystyle({bfb}-{bfa})cdot({bfb}-{bfa})=|{bfa}|^{,2}+|{bfb}|^{,2}-2,{bfa}cdot{bfb}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6313.png"
 ALT="$\displaystyle ({\bf b} - {\bf a} ) \cdot ({\bf b} - {\bf a} ) = \vert{\bf a}\vert^{ 2} + \vert{\bf b}\vert^{ 2} - 2 {\bf a} \cdot {\bf b}.$">|; 

$key = q/displaystyle{calR}_{ij}=delta_{ij}-delta{theta}_k,epsilon_{kij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6826.png"
 ALT="$\displaystyle {\cal R}_{ij} = \delta_{ij} - \delta{\theta}_k \epsilon_{kij}$">|; 

$key = q/deltar=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img714.png"
 ALT="$ \delta r= 0$">|; 

$key = q/displaystyled{calE}={calC}_V,dT=frac{c_V}{M},dT.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5307.png"
 ALT="$\displaystyle d{\cal E} ={\cal C}_V dT = \frac{c_V}{M} dT.$">|; 

$key = q/betasimeqmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6114.png"
 ALT="$ \beta\simeq \mu$">|; 

$key = q/displaystylefrac{1}{gamma-1}left(frac{Dp}{Dt}-frac{gamma,p}{rho},frac{Drho}{Dt}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="$\displaystyle \frac{1}{\gamma-1}\left(\frac{Dp}{Dt} - \frac{\gamma p}{\rho} \frac{D\rho}{Dt}\right)$">|; 

$key = q/displaystyle=left(frac{gamma+1}{4}right)left(frac{{rmMa}_1^{,2}}{{rmMa}_1^{,2}-1}right)theta_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6181.png"
 ALT="$\displaystyle =\left(\frac{\gamma+1}{4}\right)\left(\frac{{\rm Ma}_1^{ 2}}{{\rm Ma}_1^{ 2}-1}\right)\theta_1,$">|; 

$key = q/displaystyletau_i=int_Vepsilon_{ijk},x_j,F_k,dV+int_Vepsilon_{ijk}frac{partial(x_j,sigma_{kl})}{partialx_l},dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$\displaystyle \tau_i = \int_V \epsilon_{ijk} x_j F_k dV + \int_V \epsilon_{ijk}\frac{\partial (x_j \sigma_{kl})}{\partial x_l} dV,$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter09slashairfoil.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img3354.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter09/airfoil.eps}}
\end{figure}">|; 

$key = q/x=x_R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5820.png"
 ALT="$ x=x_R$">|; 

$key = q/displaystyleV>V_s=left(frac{W}{pi,rho,S,alpha_c}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3536.png"
 ALT="$\displaystyle V &gt; V_s = \left(\frac{W}{\pi \rho S \alpha_c}\right)^{1/2}.$">|; 

$key = q/displaystylefrac{partialA}{partialtheta}+frac{1}{sintheta},frac{partialB}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4780.png"
 ALT="$\displaystyle \frac{\partial A}{\partial\theta} + \frac{1}{\sin\theta} \frac{\partial B}{\partial\phi}$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter03slashfig6.06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="371" BORDER="0"
 SRC="|."$dir".q|img1219.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter03/fig6.06.eps}}
\end{figure}">|; 

$key = q/displaystyletan(beta-theta)equivfrac{tanbeta-tantheta}{1+tanbeta,tantheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5705.png"
 ALT="$\displaystyle \tan(\beta-\theta)\equiv \frac{\tan\beta-\tan\theta}{1+\tan\beta \tan\theta},$">|; 

$key = q/{bfc}equiv(c_x,,c_y,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6424.png"
 ALT="$ {\bf c} \equiv (c_x,  c_y,  0)$">|; 

$key = q/displaystyledS_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7176.png"
 ALT="$\displaystyle dS_\phi$">|; 

$key = q/H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img592.png"
 ALT="$ H$">|; 

$key = q/displaystyle=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2585.png"
 ALT="$\displaystyle = 1,$">|; 

$key = q/displaystylefrac{partial}{partialx'}=costheta,frac{partial}{partialx}+sintheta,frac{partial}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6571.png"
 ALT="$\displaystyle \frac{\partial}{\partial x'} = \cos\theta  \frac{\partial}{\partial x} + \sin\theta  \frac{\partial}{\partial y},$">|; 

$key = q/AP_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1623.png"
 ALT="$ AP_2$">|; 

$key = q/0leqyleqh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3869.png"
 ALT="$ 0\leq y\leq h$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashjet.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="339" BORDER="0"
 SRC="|."$dir".q|img1885.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/jet.eps}}
\end{figure}">|; 

$key = q/displaystylebeta=sin^{-1}left[frac{(g,d)^{1slash2}}{V}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4018.png"
 ALT="$\displaystyle \beta = \sin^{-1}\left[\frac{(g d)^{1/2}}{V}\right].$">|; 

$key = q/gammarightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img902.png"
 ALT="$ \gamma\rightarrow 0$">|; 

$key = q/displaystylephantom{=}+2sum_{m=1,n}frac{(n-m)!}{(n+m)!},P_n^{,m}(costheta),P_n^{,m}(costheta'),cos[m,(varphi-varphi')],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="481" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4259.png"
 ALT="$\displaystyle \phantom{=} + 2\sum_{m=1,n}\frac{(n-m)!}{(n+m)!} P_n^{ m}(\cos\theta) P_n^{ m}(\cos\theta') \cos[m (\varphi-\varphi')],$">|; 

$key = q/108^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3223.png"
 ALT="$ 108^\circ$">|; 

$key = q/displaystylelambda_rint_{mitOmega}{mitPhi}_r,d{mitOmega}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4811.png"
 ALT="$\displaystyle \lambda_r\int_{\mit\Omega} {\mit\Phi}_r d{\mit\Omega} = 0.$">|; 

$key = q/A(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5620.png"
 ALT="$ A(x)$">|; 

$key = q/y={rmIm}(z_0)+pislashK';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2177.png"
 ALT="$ y={\rm Im}(z_0)+\pi/K'$">|; 

$key = q/displaystylefrac{1}{xi^{,2}}frac{d}{dxi}!left(xi^{,2},frac{dtheta}{dxi}right)=-theta^{,3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5216.png"
 ALT="$\displaystyle \frac{1}{\xi^{ 2}}\frac{d}{d\xi}\!\left(\xi^{ 2} \frac{d\theta}{d\xi}\right) = -\theta^{ 3}.$">|; 

$key = q/displaystylec=b,cosh(dslashb).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1265.png"
 ALT="$\displaystyle c = b \cosh(d/b).$">|; 

$key = q/displaystyle=left(frac{r}{a}right)^nP_n^{,m}(costheta),sin(m,varphi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4277.png"
 ALT="$\displaystyle = \left(\frac{r}{a}\right)^n P_n^{ m}(\cos\theta) \sin(m \varphi),$">|; 

$key = q/displaystyleP_0(mu);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2584.png"
 ALT="$\displaystyle P_0(\mu)$">|; 

$key = q/MH';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img645.png"
 ALT="$ MH'$">|; 

$key = q/displaystyleF(z)=-frac{Q}{2pi},ln(z-{rmi},a)-frac{Q}{2pi},ln(z+{rmi},a)=-frac{Q}{2pi},ln(z^{,2}+a^{,2}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="499" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2021.png"
 ALT="$\displaystyle F(z) = - \frac{Q}{2\pi} \ln(z-{\rm i} a) - \frac{Q}{2\pi} \ln(z+{\rm i} a) = -\frac{Q}{2\pi} \ln(z^{ 2}+a^{ 2}).$">|; 

$key = q/displaystylephi(varpi,z)=-V,z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2602.png"
 ALT="$\displaystyle \phi(\varpi,z) = -V z,$">|; 

$key = q/0<k<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1168.png"
 ALT="$ 0&lt; k&lt; 1$">|; 

$key = q/r^{-m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1742.png"
 ALT="$ r^{-m}$">|; 

$key = q/s=1-s_0>1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img727.png"
 ALT="$ s = 1-s_0&gt;1/2$">|; 

$key = q/C_{p_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6043.png"
 ALT="$ C_{p_0}$">|; 

$key = q/displaystylefrac{partialV}{partialx}=A_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6556.png"
 ALT="$\displaystyle \frac{\partial V}{\partial x} = A_x,$">|; 

$key = q/V_p,t<x<V_s,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5527.png"
 ALT="$ V_p t&lt;x&lt;V_s t$">|; 

$key = q/displaystyleF(z)={rmi},frac{{mitGamma}}{2pi},lnz-V_0,z,{rme}^{-{rmi},theta_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2429.png"
 ALT="$\displaystyle F(z)={\rm i} \frac{{\mit\Gamma}}{2\pi} \ln z - V_0 z {\rm e}^{-{\rm i} \theta_0}.$">|; 

$key = q/vrightarrowV_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1777.png"
 ALT="$ v\rightarrow V_0$">|; 

$key = q/displaystylerightarrowleft(frac{gamma-1}{gamma+1}right)frac{{mitDelta}p}{p_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5518.png"
 ALT="$\displaystyle \rightarrow \left(\frac{\gamma-1}{\gamma+1}\right)\frac{{\mit\Delta}p}{p_1},$">|; 

$key = q/h_xi=h_eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2719.png"
 ALT="$ h_\xi=h_\eta$">|; 

$key = q/nu({rmMa});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5766.png"
 ALT="$ \nu({\rm Ma})$">|; 

$key = q/displaystylev_theta(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1696.png"
 ALT="$\displaystyle v_\theta(r,\theta)$">|; 

$key = q/displaystyleeta(theta,phi_pm,t)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4730.png"
 ALT="$\displaystyle \eta(\theta,\phi_\pm,t) = 0$">|; 

$key = q/displaystyledE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5329.png"
 ALT="$\displaystyle dE$">|; 

$key = q/displaystyle{bfa}cdot{bfb}=|{bfa}|,|{bfb}|,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6316.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b} = \vert{\bf a}\vert  \vert{\bf b}\vert  \cos\theta.$">|; 

$key = q/epsilonrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2419.png"
 ALT="$ \epsilon\rightarrow 0$">|; 

$key = q/x=x_L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5819.png"
 ALT="$ x=x_L$">|; 

$key = q/-F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1399.png"
 ALT="$ -F$">|; 

$key = q/displaystyleeta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6143.png"
 ALT="$\displaystyle \eta_1$">|; 

$key = q/phi(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6435.png"
 ALT="$ \phi(t)$">|; 

$key = q/dot{W}-dot{Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="$ \dot{W}-\dot{Q}$">|; 

$key = q/3pislash4-thetaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1204.png"
 ALT="$ 3\pi/4-\theta/2$">|; 

$key = q/displaystyledelta(x)=left(frac{nu,x}{U_0}right)^{1slash2}sqrt{2}int_0^infty[1-f'(eta)],deta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2988.png"
 ALT="$\displaystyle \delta(x) = \left(\frac{\nu x}{U_0}\right)^{1/2}\sqrt{2}\int_0^\infty [1-f'(\eta)] d\eta.$">|; 

$key = q/f_0=2,{mitOmega},costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5105.png"
 ALT="$ f_0=2 {\mit\Omega} \cos\theta$">|; 

$key = q/{bfa}={bfe}_x+3,{bfe}_y-{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6708.png"
 ALT="$ {\bf a} = {\bf e}_x + 3 {\bf e}_y-{\bf e}_z$">|; 

$key = q/displaystylepsi(r,theta)=frac{A}{1+m},r^{,1+m},sinleft[(1+m),(pi-theta)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1870.png"
 ALT="$\displaystyle \psi(r,\theta) = \frac{A}{1+m} r^{ 1+m} \sin\left[(1+m) (\pi-\theta)\right]$">|; 

$key = q/{bfc}slashgamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6739.png"
 ALT="$ {\bf c}/\gamma$">|; 

$key = q/hslasha<sqrt{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1041.png"
 ALT="$ h/a&lt;\sqrt{2}$">|; 

$key = q/rggaslash{rmRe};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3832.png"
 ALT="$ r\gg a/{\rm Re}$">|; 

$key = q/deltaK=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7335.png"
 ALT="$ \delta K= 0$">|; 

$key = q/displaystyleV=-u_1+left[frac{g,h_2,(h_1+h_2)}{2,h_1}right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1516.png"
 ALT="$\displaystyle V = -u_1 +\left[\frac{g h_2 (h_1+h_2)}{2 h_1}\right]^{1/2}.$">|; 

$key = q/rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="$ \rho$">|; 

$key = q/displaystylefrac{D}{Dt}!left(frac{p}{rho^{,gamma}}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="$\displaystyle \frac{D}{Dt}\!\left(\frac{p}{\rho^{ \gamma}}\right)$">|; 

$key = q/displaystyle=left(frac{gamma}{gamma-1}right){calR},T_{1,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5472.png"
 ALT="$\displaystyle = \left(\frac{\gamma}{\gamma-1}\right){\cal R} T_{1,2}.$">|; 

$key = q/displaystylesimeqtheta_1',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6201.png"
 ALT="$\displaystyle \simeq \theta_1',$">|; 

$key = q/{mitDelta}(delta{mitPhi})=left.delta{mitPhi}right|_{z=d}-left.delta{mitPhi}right|_{z=0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4482.png"
 ALT="$ {\mit\Delta} (\delta{\mit\Phi})= \left.\delta{\mit\Phi}\right\vert _{z=d}-\left.\delta{\mit\Phi}\right\vert _{z=0}$">|; 

$key = q/displaystylesimeqleft(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right)theta_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6193.png"
 ALT="$\displaystyle \simeq \left(\frac{\gamma {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right)\theta_1,$">|; 

$key = q/displaystyle(nabla^{,2}{bfA})_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7161.png"
 ALT="$\displaystyle (\nabla^{ 2}{\bf A})_\theta$">|; 

$key = q/displaystyle=-frac{1}{2},V,c^{,2},sinh^2xi_0,sin^2eta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2799.png"
 ALT="$\displaystyle = -\frac{1}{2} V c^{ 2} \sinh^2\xi_0 \sin^2\eta,$">|; 

$key = q/displaystylefrac{rho,U,d^{,2}}{mu,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3695.png"
 ALT="$\displaystyle \frac{\rho U d^{ 2}}{\mu l}$">|; 

$key = q/displaystyleF=sum_{r=1,infty}a_r,{mitPhi}_r.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4821.png"
 ALT="$\displaystyle F= \sum_{r=1,\infty} a_r {\mit\Phi}_r.$">|; 

$key = q/dS;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="$ dS$">|; 

$key = q/partialrho({bfr},t)slashpartialt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="$ \partial \rho({\bf r}, t)/\partial t$">|; 

$key = q/displaystylef_theta(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3798.png"
 ALT="$\displaystyle f_\theta(\theta)$">|; 

$key = q/{mitOmega}-2,omega_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4555.png"
 ALT="$ {\mit\Omega}-2 \omega_M$">|; 

$key = q/a,{rme}^{-,k,h};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3946.png"
 ALT="$ a {\rm e}^{- k h}$">|; 

$key = q/2pi,psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2533.png"
 ALT="$ 2\pi \psi$">|; 

$key = q/displaystyle=L_0,alpha,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3513.png"
 ALT="$\displaystyle = L_0 \alpha,$">|; 

$key = q/displaystylefrac{V}{v_y}-frac{v_y}{V}=V,frac{partialy}{partialskew{3}bar{phi}}-flambda^{,2}-1}=-frac{2,sinalpha,sinomega}{cosalpha-cosomega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="482" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2334.png"
 ALT="$\displaystyle \frac{V}{v_y}-\frac{v_y}{V}=V \frac{\partial y}{\partial\skew{3}...
...sqrt{\lambda^{ 2}-1}=-\frac{2 \sin\alpha \sin\omega}{\cos\alpha-\cos\omega}.$">|; 

$key = q/displaystylerho,(1-s),g+gamma,k^{,2}=rho,k,c^{,2}+s,rho,k,(V'-c)^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4188.png"
 ALT="$\displaystyle \rho (1-s) g + \gamma k^{ 2} = \rho k c^{ 2}+s \rho k (V'-c)^2.$">|; 

$key = q/h_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1604.png"
 ALT="$ h_c$">|; 

$key = q/dot{Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$ \dot{Q}$">|; 

$key = q/displaystylez=-frac{b}{2},cosphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3545.png"
 ALT="$\displaystyle z = -\frac{b}{2} \cos\phi.
$">|; 

$key = q/P={mitPhi}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4767.png"
 ALT="$ P={\mit\Phi}'$">|; 

$key = q/displaystyle{mitOmega}(r_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3630.png"
 ALT="$\displaystyle {\mit\Omega}(r_2)$">|; 

$key = q/13.661;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4618.png"
 ALT="$ 13.661$">|; 

$key = q/displaystyleE=K+U=frac{1}{10},M,(a_1^{,2}+a_2^{,2}),omega^{,2}-frac{3}{10},G,M^{,2},alpha_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="401" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img870.png"
 ALT="$\displaystyle E = K+U = \frac{1}{10} M (a_1^{ 2}+a_2^{ 2}) \omega^{ 2}-\frac{3}{10} G M^{ 2} \alpha_0.$">|; 

$key = q/displaystylefrac{D{mbox{boldmathomega}}}{Dt}=(mbox{boldmathomega}cdotnabla),{bfv}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1568.png"
 ALT="$\displaystyle \frac{D{\mbox{\boldmath$\omega$}}}{Dt} = (\mbox{\boldmath$\omega$}\cdot\nabla) {\bf v}.$">|; 

$key = q/displaystylephi(xi_0,eta)=-V,aleft{frac{1+(1slash2,e)lnleft[(1-e)slash(1+e)right)^{,-1}+(1slash2,e)lnleft[(1-e)slash(1+e)right]}right}coseta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="531" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2808.png"
 ALT="$\displaystyle \phi(\xi_0,\eta)= -V a\left\{\frac{1 +(1/2 e)\ln\left[(1-e)/(1+...
...ght]} {(1-e^{ 2})^{ -1}+(1/2 e)\ln\left[(1-e)/(1+e)\right]}\right\}\cos\eta.$">|; 

$key = q/displaystylefrac{1}{2},rho,omega,a^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4205.png"
 ALT="$\displaystyle \frac{1}{2} \rho \omega a^{ 2}.
$">|; 

$key = q/dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6329.png"
 ALT="$ dt$">|; 

$key = q/{bfA}(u_1,u_2,u_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7068.png"
 ALT="$ {\bf A}(u_1,u_2,u_3)$">|; 

$key = q/{rmFr}ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="$ {\rm Fr}\ll 1$">|; 

$key = q/displaystylesigma_{ij}=-p,delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="$\displaystyle \sigma_{ij} = - p \delta_{ij},$">|; 

$key = q/{rmMa}gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="$ {\rm Ma}\gg 1$">|; 

$key = q/g=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5991.png"
 ALT="$ g=0$">|; 

$key = q/displaystyle=-ointp(a,theta,varphi),a^{,2},sintheta,sinvarphi,d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2655.png"
 ALT="$\displaystyle = -\oint p(a,\theta,\varphi) a^{ 2} \sin\theta \sin\varphi d{\mit\Omega},$">|; 

$key = q/displaystyleepsilon_{ijk}'={calR}_{il},{calR}_{jm},{calR}_{ln},epsilon_{lmn}=epsilon_{ijk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6868.png"
 ALT="$\displaystyle \epsilon_{ijk}' = {\cal R}_{il} {\cal R}_{jm} {\cal R}_{ln} \epsilon_{lmn} = \epsilon_{ijk},$">|; 

$key = q/displaystyleint_Vr^{,m},dV=frac{1}{m+3}int_Sr^{,m+1},{bfe}_rcdotd{bfS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7207.png"
 ALT="$\displaystyle \int_V r^{ m} dV = \frac{1}{m+3}\int_S r^{ m+1} {\bf e}_r\cdot d{\bf S},
$">|; 

$key = q/displaystylefrac{h_1}{h_2}=frac{1}{2}left(-1+sqrt{1+8,{rmFr}_2^{,2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1489.png"
 ALT="$\displaystyle \frac{h_1}{h_2}=\frac{1}{2}\left(-1+\sqrt{1+8 {\rm Fr}_2^{ 2}}\right).$">|; 

$key = q/displaystylefrac{d}{dx}!left(frac{U,delta_2^{,2}}{nu}right)=0.47-5,frac{dU}{dx},frac{delta_2^{,2}}{nu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3206.png"
 ALT="$\displaystyle \frac{d}{dx}\!\left(\frac{U \delta_2^{ 2}}{\nu}\right) = 0.47 - 5 \frac{dU}{dx} \frac{\delta_2^{ 2}}{\nu},$">|; 

$key = q/zeta=a,{rme}^{,{rmi},theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3312.png"
 ALT="$ \zeta=a {\rm e}^{ {\rm i} \theta}$">|; 

$key = q/u,partialuslashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5351.png"
 ALT="$ u \partial u/\partial x$">|; 

$key = q/displaystylefrac{rho,Q^{,2}}{2pi,b},frac{a^{,2}}{(b^{,2}-a^{,2})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2444.png"
 ALT="$\displaystyle \frac{\rho Q^{ 2}}{2\pi b} \frac{a^{ 2}}{(b^{ 2}-a^{ 2})}
$">|; 

$key = q/displaystyledeltal=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7301.png"
 ALT="$\displaystyle \delta l = 0.$">|; 

$key = q/displaystylefrac{d^{,2}X}{dt^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1846.png"
 ALT="$\displaystyle \frac{d^{ 2} X}{dt^{ 2}}$">|; 

$key = q/displaystylefrac{x}{l}=left(k-frac{2}{k}right)tilde{F}(phi,k)+frac{2}{k},tilde{E}(phi,k),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1175.png"
 ALT="$\displaystyle \frac{x}{l} = \left(k-\frac{2}{k}\right)\tilde{F}(\phi,k) + \frac{2}{k} \tilde{E}(\phi,k),$">|; 

$key = q/p=1slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3251.png"
 ALT="$ p=1/3$">|; 

$key = q/cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2144.png"
 ALT="$ \cdots$">|; 

$key = q/displaystylerho,frac{D{calE}}{Dt}-frac{p}{rho},frac{Drho}{Dt}=chi-nablacdot{bfq},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img517.png"
 ALT="$\displaystyle \rho \frac{D{\cal E}}{Dt} - \frac{p}{\rho} \frac{D\rho}{Dt} = \chi - \nabla\cdot{\bf q},
$">|; 

$key = q/displaystylecdotd{bfS}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1546.png"
 ALT="$\displaystyle \cdot d{\bf S} =0.$">|; 

$key = q/2,H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5257.png"
 ALT="$ 2 H$">|; 

$key = q/omega_varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$ \omega _\varphi $">|; 

$key = q/displaystyle=-U_0,x^{,m},f',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2929.png"
 ALT="$\displaystyle = - U_0 x^{ m} f',$">|; 

$key = q/-{bfV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3516.png"
 ALT="$ -{\bf V}$">|; 

$key = q/{rmMa}_pgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5548.png"
 ALT="$ {\rm Ma}_p\gg 1$">|; 

$key = q/displaystyle=-C_{p0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6061.png"
 ALT="$\displaystyle =-C_{p0},$">|; 

$key = q/p=p(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1720.png"
 ALT="$ p=p(r)$">|; 

$key = q/xrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3928.png"
 ALT="$ x\rightarrow \infty$">|; 

$key = q/displaystylesimeq3.2times10^{-1},V_0({rmm,s^{-1}})slash[L({rmm})]^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="$\displaystyle \simeq 3.2\times 10^{-1} V_0({\rm m s^{-1}})/[L({\rm m})]^{1/2},$">|; 

$key = q/displaystyle=frac{partialP'}{partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4748.png"
 ALT="$\displaystyle =\frac{\partial P'}{\partial\theta},$">|; 

$key = q/theta_2ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6178.png"
 ALT="$ \theta_2\ll 1$">|; 

$key = q/a,(1-hslashd);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3962.png"
 ALT="$ a (1-h/d)$">|; 

$key = q/r';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4253.png"
 ALT="$ r'$">|; 

$key = q/displaystyle_S={bfn}times{bfU},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3412.png"
 ALT="$\displaystyle _S = {\bf n}\times {\bf U},$">|; 

$key = q/M_0=rho_0,V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img587.png"
 ALT="$ M_0 = \rho_0 V_0$">|; 

$key = q/displaystyle{mitPsi}_r''(theta,phi_pm);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4845.png"
 ALT="$\displaystyle {\mit\Psi}_r''(\theta,\phi_\pm)$">|; 

$key = q/displaystyle=1-(1-eta)^{,3},(1+eta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3168.png"
 ALT="$\displaystyle =1 - (1-\eta)^{ 3} (1+\eta),$">|; 

$key = q/displaystylesim{mitDelta}theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5749.png"
 ALT="$\displaystyle \sim {\mit\Delta}\theta,$">|; 

$key = q/displaystyle{bfF}=-rho,{mitGamma},{bfV}times{bfe}_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2432.png"
 ALT="$\displaystyle {\bf F} = -\rho {\mit\Gamma} {\bf V}\times{\bf e}_z.
$">|; 

$key = q/{calC}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img548.png"
 ALT="$ {\cal C}(t)$">|; 

$key = q/displaystyle=omega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1925.png"
 ALT="$\displaystyle =\omega,$">|; 

$key = q/displaystylezeta_c(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5023.png"
 ALT="$\displaystyle \zeta_c(\theta,\phi)$">|; 

$key = q/displaystyleint!int_S{bfv}cdotd{bfS}=lim_{dSrightarrow0}left[sumv,costheta,dSright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6500.png"
 ALT="$\displaystyle \int\!\int_S {\bf v}\cdot d{\bf S} = \lim_{dS\rightarrow 0}\left[ \sum v \cos\theta  dS\right],$">|; 

$key = q/displaystyle=p_0+g,rho,cosalpha,(h-y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3873.png"
 ALT="$\displaystyle = p_0 + g \rho \cos\alpha (h-y),$">|; 

$key = q/displaystyle=left[1-frac{1}{2},(gamma-1),frac{V_p}{c_1}right]^{,2,gammaslash(gamma-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5562.png"
 ALT="$\displaystyle = \left[1-\frac{1}{2} (\gamma-1) \frac{V_p}{c_1}\right]^{ 2 \gamma/(\gamma-1)}.$">|; 

$key = q/displaystyleleft.sigma_{xy}right|_{y=0}=rho,nuleft.frac{partialv_x}{partialy}rig-rho,nuleft.frac{partial^{,2}psi}{partialy^{,2}}right|_{y=0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2912.png"
 ALT="$\displaystyle \left.\sigma_{xy}\right\vert _{y=0} = \rho \nu\left.\frac{\parti...
... -\rho \nu\left.\frac{\partial^{ 2}\psi}{\partial y^{ 2}}\right\vert _{y=0}.$">|; 

$key = q/zeta=f(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2053.png"
 ALT="$ \zeta=f(z)$">|; 

$key = q/left.sigma_{xy}right|_{y=h}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3594.png"
 ALT="$ \left.\sigma_{xy}\right\vert _{y=h}=0$">|; 

$key = q/v_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4028.png"
 ALT="$ v_g$">|; 

$key = q/z=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img795.png"
 ALT="$ z=1$">|; 

$key = q/displaystyle{mitDelta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="$\displaystyle {\mit\Delta}$">|; 

$key = q/displaystylepsi(r,theta)=-frac{Q}{2pi},tan^{-1}left[frac{(rslasha-aslashr),sintheta}{(rslasha+aslashr),costheta-(bslasha+aslashb)}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2047.png"
 ALT="$\displaystyle \psi(r,\theta) = -\frac{Q}{2\pi} \tan^{-1}\left[\frac{(r/a-a/r) \sin\theta}{(r/a+a/r) \cos\theta-(b/a+a/b)}\right].$">|; 

$key = q/mu=cot^{-1}[({rmMa}_infty^{,2}-1)^{,1slash2}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5990.png"
 ALT="$ \mu=\cot^{-1}[({\rm Ma}_\infty^{ 2}-1)^{ 1/2}]$">|; 

$key = q/tslashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6214.png"
 ALT="$ t/c$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter05slashstream1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="279" BORDER="0"
 SRC="|."$dir".q|img1628.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter05/stream1.eps}}
\end{figure}">|; 

$key = q/c_f(x)proptox^{,-(m+1)slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2950.png"
 ALT="$ c_f(x)\propto x^{ -(m+1)/2}$">|; 

$key = q/displaystyle=-frac{costheta}{sintheta},frac{partial{mitPsi}_r(theta,phi_pm)}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4858.png"
 ALT="$\displaystyle =- \frac{\cos\theta}{\sin\theta} \frac{\partial {\mit\Psi}_r(\theta,\phi_\pm)}{\partial\phi},$">|; 

$key = q/{bfB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img584.png"
 ALT="$ {\bf B}$">|; 

$key = q/alpha>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1313.png"
 ALT="$ \alpha&gt;0$">|; 

$key = q/displaystylefrac{v_y}{v_x}sim{calO}left(frac{d}{l}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3653.png"
 ALT="$\displaystyle \frac{v_y}{v_x}\sim {\cal O}\left(\frac{d}{l}\right).$">|; 

$key = q/z=a,(0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2103.png"
 ALT="$ z=a (0$">|; 

$key = q/displaystyleK=-pi,rholeft.int_{0}^piphi,frac{partialpsi}{partialeta},detaright|_{xi=xi_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2721.png"
 ALT="$\displaystyle K= -\pi \rho\left.\int_{0}^\pi \phi \frac{\partial\psi}{\partial\eta} d\eta\right\vert _{\xi=\xi_0},$">|; 

$key = q/zeta_2=zeta_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4577.png"
 ALT="$ \zeta_2=\zeta_M$">|; 

$key = q/skew{3}bar{C}_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5313.png"
 ALT="$ \skew{3}\bar{C}_p$">|; 

$key = q/displaystyleU,frac{d}{dx}!left(frac{lambda}{dUslashdx}right)=2left[F_2(lambda)-lambdaleft{2+F_1(lambda)right}right]=F(lambda),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3188.png"
 ALT="$\displaystyle U \frac{d}{dx}\!\left(\frac{\lambda}{dU/dx}\right) = 2\left[F_2(\lambda)-\lambda\left\{2+F_1(\lambda)\right\}\right] = F(\lambda),$">|; 

$key = q/{bfv}=-nablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1589.png"
 ALT="$ {\bf v}=-\nabla\phi$">|; 

$key = q/y_pm=(deltaslash2),sintheta_{pm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3368.png"
 ALT="$ y_\pm = (\delta/2) \sin\theta_{\pm}$">|; 

$key = q/displaystyle=-frac{1}{a,sintheta},frac{partial}{partialphi}left[g,(zeta_b+zeta)+delta{mitPhi}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4493.png"
 ALT="$\displaystyle = - \frac{1}{a \sin\theta} \frac{\partial}{\partial\phi}\left[g (\zeta_b+\zeta)+ \delta{\mit\Phi}\right].$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashairf.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="493" HEIGHT="284" BORDER="0"
 SRC="|."$dir".q|img5781.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/airf.eps}}
\end{figure}">|; 

$key = q/nabla^{,2}=partial^{,2}slashpartialx^{,2}+partial^{,2}slashpartialy^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img541.png"
 ALT="$ \nabla^{ 2}= \partial^{ 2}/\partial x^{ 2}+ \partial^{ 2}/\partial y^{ 2}$">|; 

$key = q/P={mitPhi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4757.png"
 ALT="$ P={\mit\Phi}$">|; 

$key = q/displaystyle=-rho,{mitDelta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="$\displaystyle =-\rho {\mit\Delta},$">|; 

$key = q/F(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6466.png"
 ALT="$ F(x,y)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter12slashampmf.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5044.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter12/ampmf.eps}}
\end{figure}">|; 

$key = q/displaystylerho,g,zeta-left.p_1right|_{z=0}=gamma,frac{partial^{,2}zeta}{partialx^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4143.png"
 ALT="$\displaystyle \rho g \zeta - \left.p_1\right\vert _{z=0} = \gamma \frac{\partial^{ 2}\zeta}{\partial x^{ 2}},$">|; 

$key = q/displaystylefrac{v_p}{v_c}=left[frac{1}{2}left(k,l+frac{1}{k,l}right)right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4153.png"
 ALT="$\displaystyle \frac{v_p}{v_c} = \left[\frac{1}{2}\left(k l+\frac{1}{k l}\right)\right]^{1/2}.$">|; 

$key = q/dllH;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1452.png"
 ALT="$ d\ll H$">|; 

$key = q/displaystylealpha_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1004.png"
 ALT="$\displaystyle \alpha_3$">|; 

$key = q/6.0221times10^{24};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="$ 6.0221\times 10^{24}$">|; 

$key = q/displaystyleointG_F(theta,phi;theta',phi'),zeta_j(theta',phi',t),d{mitOmega}'=sulpha_n,z_n,P_n^{,m_j}(mu),{rme}^{,{rmi},(m_j,phi+sigma_j,t)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="641" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4683.png"
 ALT="$\displaystyle \oint G_F(\theta,\phi;\theta',\phi') \zeta_j(\theta',\phi',t) d...
...,\alpha_n z_n P_n^{ m_j}(\mu) {\rm e}^{ {\rm i} (m_j \phi+\sigma_j t)}.$">|; 

$key = q/displaystylev_x,frac{partialv_x}{partialx}+v_y,frac{partialv_x}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2861.png"
 ALT="$\displaystyle v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y}$">|; 

$key = q/displaystyledelta(x)=2left(frac{nu,x}{U_0}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3027.png"
 ALT="$\displaystyle \delta(x) =2\left(\frac{\nu x}{U_0}\right)^{1/2},$">|; 

$key = q/c(-z)=c(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3397.png"
 ALT="$ c(-z)=c(z)$">|; 

$key = q/displaystylephantom{=}+frac{mu}{rho}left(nabla^{,2}v_r-frac{v_r}{r^{,2}}-frac{2}ialtheta}+frac{1}{3},frac{partial{mitDelta}}{partialr}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="$\displaystyle \phantom{=}+ \frac{\mu}{\rho}\left(\nabla^{ 2} v_r -\frac{v_r}{r...
...}{\partial\theta}+ \frac{1}{3} \frac{\partial{\mit\Delta}}{\partial r}\right),$">|; 

$key = q/displaystyleX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2388.png"
 ALT="$\displaystyle X$">|; 

$key = q/xi_{i,n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4423.png"
 ALT="$ \xi_{i n}$">|; 

$key = q/x=-d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1196.png"
 ALT="$ x=-d$">|; 

$key = q/H_1=H_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1501.png"
 ALT="$ H_1=H_2$">|; 

$key = q/displaystyletheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2138.png"
 ALT="$\displaystyle \theta$">|; 

$key = q/dS=dy,dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6634.png"
 ALT="$ dS=dy dz$">|; 

$key = q/V_0'=A_0,(delta_0+delta_1)=V_0+A_0,delta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img613.png"
 ALT="$ V_0'=A_0 (\delta_0+\delta_1)= V_0 + A_0 \delta_1$">|; 

$key = q/displaystyle{mitOmega}=lnleft(frac{dz}{dbar{F}}right)=lnleft(frac{1}{v_x-{rmi},v_y}right)=lnleft(frac{1}{v}right)+{rmi},theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2179.png"
 ALT="$\displaystyle {\mit\Omega} =\ln\left(\frac{dz}{d\bar{F}}\right)= \ln\left(\frac{1}{v_x-{\rm i} v_y}\right)=\ln \left(\frac{1}{v}\right)+ {\rm i} \theta,$">|; 

$key = q/displaystyleA_{ijkl}=alpha,delta_{ij},delta_{kl}+beta,delta_{ik},delta_{jl}+gamma,delta_{il},delta_{jk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="$\displaystyle A_{ijkl}= \alpha \delta_{ij} \delta_{kl} + \beta \delta_{ik} \delta_{jl}+\gamma \delta_{il} \delta_{jk},$">|; 

$key = q/D>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$ D&gt;0$">|; 

$key = q/p_0+rho,g,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3898.png"
 ALT="$ p_0+\rho g h$">|; 

$key = q/displaystyleomega_varphi(r,theta)=frac{1}{r},frac{partial,(r,v_theta)}{partialr}frac{partialv_r}{partialtheta}=frac{{calL}(psi)}{r,sintheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3735.png"
 ALT="$\displaystyle \omega_\varphi(r,\theta) = \frac{1}{r} \frac{\partial (r v_\th...
...r} \frac{\partial v_r}{\partial\theta} = \frac{{\cal L}(\psi)}{r \sin\theta},$">|; 

$key = q/(1slash2),rho,g,a^{,2},w,V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3985.png"
 ALT="$ (1/2) \rho g a^{ 2} w V$">|; 

$key = q/nablacdot{bfA}neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6592.png"
 ALT="$ \nabla\cdot{\bf A}\neq 0 $">|; 

$key = q/displaystylefrac{Dv_r}{Dt}-frac{v_theta^{,2}}{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="$\displaystyle \frac{Dv_r}{Dt}-\frac{v_\theta^{ 2}}{r}$">|; 

$key = q/displaystyle=rhoint_S{bfv}_Stimes;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3443.png"
 ALT="$\displaystyle =\rho\int_S {\bf v}_S\times$">|; 

$key = q/theta({bfr},t),{bfv}({bfr},t)cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="$ \theta({\bf r},t) {\bf v}({\bf r}, t)\cdot d{\bf S}$">|; 

$key = q/displaystyle=U_0-u(x,y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3010.png"
 ALT="$\displaystyle = U_0 - u(x,y),$">|; 

$key = q/displaystyleu(x,y)=frac{Q}{sqrt{pi},delta},expleft(-frac{y^{,2}}{delta^{,2}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3026.png"
 ALT="$\displaystyle u(x,y) = \frac{Q}{\sqrt{\pi} \delta} \exp\left(-\frac{y^{ 2}}{\delta^{ 2}}\right),$">|; 

$key = q/displaystyleQ=Uleft(frac{d_1,d_2}{d_1+d_2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3704.png"
 ALT="$\displaystyle Q = U\left(\frac{d_1 d_2}{d_1+d_2}\right),$">|; 

$key = q/omegall(G,Mslasha_0^{,3})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1072.png"
 ALT="$ \omega\ll (G M/a_0^{ 3})^{1/2}$">|; 

$key = q/displaystyle=-left(1+frac{1}{{rmFr}^{,2}}+frac{1}{{rmRe}}right)overline{nablap}-}{{rmFr}^{,2}}+frac{overline{nabla}^{,2}overline{bfv}}{rmRe},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="$\displaystyle = - \left(1+\frac{1}{{\rm Fr}^{ 2}}+\frac{1}{{\rm Re}}\right)\ov...
...i}}}{{\rm Fr}^{ 2}} + \frac{\overline{\nabla}^{ 2} \overline{\bf v}}{\rm Re},$">|; 

$key = q/displaystylefrac{p^{,(gamma-1)slash2,gamma}-p_2^{,(gamma-1)slash2,gamma}}{p_1^{,sh2,gamma}-p_2^{,(gamma-1)slash2,gamma}}=frac{x-x_2}{x_1-x_2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5572.png"
 ALT="$\displaystyle \frac{p^{ (\gamma-1)/2 \gamma}-p_2^{ (\gamma-1)/2 \gamma}}{p_1^{ (\gamma-1)/2 \gamma}-p_2^{ (\gamma-1)/2 \gamma}}=\frac{x-x_2}{x_1-x_2}$">|; 

$key = q/deltatheta_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6824.png"
 ALT="$ \delta\theta_3$">|; 

$key = q/displaystyle=frac{1}{2}left(frac{1}{r},frac{partialA_r}{partialtheta}+frac{partialA_theta}{partialr}-frac{A_theta}{r}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7154.png"
 ALT="$\displaystyle = \frac{1}{2}\left(\frac{1}{r} \frac{\partial A_r}{\partial\theta} + \frac{\partial A_\theta}{\partial r} - \frac{A_\theta}{r}\right),$">|; 

$key = q/displaystyle=intdV=frac{4}{3},pi,a_1,a_2,a_3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img847.png"
 ALT="$\displaystyle = \int dV = \frac{4}{3} \pi a_1 a_2 a_3,$">|; 

$key = q/+1.93times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4610.png"
 ALT="$ +1.93\times 10^{-2}$">|; 

$key = q/displaystylefrac{rho}{rho_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5405.png"
 ALT="$\displaystyle \frac{\rho}{\rho_1}$">|; 

$key = q/G(u_0)<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1275.png"
 ALT="$ G(u_0)&lt;0$">|; 

$key = q/{mitDelta}J_r={mitDelta}J_i=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2372.png"
 ALT="$ {\mit\Delta} J_r={\mit\Delta} J_i=0$">|; 

$key = q/displaystyle-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6806.png"
 ALT="$\displaystyle -1$">|; 

$key = q/displaystylezeta_n(theta,phi)=sum_{m=0,n}c_{n}^{,m},P_n^{,m}(costheta),{rme}^{,{rmi},m,phi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4512.png"
 ALT="$\displaystyle \zeta_n(\theta,\phi) = \sum_{m=0,n}c_{n}^{ m} P_n^{ m}(\cos\theta) {\rm e}^{ {\rm i} m \phi},$">|; 

$key = q/displaystyle=nabla^{,2}A_phi-frac{A_phi}{r^{,2},sin^2theta}+frac{2}{r^{,2},sin^22cottheta}{r^{,2},sintheta},frac{partialA_theta}{partialphi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="434" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7198.png"
 ALT="$\displaystyle =\nabla^{ 2} A_\phi-\frac{A_\phi}{r^{ 2} \sin^2\theta} + \frac...
...rac{2 \cot\theta}{r^{ 2} \sin\theta} \frac{\partial A_\theta}{\partial\phi}.$">|; 

$key = q/t<2,s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img768.png"
 ALT="$ t&lt;2 s$">|; 

$key = q/lambda=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2256.png"
 ALT="$ \lambda=-1$">|; 

$key = q/{rmRe}=(aslash6times10^{-5})^{,3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3818.png"
 ALT="$ {\rm Re} = (a/6\times 10^{-5})^{ 3}$">|; 

$key = q/nablacdot(nablaphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6603.png"
 ALT="$ \nabla\cdot(\nabla\phi)$">|; 

$key = q/S_{ji};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7145.png"
 ALT="$ S_{ji}$">|; 

$key = q/V=a,b,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img663.png"
 ALT="$ V=a b c$">|; 

$key = q/displaystyle=frac{partialpsi}{partialx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1636.png"
 ALT="$\displaystyle =\frac{\partial\psi}{\partial x}.$">|; 

$key = q/displaystyle=V_x(t),frac{a^{,2}}{r^{,2}},costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1805.png"
 ALT="$\displaystyle = V_x(t) \frac{a^{ 2}}{r^{ 2}} \cos\theta,$">|; 

$key = q/displaystyle=frac{1}{2}left(3,mu^{,2}-1right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2589.png"
 ALT="$\displaystyle = \frac{1}{2}\left(3 \mu^{ 2}-1\right),$">|; 

$key = q/displaystylembox{boldmath{{tau{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6385.png"
 ALT="$\displaystyle \mbox{\boldmath$\tau$}$">|; 

$key = q/r'+r''=-BslashA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7246.png"
 ALT="$ r'+r''=-B/A$">|; 

$key = q/h(x_L)=h(x_R)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5826.png"
 ALT="$ h(x_L)=h(x_R)=0$">|; 

$key = q/displaystyle=-frac{partialphi}{partialy}=U,epsilon,alpha,cos(alpha,x),{rme}^{-m,alpha,y}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5977.png"
 ALT="$\displaystyle =-\frac{\partial\phi}{\partial y} = U \epsilon \alpha \cos(\alpha x) {\rm e}^{-m \alpha y}.$">|; 

$key = q/bar{~};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2398.png"
 ALT="$ \bar{&nbsp;}$">|; 

$key = q/{mitGamma}_C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1794.png"
 ALT="$ {\mit\Gamma}_C$">|; 

$key = q/a,bsintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6380.png"
 ALT="$ a b \sin\theta$">|; 

$key = q/d_n^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4318.png"
 ALT="$ d_n^{ m}$">|; 

$key = q/displaystylenabla^{,2}equivfrac{partial^{,2}}{partialx^{,2}}+frac{partial^{,2}}{partialy^{,2}}+frac{partial^{,2}}{partialz^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6615.png"
 ALT="$\displaystyle \nabla^{ 2} \equiv \frac{\partial^{ 2}}{\partial x^{ 2}}+ \frac{\partial^{ 2}}{\partial y^{ 2}}+ \frac{\partial^{ 2}}{\partial z^{ 2}}$">|; 

$key = q/displaystyleV_y(X,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2903.png"
 ALT="$\displaystyle V_y(X,0)$">|; 

$key = q/displaystyleC_p=frac{2}{U},f'(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6056.png"
 ALT="$\displaystyle C_p = \frac{2}{U} f'(x)$">|; 

$key = q/displaystyle{bfv}=v_z(r),{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3609.png"
 ALT="$\displaystyle {\bf v} = v_z(r) {\bf e}_z$">|; 

$key = q/u_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7029.png"
 ALT="$ u_3$">|; 

$key = q/displaystylenabla^{,2}psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1924.png"
 ALT="$\displaystyle \nabla^{ 2}\psi$">|; 

$key = q/displaystylephantom{-frac{G,m',a^{2}}{r'^{,3}}}left.+frac{1}{12},P_2^{,2},(costheta'),sin(2,varphi'),{calR}_2^{,(+2)}(r,theta,varphi)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="415" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4284.png"
 ALT="$\displaystyle \phantom{ - \frac{G m' a^{2}}{r'^{ 3}}}\left.+ \frac{1}{12} P...
...(\cos\theta') \sin(2 \varphi') {\cal R}_2^{ (+2)}(r,\theta,\varphi)\right].$">|; 

$key = q/displaystyle=frac{1}{r,sintheta},frac{partial^{,2}psi}{partialr^{,2}}+frac{1}{r^}left(frac{1}{sintheta},frac{partialpsi}{partialtheta}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2569.png"
 ALT="$\displaystyle = \frac{1}{r \sin\theta} \frac{\partial^{ 2}\psi}{\partial r^{...
...\theta}\left( \frac{1}{\sin\theta} \frac{\partial\psi}{\partial\theta}\right).$">|; 

$key = q/displaystyle=zeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4760.png"
 ALT="$\displaystyle =\zeta,$">|; 

$key = q/displaystyled=frac{h}{2},tanalpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1898.png"
 ALT="$\displaystyle d = \frac{h}{2} \tan\alpha.$">|; 

$key = q/displaystyle{mitDelta}J_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2368.png"
 ALT="$\displaystyle {\mit\Delta} J_r$">|; 

$key = q/V_0sim300,{rmm,s^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="$ V_0\sim 300 {\rm m s^{-1}}$">|; 

$key = q/displaystylee_{12}^{,2}simeqe_{13}^{,2}simeqfrac{15}{2},mu.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1026.png"
 ALT="$\displaystyle e_{12}^{ 2}\simeq e_{13}^{ 2}\simeq \frac{15}{2} \mu.$">|; 

$key = q/rho=rho_c,theta^{,n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5272.png"
 ALT="$ \rho=\rho_c \theta^{ n}$">|; 

$key = q/displaystylefrac{partialphi(x,0)}{partialy}=-U,frac{dy}{dx}=-U,epsilon,alpha,cos(alpha,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5955.png"
 ALT="$\displaystyle \frac{\partial\phi(x,0)}{\partial y} = -U \frac{dy}{dx} = -U \epsilon \alpha \cos(\alpha x).$">|; 

$key = q/displaystyleC_p=frac{2}{U},g'(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6057.png"
 ALT="$\displaystyle C_p = \frac{2}{U} g'(x)$">|; 

$key = q/displaystyle|{bfa}times{bfb}|=|{bfa}|,|{bfb}|,sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6377.png"
 ALT="$\displaystyle \vert{\bf a}\times{\bf b}\vert = \vert{\bf a}\vert \vert{\bf b}\vert \sin\theta,$">|; 

$key = q/displaystyle=sum_{k=1,3}nablatimes(A_k,{bfe}_k)=sum_{k=1,3}nablatimes(h_k,A_k,nablau_k)=sum_{k=1,3}nabla(h_k,u_k),timesnablau_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="556" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7074.png"
 ALT="$\displaystyle = \sum_{k=1,3}\nabla\times (A_k {\bf e}_k)=\sum_{k=1,3} \nabla\times (h_k A_k \nabla u_k) =\sum_{k=1,3}\nabla (h_k u_k) \times \nabla u_k$">|; 

$key = q/displaystylerho,({bfv}cdotnabla),{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3644.png"
 ALT="$\displaystyle \rho ({\bf v}\cdot \nabla) {\bf v}$">|; 

$key = q/chi_2({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4383.png"
 ALT="$ \chi_2({\bf r})$">|; 

$key = q/displaystyle=E(pislash2,k)-E(phi,k),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1177.png"
 ALT="$\displaystyle = E(\pi/2,k)- E(\phi,k),$">|; 

$key = q/displaystyleAP';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img740.png"
 ALT="$\displaystyle AP'$">|; 

$key = q/{bfL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3301.png"
 ALT="$ {\bf L}$">|; 

$key = q/skew{3}bar{psi}=+skew{3}bar{psi}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2187.png"
 ALT="$ \skew{3}\bar{\psi}=+\skew{3}\bar{\psi}_1$">|; 

$key = q/displaystyleH=h+d+frac{v^{,2}}{2,g},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1523.png"
 ALT="$\displaystyle H = h+d + \frac{v^{ 2}}{2 g},$">|; 

$key = q/beta<beta_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2973.png"
 ALT="$ \beta&lt;\beta_\ast$">|; 

$key = q/skew{3}bar{phi}(xi,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6008.png"
 ALT="$ \skew{3}\bar{\phi}(\xi,\eta)$">|; 

$key = q/displaystyle=frac{1}{6},eta,(1-eta)^{,3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3170.png"
 ALT="$\displaystyle = \frac{1}{6} \eta (1-\eta)^{ 3}.$">|; 

$key = q/displaystyleR=xi_1,a.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5214.png"
 ALT="$\displaystyle R = \xi_1 a.$">|; 

$key = q/nabla^{,2}{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7096.png"
 ALT="$ \nabla^{ 2}{\bf A}$">|; 

$key = q/displaystylezeta_t(theta,phi)=sum_{n=1,infty}zeta_{t,n}(theta,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4345.png"
 ALT="$\displaystyle \zeta_t(\theta,\phi)=\sum_{n=1,\infty}\zeta_{t n}(\theta,\phi),$">|; 

$key = q/displaystyleu=int_{rho_0}^rhoc,frac{drho}{rho}=frac{2,c_0}{gamma-1}left[left(fra0}right)^{,(gamma-1)slash2}-1right]=frac{2}{gamma-1},(c-c_0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="479" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5369.png"
 ALT="$\displaystyle u=\int_{\rho_0}^\rho c \frac{d\rho}{\rho} =\frac{2 c_0}{\gamma-...
...ac{\rho}{\rho_0}\right)^{ (\gamma-1)/2}-1\right]= \frac{2}{\gamma-1} (c-c_0),$">|; 

$key = q/{rmMa}>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5418.png"
 ALT="$ {\rm Ma}&gt;1$">|; 

$key = q/displaystyle{bfa}times({bfb}times{bfc});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6673.png"
 ALT="$\displaystyle {\bf a}\times ({\bf b}\times {\bf c})$">|; 

$key = q/c=c_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4179.png"
 ALT="$ c=c_0$">|; 

$key = q/cslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img668.png"
 ALT="$ c/2$">|; 

$key = q/displaystyleg(theta,phi_pm);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4802.png"
 ALT="$\displaystyle g(\theta,\phi_\pm)$">|; 

$key = q/displaystyleepsilon_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6182.png"
 ALT="$\displaystyle \epsilon_2$">|; 

$key = q/displaystyle(nabla{bfA})_{ij}=frac{partialA_i}{partialx_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7092.png"
 ALT="$\displaystyle (\nabla {\bf A})_{ij} = \frac{\partial A_i}{\partial x_j}.$">|; 

$key = q/displaystyle=-frac{Q}{2pi},theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1979.png"
 ALT="$\displaystyle =-\frac{Q}{2\pi} \theta.$">|; 

$key = q/-Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2041.png"
 ALT="$ -Q$">|; 

$key = q/displaystylesimeq1+left(frac{gamma+1}{4}right){rmMa}_p,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5543.png"
 ALT="$\displaystyle \simeq 1+\left(\frac{\gamma+1}{4}\right){\rm Ma}_p,$">|; 

$key = q/displaystylefrac{z-{rmi},d}{b}=frac{2}{pi}int_lambda^1frac{dk}{k}left[-{rmi},k+(1-k^{,2})^{1slash2}right]^{,2,alphaslashpi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="365" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2246.png"
 ALT="$\displaystyle \frac{z-{\rm i} d}{b} = \frac{2}{\pi}\int_\lambda^1 \frac{dk}{k}\left[-{\rm i} k + (1-k^{ 2})^{1/2}\right]^{ 2 \alpha/\pi}.$">|; 

$key = q/displaystylefrac{c_0^{,2}}{c^{,2}}=1+left(frac{gamma-1}{2}right){rmMa}^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5769.png"
 ALT="$\displaystyle \frac{c_0^{ 2}}{c^{ 2}} = 1+\left(\frac{\gamma-1}{2}\right){\rm Ma}^{ 2},$">|; 

$key = q/{mitDelta}pslashp_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5730.png"
 ALT="$ {\mit\Delta}p/p_1$">|; 

$key = q/theta_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1749.png"
 ALT="$ \theta_m$">|; 

$key = q/displaystyleK=left[left(frac{k}{m_p}right)^4,frac{3}{alpha},frac{beta}{mu^{,4},(1-beta)^4}right]^{1slash3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5193.png"
 ALT="$\displaystyle K = \left[\left(\frac{k}{m_p}\right)^4 \frac{3}{\alpha} \frac{\beta}{\mu^{ 4} (1-\beta)^4}\right]^{1/3}.$">|; 

$key = q/displaystyle=dr,dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7124.png"
 ALT="$\displaystyle = dr dz,$">|; 

$key = q/displaystyle{mitPhi}_E=oint_Srholeft({calE}+frac{1}{2},v_i,v_iright)v_j,dS_j=intx_j}!left[rholeft({calE}+frac{1}{2},v_i,v_iright)v_jright]dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="529" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="$\displaystyle {\mit\Phi}_E = \oint_S \rho\left({\cal E} + \frac{1}{2} v_i v_i...
...ial x_j}\! \left[\rho\left({\cal E} + \frac{1}{2} v_i v_i\right)v_j\right]dV,$">|; 

$key = q/displaystylefrac{d{mitGamma}_C}{dt}=oint_Cfrac{partial{bfv}}{partialt}cdotd{bfr}ght]cdotd{bfr}=oint_C{bfv}timesmbox{boldmathomega}cdotd{bfr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="594" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1789.png"
 ALT="$\displaystyle \frac{d{\mit\Gamma}_C}{dt} = \oint_C\frac{\partial {\bf v}}{\part...
...t]\cdot d{\bf r}= \oint_C {\bf v}\times \mbox{\boldmath$\omega$}\cdot d{\bf r},$">|; 

$key = q/displaystyle{bfA}=frac{1}{2},rhoint_SU^{,2},{bfn},dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3432.png"
 ALT="$\displaystyle {\bf A} =\frac{1}{2} \rho\int_S U^{ 2} {\bf n} dS,$">|; 

$key = q/displaystylefrac{partialwidehat{P}}{partialY}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2894.png"
 ALT="$\displaystyle \frac{\partial \widehat{P}}{\partial Y} = 0.$">|; 

$key = q/displaystylep(r)=-frac{kappa^{,2},rho}{2,r^{,2}}+p_infty,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1911.png"
 ALT="$\displaystyle p(r) = -\frac{\kappa^{ 2} \rho}{2 r^{ 2}}+ p_\infty,
$">|; 

$key = q/displaystyle=-frac{1}{2},V,a^{,2},sin^2eta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2746.png"
 ALT="$\displaystyle = -\frac{1}{2} V a^{ 2} \sin^2\eta,$">|; 

$key = q/displaystyle=-g,frac{partial(zeta-skew{5}bar{zeta}')}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5066.png"
 ALT="$\displaystyle = -g \frac{\partial(\zeta-\skew{5}\bar{\zeta}')}{\partial y},$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter04slashbore.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="342" BORDER="0"
 SRC="|."$dir".q|img1510.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter04/bore.eps}}
\end{figure}">|; 

$key = q/displaystylesum_{n=1,infty}left{{mitPsi}_nleft[-f_j,n,(n+1)+m_jright]P_n^{,m_j}(-mu^{,2})}{n,(n+1)},frac{d}{dmu}right]P_n^{,m_j}(mu)right}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="636" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4684.png"
 ALT="$\displaystyle \sum_{n=1,\infty}\left\{{\mit\Psi}_n\left[-f_j n (n+1)+m_j\righ...
...ac{(1-\mu^{ 2})}{n (n+1)} \frac{d}{d\mu}\right]P_n^{ m_j}(\mu)\right\} = 0,$">|; 

$key = q/displaystyleu(x,0)=v(x,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3078.png"
 ALT="$\displaystyle u(x,0)=v(x,0)$">|; 

$key = q/displaystylepsi(r,theta)=-frac{1}{2},V,r^{,2},sin^2theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2601.png"
 ALT="$\displaystyle \psi(r,\theta)= -\frac{1}{2} V r^{ 2} \sin^2\theta.$">|; 

$key = q/displaystylepsi(xirightarrowinfty,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2732.png"
 ALT="$\displaystyle \psi(\xi\rightarrow\infty,\eta)$">|; 

$key = q/displaystylea_i'=a_i+epsilon_{ijk},deltatheta_j,a_k.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6820.png"
 ALT="$\displaystyle a_i' = a_i + \epsilon_{ijk}  \delta\theta_j a_k.$">|; 

$key = q/displaystylev_r(r)=frac{Q}{2pi,r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1672.png"
 ALT="$\displaystyle v_r(r) = \frac{Q}{2\pi r}.$">|; 

$key = q/displaystyle{bfv}={bfV}+{bfv}_{mitSigma}+{bfv}_S,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3413.png"
 ALT="$\displaystyle {\bf v} = {\bf V} + {\bf v}_{\mit\Sigma}+ {\bf v}_S,$">|; 

$key = q/displaystylesimeq7.2times10^{-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="$\displaystyle \simeq 7.2\times 10^{-1},$">|; 

$key = q/displaystyle={calR}_{il},{calR}_{jm},{calR}_{kn},c_{lmn},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6854.png"
 ALT="$\displaystyle ={\cal R}_{il} {\cal R}_{jm} {\cal R}_{kn} c_{lmn},$">|; 

$key = q/displaystyleW,aleft(1-frac{2}{3},frac{a^{,2}}{R^{,2}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1054.png"
 ALT="$\displaystyle W a\left(1-\frac{2}{3} \frac{a^{ 2}}{R^{ 2}}\right),
$">|; 

$key = q/displaystylepsi(xi,eta)=frac{Q,sinh(pi,xislasha),sin(pi,etaslasha)}{2pi,[sinh^2(pi,xislasha)+cos^2(pi,etaslasha)]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2497.png"
 ALT="$\displaystyle \psi(\xi,\eta) = \frac{Q \sinh(\pi \xi/a) \sin(\pi \eta/a)}{2\pi [\sinh^2(\pi \xi/a)+\cos^2(\pi \eta/a)]}.
$">|; 

$key = q/r'=|PQ|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7233.png"
 ALT="$ r'=\vert PQ\vert$">|; 

$key = q/displaystyle=z'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6569.png"
 ALT="$\displaystyle = z'.$">|; 

$key = q/{mitLambda}=12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$ {\mit \Lambda }=12$">|; 

$key = q/u_3=z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7107.png"
 ALT="$ u_3=z$">|; 

$key = q/displaystylembox{boldmath{{eta{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1581.png"
 ALT="$\displaystyle \mbox{\boldmath$\eta$}$">|; 

$key = q/b_{ijkl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6768.png"
 ALT="$ b_{ijkl}$">|; 

$key = q/yslasha=-zeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2002.png"
 ALT="$ y/a=-\zeta$">|; 

$key = q/dfslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6518.png"
 ALT="$ df/dx$">|; 

$key = q/displaystylep_0=p_infty-rholeft(frac{mitGamma}{2pi,a}right)^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1725.png"
 ALT="$\displaystyle p_0= p_\infty-\rho\left(\frac{\mit\Gamma}{2\pi a}\right)^2.$">|; 

$key = q/U=Qslash(pi,a^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3620.png"
 ALT="$ U=Q/(\pi a^{ 2})$">|; 

$key = q/omega_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$ \omega _\phi $">|; 

$key = q/r_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2620.png"
 ALT="$ r_2$">|; 

$key = q/displaystyle=lambda_rint_{mitOmega}{mitPhi}',{mitPhi}_r,d{mitOmega}=-int_{mitOmega}{mitPhi}',D,{mitPhi}_r,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4897.png"
 ALT="$\displaystyle = \lambda_r\int_{\mit\Omega}{\mit\Phi}' {\mit\Phi}_r d{\mit\Omega} = - \int_{\mit\Omega}{\mit\Phi}' D {\mit\Phi}_r d{\mit\Omega}$">|; 

$key = q/_z({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="$ _z({\bf r},t)$">|; 

$key = q/displaystyle(1-{rmMa}_infty^{,2}),frac{partialu'}{partialx}gg{rmMa}_infty^{,2},(gamma+1),frac{u'}{U},frac{partialu'}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5981.png"
 ALT="$\displaystyle (1-{\rm Ma}_\infty^{ 2}) \frac{\partial u'}{\partial x} \gg {\rm Ma}_\infty^{ 2} (\gamma+1) \frac{u'}{U} \frac{\partial u'}{\partial x},$">|; 

$key = q/displaystylenablacdot{bfA}=sum_{i=1,3}frac{1}{h},frac{partial}{partialu_i}!left(frac{h}{h_i},A_iright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7072.png"
 ALT="$\displaystyle \nabla\cdot{\bf A} =\sum_{i=1,3} \frac{1}{h} \frac{\partial}{\partial u_i}\!\left(\frac{h}{h_i} A_i\right).$">|; 

$key = q/U_0,x^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2922.png"
 ALT="$ U_0 x^{ m}$">|; 

$key = q/displaystylefrac{1}{2},v^{,2}+g,y={calE},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3561.png"
 ALT="$\displaystyle \frac{1}{2} v^{ 2} + g y = {\cal E},
$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter02slashfig3.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="313" BORDER="0"
 SRC="|."$dir".q|img619.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter02/fig3.01.eps}}
\end{figure}">|; 

$key = q/(xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2056.png"
 ALT="$ (\xi$">|; 

$key = q/h_xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7225.png"
 ALT="$ h_\xi$">|; 

$key = q/V^{,4slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="$ V^{ 4/3}$">|; 

$key = q/{rmMa}_2<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$ {\rm Ma}_2&lt;1$">|; 

$key = q/stackrel{displaystylerightarrow}{PQ};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6221.png"
 ALT="$ \stackrel{\displaystyle \rightarrow}{PQ}$">|; 

$key = q/displaystyle=mu,delta_{jl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6952.png"
 ALT="$\displaystyle =\mu \delta_{jl},$">|; 

$key = q/nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="$ \nu$">|; 

$key = q/+(1slash2),zeta_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4565.png"
 ALT="$ +(1/2) \zeta_M$">|; 

$key = q/displaystyle=-frac{cos^2theta}{{rmMa}_1^{,4}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6093.png"
 ALT="$\displaystyle = -\frac{\cos^2\theta}{{\rm Ma}_1^{ 4}}.$">|; 

$key = q/nabla^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4315.png"
 ALT="$ \nabla^{ 2}$">|; 

$key = q/displaystylenablacdot(phi,{bfA});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6688.png"
 ALT="$\displaystyle \nabla\cdot (\phi {\bf A})$">|; 

$key = q/y';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2323.png"
 ALT="$ y'$">|; 

$key = q/displaystyle=phi,nablapsi+psi,nablaphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6682.png"
 ALT="$\displaystyle =\phi \nabla\psi+\psi \nabla\phi,$">|; 

$key = q/displaystyle-rho,nabla({mitPsi}'+{mitPsi}'')=rho,lambda,(3,x_1,,0,,-x_3).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img989.png"
 ALT="$\displaystyle -\rho \nabla({\mit\Psi}'+{\mit\Psi}'') = \rho \lambda (3 x_1, 0, -x_3).$">|; 

$key = q/displaystyle=left(frac{partial{calG}}{partialp}right)_{T},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5592.png"
 ALT="$\displaystyle =\left(\frac{\partial {\cal G}}{\partial p}\right)_{T},$">|; 

$key = q/F(phi,k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1310.png"
 ALT="$ F(\phi,k)$">|; 

$key = q/V,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3746.png"
 ALT="$ V {\bf e}_z$">|; 

$key = q/displaystyle=frac{2}{a_1^{,2},sin^3gamma}left[frac{E(gamma,alpha)}{sin^2alpha,coin^2alpha}-frac{cosgamma,singamma}{cos^2alpha,cosbeta}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1003.png"
 ALT="$\displaystyle =\frac{2}{a_1^{ 2} \sin^3\gamma}\left[\frac{E(\gamma,\alpha)}{\...
...{\sin^2\alpha} - \frac{\cos\gamma \sin\gamma}{\cos^2\alpha \cos\beta}\right],$">|; 

$key = q/mu_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6123.png"
 ALT="$ \mu_2$">|; 

$key = q/{mitPi}_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4593.png"
 ALT="$ {\mit\Pi}_M$">|; 

$key = q/v_g=domegaslashdk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4154.png"
 ALT="$ v_g=d\omega/dk$">|; 

$key = q/displaystyle{rmtr}(A);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7010.png"
 ALT="$\displaystyle {\rm tr}(A)$">|; 

$key = q/displaystyle{bfa}cdot{bfb}=|{bfa}|,|{bfb}|,costhetaleq|{bfa}|,|{bfb}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6706.png"
 ALT="$\displaystyle {\bf a}\cdot{\bf b}= \vert{\bf a}\vert \vert{\bf b}\vert \cos\theta\leq \vert{\bf a}\vert \vert{\bf b}\vert
$">|; 

$key = q/0leqxileqinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7223.png"
 ALT="$ 0\leq \xi\leq \infty$">|; 

$key = q/displaystylefrac{omega^{,2}}{2pi,G,rho}=frac{a_1,a_3}{a_2},(a_2^{,2}-a_3^{,2})int_0^inftyfrac{u,du}{(a_2^{,2}+u),(a_3^{,2}+u),{mitDelta}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img867.png"
 ALT="$\displaystyle \frac{\omega^{ 2}}{2\pi G \rho} = \frac{a_1 a_3}{a_2} (a_2^{...
...3^{ 2})\int_0^\infty \frac{u du}{(a_2^{ 2}+u) (a_3^{ 2}+u) {\mit\Delta}},$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{Chapter04slashjump.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="497" HEIGHT="169" BORDER="0"
 SRC="|."$dir".q|img1464.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{Chapter04/jump.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{partialsigma_{ij}}{partialx_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7017.png"
 ALT="$\displaystyle \frac{\partial\sigma_{ij}}{\partial x_j}$">|; 

$key = q/displaystylef''(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3164.png"
 ALT="$\displaystyle f''(1)$">|; 

$key = q/epsilon_{ijk}=epsilon_{iml}=-epsilon_{ilm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6815.png"
 ALT="$ \epsilon_{ijk}=\epsilon_{iml}=-\epsilon_{ilm}$">|; 

$key = q/z=3,H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5136.png"
 ALT="$ z=3 H$">|; 

$key = q/nablacdotnablatimes{bfA}equiv{bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1566.png"
 ALT="$ \nabla\cdot\nabla\times {\bf A}\equiv {\bf0}$">|; 

$key = q/displaystylea=left(frac{Q}{4pi,V}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2816.png"
 ALT="$\displaystyle a = \left(\frac{Q}{4\pi V}\right)^{1/2},
$">|; 

$key = q/p=p_0-rho,g,z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1117.png"
 ALT="$ p=p_0-\rho g z$">|; 

$key = q/v_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1354.png"
 ALT="$ v_2$">|; 

$key = q/h_1=h_2=h_3=h=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7104.png"
 ALT="$ h_1=h_2=h_3=h=1$">|; 

$key = q/displaystylephi=varphi-{mitOmega},t.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4339.png"
 ALT="$\displaystyle \phi = \varphi - {\mit\Omega} t.$">|; 

$key = q/{mitDelta}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5032.png"
 ALT="$ {\mit\Delta}t$">|; 

$key = q/displaystyle=-frac{1}{v}left(frac{partialv}{partialp}right)_T,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5599.png"
 ALT="$\displaystyle = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_T,$">|; 

$key = q/A_{ji}=A_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7014.png"
 ALT="$ A_{ji}=A_{ij}$">|; 

$key = q/displaystylep=K,rho^{,4slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5192.png"
 ALT="$\displaystyle p = K \rho^{ 4/3},$">|; 

$key = q/displaystyle=sum_{j,k=1,3}epsilon_{ijk},A_j,B_k,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7087.png"
 ALT="$\displaystyle =\sum_{j,k=1,3} \epsilon_{ijk} A_j B_k,$">|; 

$key = q/displaystylefrac{v_p}{v_c}simeqleft(frac{k,l}{2}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4162.png"
 ALT="$\displaystyle \frac{v_p}{v_c}\simeq \left(\frac{k l}{2}\right)^{1/2},$">|; 

$key = q/displaystyle{mitPhi}_i=oint_Srho,v_i,v_j,dS_j.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="$\displaystyle {\mit\Phi}_i = \oint_S \rho v_i v_j dS_j.$">|; 

$key = q/displaystylef(theta,phi_pm);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4801.png"
 ALT="$\displaystyle f(\theta,\phi_\pm)$">|; 

$key = q/zeta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4726.png"
 ALT="$ \zeta=0$">|; 

$key = q/displaystyle=frac{mu}{rho_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="$\displaystyle =\frac{\mu}{\rho_0},$">|; 

$key = q/displaystylealpha=sin^{-1}(sinbetaslashsingamma).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img934.png"
 ALT="$\displaystyle \alpha = \sin^{-1}(\sin\beta/\sin\gamma).$">|; 

$key = q/displaystyleeta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6145.png"
 ALT="$\displaystyle \eta_2$">|; 

$key = q/W_0=M_0,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img591.png"
 ALT="$ W_0=M_0 g$">|; 

$key = q/{mitGamma}_nu=pi,V,alpha,c_nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3547.png"
 ALT="$ {\mit\Gamma}_\nu = \pi V \alpha c_\nu$">|; 

$key = q/2.73times10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4611.png"
 ALT="$ 2.73\times10^{-3}$">|; 

$key = q/displaystyle{bfr}=r,sintheta,cosvarphi,{bfe}_r+r,sintheta,sinvarphi,{bfe}_theta+r,costheta,{bfe}_varphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4251.png"
 ALT="$\displaystyle {\bf r} = r \sin\theta \cos\varphi {\bf e}_r + r \sin\theta \sin\varphi {\bf e}_\theta + r \cos\theta {\bf e}_\varphi,$">|; 

$key = q/displaystylezeta=b+{rme}^{,K',(z-z_0)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2174.png"
 ALT="$\displaystyle \zeta = b + {\rm e}^{ K' (z-z_0)}.$">|; 

$key = q/n-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4328.png"
 ALT="$ n-1$">|; 

$key = q/displaystylemu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1007.png"
 ALT="$\displaystyle \mu$">|; 

$key = q/epsilon_{113}=epsilon_{212}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6788.png"
 ALT="$ \epsilon_{113}=\epsilon_{212}=0$">|; 

$key = q/displaystyledelta_1(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3181.png"
 ALT="$\displaystyle \delta_1(x)$">|; 

$key = q/stackrel{displaystylerightarrow}{PR};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="30" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6228.png"
 ALT="$ \stackrel{\displaystyle \rightarrow}{PR}$">|; 

$key = q/displaystylem=sqrt{1-{rmMa}_infty^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6005.png"
 ALT="$\displaystyle m = \sqrt{1-{\rm Ma}_\infty^{ 2}}.$">|; 

$key = q/displaystylefrac{b}{d}=left(1+frac{2}{pi}int_0^{pislash2}frac{sin[(2,alphaslashpi),beta]}{tanbeta},dbetaright)^{-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="81" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2251.png"
 ALT="$\displaystyle \frac{b}{d} =\left(1+\frac{2}{\pi}\int_0^{\pi/2}\frac{\sin[(2 \alpha/\pi) \beta]}{\tan\beta} d\beta\right)^{-1}.$">|; 

$key = q/displaystyleD,{mitPsi}_r+mu_r,{mitPsi}_r=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4827.png"
 ALT="$\displaystyle D {\mit\Psi}_r +\mu_r {\mit\Psi}_r=0,$">|; 

$key = q/displaystyle=nablatimes{bfv}={calL}(psi),nablavarphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3738.png"
 ALT="$\displaystyle =\nabla\times{\bf v} = {\cal L}(\psi) \nabla\varphi.$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partialF(theta,phi_pm)}{partialphi}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4820.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial F(\theta,\phi_\pm)}{\partial \phi} =0.$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{AppendixEslashfigD.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="279" BORDER="0"
 SRC="|."$dir".q|img7300.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{AppendixE/figD.01.eps}}
\end{figure}">|; 

$key = q/j=9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$ j=9$">|; 

$key = q/rho,S_2,v_2^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1386.png"
 ALT="$ \rho S_2 v_2^{ 2}$">|; 

$key = q/displaystylefrac{p_1}{p_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5397.png"
 ALT="$\displaystyle \frac{p_1}{p_0}$">|; 

$key = q/displaystylefrac{v_g}{v_p}=frac{1}{2}left[frac{1+3,(k,l)^2}{1+(k,l)^2}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4155.png"
 ALT="$\displaystyle \frac{v_g}{v_p} =\frac{1}{2}\left[\frac{1+3 (k l)^2}{1+(k l)^2}\right].$">|; 

$key = q/3^{1slash2},k_c^{,-1}simeq4.8,{rmmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4216.png"
 ALT="$ 3^{1/2} k_c^{ -1}\simeq 4.8 {\rm mm}$">|; 

$key = q/displaystyleu(x,y)=-frac{partialpsi}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3243.png"
 ALT="$\displaystyle u(x,y)= -\frac{\partial\psi}{\partial y},
$">|; 

$key = q/mu_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6122.png"
 ALT="$ \mu_1$">|; 

$key = q/displaystyleI_{33}=frac{1}{5},M,(a_1^{,2}+a_2^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img852.png"
 ALT="$\displaystyle I_{33} = \frac{1}{5} M (a_1^{ 2}+a_2^{ 2}),$">|; 

$key = q/{bfa}cdot{bfb}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6317.png"
 ALT="$ {\bf a} \cdot {\bf b} =0$">|; 

$key = q/Oz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$ Oz$">|; 

$key = q/P=(0,,0,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3472.png"
 ALT="$ P=(0, 0, z)$">|; 

$key = q/displaystyleAP=BQ=2,a,s.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img731.png"
 ALT="$\displaystyle AP=BQ=2 a s.$">|; 

$key = q/nu>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3553.png"
 ALT="$ \nu&gt;1$">|; 

$key = q/displaystylepsi(r,theta)=Vleft(r-frac{a^{,2}}{r}right)sin(theta+alpha)+frac{{mitGamma}}{2pi},lnleft(frac{r}{a}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3294.png"
 ALT="$\displaystyle \psi(r,\theta) = V\left(r-\frac{a^{ 2}}{r}\right)\sin(\theta+\alpha) + \frac{{\mit\Gamma}}{2\pi} \ln\left(\frac{r}{a}\right),$">|; 

$key = q/displaystyle=-frac{1}{rho},frac{partialp}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5878.png"
 ALT="$\displaystyle = -\frac{1}{\rho} \frac{\partial p}{\partial y},$">|; 

$key = q/y=y_C(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5818.png"
 ALT="$ y=y_C(x)$">|; 

$key = q/t=tantheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img754.png"
 ALT="$ t=\tan\theta$">|; 

$key = q/displaystyle=frac{pi}{4},rho,V,b,{mitGamma}_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3549.png"
 ALT="$\displaystyle = \frac{\pi}{4} \rho V b {\mit\Gamma}_1,$">|; 

$key = q/theta,kappa_x^{,2},A_0slashV_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img641.png"
 ALT="$ \theta \kappa_x^{ 2} A_0/V_0$">|; 

$key = q/displaystylev_x(x,y)=frac{G(x)}{2,mu},yleft[d(x)-yright]+Uleft[frac{d(x)-y}{d}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3696.png"
 ALT="$\displaystyle v_x(x,y) = \frac{G(x)}{2 \mu} y\left[d(x)-y\right]+ U\left[\frac{d(x)-y}{d}\right],$">|; 

$key = q/displaystyleK=-pi,rholeft.int_{eta=0}^{eta=pi}phi,dpsiright|_{xi=xi_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2722.png"
 ALT="$\displaystyle K = -\pi \rho\left.\int_{\eta=0}^{\eta=\pi} \phi d\psi\right\vert _{\xi=\xi_0}.$">|; 

$key = q/x=x_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5674.png"
 ALT="$ x=x_\ast$">|; 

$key = q/{mitOmega}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3885.png"
 ALT="$ {\mit\Omega}_0$">|; 

$key = q/displaystylefrac{partialx_i'}{partialx_j}={calR}_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6876.png"
 ALT="$\displaystyle \frac{\partial x_i'}{\partial x_j} = {\cal R}_{ij}.$">|; 

$key = q/alpha_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6335.png"
 ALT="$ \alpha_x$">|; 

$key = q/displaystyleq=T,dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5320.png"
 ALT="$\displaystyle q = T dS,$">|; 

$key = q/displaystylefrac{1}{r^{,2}},frac{d}{dr}!left(r^{,3},frac{d{mitOmega}}{dr}right)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3636.png"
 ALT="$\displaystyle \frac{1}{r^{ 2}} \frac{d}{dr}\!\left(r^{ 3} \frac{d{\mit\Omega}}{dr}\right) = 0,$">|; 

$key = q/v={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7219.png"
 ALT="$ v={\rm constant}$">|; 

$key = q/|z|<bslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3492.png"
 ALT="$ \vert z\vert&lt;b/2$">|; 

$key = q/omega_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1904.png"
 ALT="$ \omega_z$">|; 

$key = q/erightarrow1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2787.png"
 ALT="$ e\rightarrow 1$">|; 

$key = q/displaystyle=-frac{{mitDelta}P}{l},frac{1}{a^{,4}(z),langlea^{,-4}rangle},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3680.png"
 ALT="$\displaystyle = - \frac{{\mit\Delta} P}{l} \frac{1}{a^{ 4}(z) \langle a^{ -4}\rangle},$">|; 

$key = q/{rmMa}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$ {\rm Ma}_2$">|; 

$key = q/x_1slashR;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img986.png"
 ALT="$ x_1/R$">|; 

$key = q/displaystyle=a_x,b_x+a_y,b_y+a_z,b_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6305.png"
 ALT="$\displaystyle = a_x  b_x + a_y  b_y + a_z  b_z$">|; 

$key = q/displaystyle{mitDelta}u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4499.png"
 ALT="$\displaystyle {\mit\Delta} u$">|; 

$key = q/displaystyle=a_{ijss},delta_{kl}+a_{sjsl},delta_{ik}+a_{issl},delta_{jk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6943.png"
 ALT="$\displaystyle =a_{ijss} \delta_{kl}+a_{sjsl} \delta_{ik} + a_{issl} \delta_{jk},$">|; 

$key = q/deltatheta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6822.png"
 ALT="$ \delta\theta_1$">|; 

$key = q/displaystylefrac{F''}{F}=-frac{1}{m^{,2}},frac{G''}{G},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5960.png"
 ALT="$\displaystyle \frac{F''}{F} =-\frac{1}{m^{ 2}} \frac{G''}{G},$">|; 

$key = q/displaystyle{mitDeltaT}=T_0-T=frac{v^{,2}}{2,{calC}_p},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5379.png"
 ALT="$\displaystyle {\mit\Delta T} =T_0-T= \frac{v^{ 2}}{2 {\cal C}_p},$">|; 

$key = q/displaystylec=c_0+left(frac{gamma-1}{2}right)u.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5370.png"
 ALT="$\displaystyle c = c_0 +\left(\frac{\gamma-1}{2}\right)u.$">|; 

$key = q/{mitPhi}_rrightarrow{mitPhi}_n^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4934.png"
 ALT="$ {\mit\Phi}_r\rightarrow {\mit\Phi}_n^{ m}$">|; 

$key = q/{bfv}cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6499.png"
 ALT="$ {\bf v} \cdot d{\bf S}$">|; 

$key = q/displaystyle=-frac{g,d}{a^{,2}},frac{partial(zeta-F)}{partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4656.png"
 ALT="$\displaystyle =- \frac{g d}{a^{ 2}} \frac{\partial (\zeta-F)}{\partial\theta},$">|; 

$key = q/displaystylefrac{partial^{,2}psi}{partialx^{,2}}+frac{partial^{,2}psi}{partialy^{,2}}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1661.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\psi}{\partial x^{ 2}}+\frac{\partial^{ 2}\psi}{\partial y^{ 2}} = 0.$">|; 

$key = q/displaystyle=int_{mitOmega}frac{costheta}{sintheta}left(frac{partial{mitPhi}_r}{altheta},frac{partial{mitPhi}_s}{partialphi}right)d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4867.png"
 ALT="$\displaystyle = \int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\left(\frac{\part...
...{\partial\theta} \frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}$">|; 

$key = q/displaystyle=v_c=(Q,g)^{1slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1531.png"
 ALT="$\displaystyle = v_c = (Q g)^{1/3},$">|; 

$key = q/displaystyle{bfv}=v_r(r),{bfe}_r,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1669.png"
 ALT="$\displaystyle {\bf v} = v_r(r) {\bf e}_r,$">|; 

$key = q/displaystylea_{ij}=b_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6845.png"
 ALT="$\displaystyle a_{ij} = b_{ij}$">|; 

$key = q/displaystyle{mitPhi}(r,theta,phi)={mitPhi}_0(r)+sum_{n=1,infty}{mitPhi}_n(r,theta,phi)+{mitPhi}_{rmtide}(r,theta,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4354.png"
 ALT="$\displaystyle {\mit\Phi}(r,\theta,\phi) = {\mit\Phi}_0(r) +\sum_{n=1,\infty}{\mit\Phi}_n(r,\theta,\phi) + {\mit\Phi}_{\rm tide}(r,\theta,\phi),$">|; 

$key = q/displaystyleE=K+U=frac{1}{2},rho,g,a^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3981.png"
 ALT="$\displaystyle E = K+ U = \frac{1}{2} \rho g a^{ 2}.$">|; 

$key = q/0.2,{rmmslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3825.png"
 ALT="$ 0.2 {\rm m/s}$">|; 

$key = q/s>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1039.png"
 ALT="$ s&gt;1$">|; 

$key = q/displaystylem=frac{alpha}{2-alpha}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1872.png"
 ALT="$\displaystyle m=\frac{\alpha}{2-\alpha}.$">|; 

$key = q/displaystyleQ=h,v,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1524.png"
 ALT="$\displaystyle Q = h v,$">|; 

$key = q/P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$ P$">|; 

$key = q/m+m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1822.png"
 ALT="$ m+m'$">|; 

$key = q/displaystylephantom{-frac{G,m',a^{2}}{r'^{,3}}}+frac{1}{12},P_2^{,2},(costheta'),cos(2,varphi'),{calR}_2^{,(-2)}(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4283.png"
 ALT="$\displaystyle \phantom{ - \frac{G m' a^{2}}{r'^{ 3}}} +\frac{1}{12} P_2^{ 2} (\cos\theta') \cos(2 \varphi') {\cal R}_2^{ (-2)}(r,\theta,\varphi)$">|; 

$key = q/v=c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5392.png"
 ALT="$ v=c$">|; 

$key = q/|rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1583.png"
 ALT="$ \vert\rightarrow 0$">|; 

$key = q/t=t_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5669.png"
 ALT="$ t=t_\ast$">|; 

$key = q/displaystylenablacdot{bfn}=-frac{f_{xx}}{(1+f_x^{,2})^{3slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1158.png"
 ALT="$\displaystyle \nabla\cdot {\bf n} = - \frac{f_{xx}}{(1+f_x^{ 2})^{3/2}},$">|; 

$key = q/skew{5}tilde{phi}=pislash2-phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1211.png"
 ALT="$ \skew{5}\tilde{\phi}=\pi/2-\phi$">|; 

$key = q/F(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4819.png"
 ALT="$ F(\theta,\phi)$">|; 

$key = q/displaystyle{calS}_2-{calS}_1={calC}_Vlnleft[G({rmMa}_1^{,2})right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5494.png"
 ALT="$\displaystyle {\cal S}_2-{\cal S}_1 = {\cal C}_V\ln\left[G({\rm Ma}_1^{ 2})\right],$">|; 

$key = q/displaystyle={rmconstant},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4792.png"
 ALT="$\displaystyle ={\rm constant},$">|; 

$key = q/displaystyle=a,x+b,y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1899.png"
 ALT="$\displaystyle = a x+b y,$">|; 

$key = q/{calS}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5692.png"
 ALT="$ {\cal S}_1$">|; 

$key = q/displaystyle=n,{bfa}+n,{bfb}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6251.png"
 ALT="$\displaystyle = n {\bf a} + n {\bf b}.$">|; 

$key = q/displaystylefrac{Dv_theta}{Dt}+frac{v_r,v_theta-cottheta,v_varphi^{,2}}{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img505.png"
 ALT="$\displaystyle \frac{Dv_\theta}{Dt}+\frac{v_r v_\theta-\cot\theta v_\varphi^{ 2}}{r}$">|; 

$key = q/displaystyle{mitTheta}=int_Vtheta,dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="$\displaystyle {\mit\Theta} = \int_V \theta dV,$">|; 

$key = q/displaystylesimeqfrac{gamma,(gamma+1)}{4},frac{{rmMa}_1^{,6}}{({rmMa}_1^{,2}-1)^{3slash2}},theta_1,theta_2,(theta_1+theta_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6190.png"
 ALT="$\displaystyle \simeq \frac{\gamma (\gamma+1)}{4} \frac{{\rm Ma}_1^{ 6}}{({\rm Ma}_1^{ 2}-1)^{3/2}} \theta_1 \theta_2 (\theta_1+\theta_2),$">|; 

$key = q/displaystylesimeq-frac{partialzeta}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4137.png"
 ALT="$\displaystyle \simeq -\frac{\partial\zeta}{\partial x},$">|; 

$key = q/P,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2380.png"
 ALT="$ P dx$">|; 

$key = q/displaystyle=frac{delta}{2},sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3310.png"
 ALT="$\displaystyle = \frac{\delta}{2} \sin\theta.$">|; 

$key = q/displaystylepsi(xi,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2788.png"
 ALT="$\displaystyle \psi(\xi,\eta)$">|; 

$key = q/1.81times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4614.png"
 ALT="$ 1.81\times 10^{-2}$">|; 

$key = q/displaystyleQ=int_0^a,2pi,r,v_z,dr=frac{pi,G,a^{,4}}{8,mu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3616.png"
 ALT="$\displaystyle Q = \int_0^a 2\pi r v_z dr = \frac{\pi G a^{ 4}}{8 \mu}.$">|; 

$key = q/displaystyle=nablatimes{bfv}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1535.png"
 ALT="$\displaystyle = \nabla\times {\bf v}.$">|; 

$key = q/displaystyleF_z=M,g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3809.png"
 ALT="$\displaystyle F_z = M g,$">|; 

$key = q/displaystyleoint_Sa,dS_i=int_Vfrac{partiala}{partialx_i},dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6895.png"
 ALT="$\displaystyle \oint_S a dS_i = \int_V \frac{\partial a}{\partial x_i} dV.$">|; 

$key = q/displaystylephantom{=}+{rmMa}_infty^{,2},frac{v'}{U}left(1+frac{u'}{U}right)left(frac{partialu'}{partialy}+frac{partialv'}{partialx}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5906.png"
 ALT="$\displaystyle \phantom{=} + {\rm Ma}_\infty^{ 2} \frac{v'}{U}\left(1+\frac{u'...
...ght)\left(\frac{\partial u'}{\partial y}+\frac{\partial v'}{\partial x}\right).$">|; 

$key = q/skew{3}bar{rho};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4343.png"
 ALT="$ \skew{3}\bar{\rho}$">|; 

$key = q/displaystyle{rmMa}_2=left[frac{2+(gamma-1),{rmMa}_1^{,2}}{1-gamma+2,gamma,{rmMa}_1^{,2}}right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5478.png"
 ALT="$\displaystyle {\rm Ma}_2=\left[\frac{2+(\gamma-1) {\rm Ma}_1^{ 2}}{1-\gamma+2 \gamma {\rm Ma}_1^{ 2}}\right]^{1/2}.$">|; 

$key = q/displaystylek,dgg1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3924.png"
 ALT="$\displaystyle k d\gg 1,$">|; 

$key = q/'=dslashdeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2936.png"
 ALT="$ '= d/d\eta$">|; 

$key = q/displaystylefrac{z}{l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1186.png"
 ALT="$\displaystyle \frac{z}{l}$">|; 

$key = q/t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img763.png"
 ALT="$ t=0$">|; 

$key = q/displaystyle{bfV}(t)=V_x(t),{bfe}_x+V_y(t),{bfe}_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1827.png"
 ALT="$\displaystyle {\bf V}(t) = V_x(t) {\bf e}_x+ V_y(t) {\bf e}_y.$">|; 

$key = q/displaystylerho,h,v^{,2},sinalpha=rho,h_1,v^{,2}-rho,h_2,v^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1888.png"
 ALT="$\displaystyle \rho h v^{ 2} \sin\alpha = \rho h_1 v^{ 2} - \rho h_2 v^{ 2},$">|; 

$key = q/displaystyletheta_{n,,n'}^{,m,,m'}=sum_{n''=1,infty}sum_{m''=0,n''}^{m'',{rmeven{,m''})^{-2},kappa_{n'',n}^{,m'',m},kappa_{n'',n'}^{,m'',m'};;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="526" HEIGHT="79" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4990.png"
 ALT="$\displaystyle \theta_{n, n'}^{ m, m'}= \sum_{n''=1,\infty}\sum_{m''=0,n''}^{...
... (a_{n''}^{ m''})^{-2} \kappa_{n'',n}^{ m'',m} \kappa_{n'',n'}^{ m'',m'};$">|; 

$key = q/displaystylephantom{simeqfrac{G,m',a^{2}}{R^{,3}}}-frac{7,e}{8},{calR}_2^{,(-2)}(r,theta,varphi-[(3slash2),omega_1-(1slash2),omega_2],tright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4306.png"
 ALT="$\displaystyle \phantom{\simeq \frac{G m' a^{2}}{R^{ 3}}} - \frac{7 e}{8} {...
...R}_2^{ (-2)}\left(r,\theta,\varphi-[(3/2) \omega_1-(1/2) \omega_2] t\right)$">|; 

$key = q/displaystylez+{rmi},varpi=c,sinh(xi+{rmi},eta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2752.png"
 ALT="$\displaystyle z+{\rm i} \varpi = c \sinh(\xi+{\rm i} \eta),$">|; 

$key = q/displaystyle=-frac{1}{sintheta},frac{partial{mitPhi}'}{partialphi}+frac{partial{mitPsi}'}{partialtheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4779.png"
 ALT="$\displaystyle = -\frac{1}{\sin\theta} \frac{\partial{\mit\Phi}'}{\partial\phi} +\frac{\partial{\mit\Psi}'}{\partial\theta}.$">|; 

$key = q/displaystylep_g=frac{rho,k,T}{mu,m_p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5180.png"
 ALT="$\displaystyle p_g = \frac{\rho k T}{\mu m_p}$">|; 

$key = q/F(z)=phi(x,y)+{rmi},psi(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2342.png"
 ALT="$ F(z)=\phi(x,y)+{\rm i} \psi(x,y)$">|; 

$key = q/F(zeta)=phi(xi,eta)+{rmi},psi(xi,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2082.png"
 ALT="$ F(\zeta) = \phi(\xi,\eta)+{\rm i} \psi(\xi,\eta)$">|; 

$key = q/P=dWslashdt={bff}cdotd{bfr}slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6331.png"
 ALT="$ P = dW/dt = {\bf f}\cdot d{\bf r}/dt$">|; 

$key = q/displaystyle=-frac{nablap_1}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4072.png"
 ALT="$\displaystyle = -\frac{\nabla p_1}{\rho},$">|; 

$key = q/p(x)-p_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3707.png"
 ALT="$ p(x)-p_0$">|; 

$key = q/{mitDelta}pslashp_1gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5509.png"
 ALT="$ {\mit\Delta}p/p_1\gg 1$">|; 

$key = q/ysimeqdelta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2858.png"
 ALT="$ y\simeq \delta$">|; 

$key = q/t+deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1387.png"
 ALT="$ t+\delta t$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashvortexa.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img2036.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/vortexa.eps}}
\end{figure}">|; 

$key = q/displaystyleh(theta)=(c^{,2},sin^2theta+delta^{,2},cos^2theta)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3357.png"
 ALT="$\displaystyle h(\theta) = (c^{ 2} \sin^2\theta+\delta^{ 2} \cos^2\theta)^{1/2}.$">|; 

$key = q/dS=pi,r^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6655.png"
 ALT="$ dS = \pi  r^2$">|; 

$key = q/displaystylefrac{sina}{A}=frac{sinb}{B}=frac{sinc}{C};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6705.png"
 ALT="$\displaystyle \frac{\sin a}{A} = \frac{\sin b}{B} = \frac{\sin c}{C}
$">|; 

$key = q/muslash(rho,g,a)=0.35;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5033.png"
 ALT="$ \mu/(\rho g a)=0.35$">|; 

$key = q/0leqphileqpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3484.png"
 ALT="$ 0\leq \phi\leq \pi$">|; 

$key = q/displaystylef_i=oint_Ssigma_{ij},dS_j+int_VF_i,dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img574.png"
 ALT="$\displaystyle f_i = \oint_S \sigma_{ij} dS_j + \int_V F_i dV,$">|; 

$key = q/displaystyle=v,dp+T,d{calS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5336.png"
 ALT="$\displaystyle = v dp + T d{\cal S}.$">|; 

$key = q/I(z)=-{mitDelta}v_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3407.png"
 ALT="$ I(z)=-{\mit\Delta} v_z$">|; 

$key = q/displaystylek=frac{2,b,sin^4alpha}{pi,sinalpha+4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2315.png"
 ALT="$\displaystyle k =\frac{2 b \sin^4\alpha}{\pi \sin\alpha+4}.$">|; 

$key = q/displaystyle=lambda_sint_{mitOmega}{mitPhi}_r',{mitPhi}_s,d{mitOmega}=-int_{mitOmega}{mitPhi}_r',D,{mitPhi}_s,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4865.png"
 ALT="$\displaystyle =\lambda_s\int_{\mit\Omega} {\mit\Phi}_r' {\mit\Phi}_s d{\mit\Omega} = -\int_{\mit\Omega} {\mit\Phi}_r' D {\mit\Phi}_s d{\mit\Omega}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter11slashshock.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img4024.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter11/shock.eps}}
\end{figure}">|; 

$key = q/{rmFr}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1606.png"
 ALT="$ {\rm Fr}_1$">|; 

$key = q/displaystylefrac{partial^{,2}xi_i}{partialt^{,2}}+2,epsilon_{ijk},{mitOmega}_j,fac{partial{mitPhi}}{partialx_i}-frac{partialchi}{partialx_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4376.png"
 ALT="$\displaystyle \frac{\partial^{ 2} \xi_i}{\partial t^{ 2}} + 2 \epsilon_{ijk}...
...} -\frac{\partial {\mit\Phi}}{\partial x_i}-\frac{\partial \chi}{\partial x_i},$">|; 

$key = q/displaystyleD=frac{1}{2},,rho,g,a^{,2},wleft(1-frac{v_g}{v_p}right)=frac{1}{4},rho,g,a^{,2},wleft[1-frac{2,k,d}{sinh(2,k,d)}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3991.png"
 ALT="$\displaystyle D = \frac{1}{2}  \rho g a^{ 2} w\left(1-\frac{v_g}{v_p}\right)=\frac{1}{4} \rho g a^{ 2} w\left[1-\frac{2 k d}{\sinh(2 k d)}\right],$">|; 

$key = q/s<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4174.png"
 ALT="$ s&lt;1$">|; 

$key = q/displaystyle=-ointp(a,theta,varphi),a^{,2},sintheta,cosvarphi,d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2654.png"
 ALT="$\displaystyle = -\oint p(a,\theta,\varphi) a^{ 2} \sin\theta \cos\varphi d{\mit\Omega},$">|; 

$key = q/rho>rho';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4114.png"
 ALT="$ \rho&gt;\rho'$">|; 

$key = q/displaystyle=sigma_{rr}(a,theta)=left(-p+2,mu,frac{partialv_r}{partialr}right)_{r=a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3797.png"
 ALT="$\displaystyle = \sigma_{rr}(a,\theta) = \left(-p + 2 \mu \frac{\partial v_r}{\partial r}\right)_{r=a},$">|; 

$key = q/displaystyle=frac{partialpsi}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1657.png"
 ALT="$\displaystyle =\frac{\partial\psi}{\partial y},$">|; 

$key = q/epsilon_{ijk}=epsilon_{ilm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6813.png"
 ALT="$ \epsilon_{ijk}=\epsilon_{ilm}$">|; 

$key = q/K;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2146.png"
 ALT="$ K$">|; 

$key = q/displaystyle=-frac{costheta}{sintheta},frac{partial{mitPhi}_r}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4835.png"
 ALT="$\displaystyle = -\frac{\cos\theta}{\sin\theta} \frac{\partial{\mit\Phi}_r}{\partial \phi},$">|; 

$key = q/epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4590.png"
 ALT="$ \epsilon$">|; 

$key = q/displaystyle=~_{lim|deltay|rightarrow0}frac{phi(x,y+deltay)+{rmi},psi(x,y+deltay)-phi(x,y)-{rmi},psi(x,y)}{{rmi},deltay};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="489" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1965.png"
 ALT="$\displaystyle =&nbsp;_{\lim \vert\delta y\vert\rightarrow 0} \frac{\phi(x, y+\delta...
...i}  \psi(x, y+\delta y) - \phi(x, y) - {\rm i}  \psi(x,y)}{{\rm i} \delta y}$">|; 

$key = q/displaystylep(a,theta,t)=p_0-left.frac{1}{2},rho,v^{,2}right|_{r=a}-rho,g,a,sint{partialt}right|_{r=a}-rholeft.{bfV}cdotnablaphiright|_{r=a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="548" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1835.png"
 ALT="$\displaystyle p(a,\theta,t) = p_0 -\left.\frac{1}{2} \rho v^{ 2}\right\vert ...
...ial t}\right\vert _{r=a} - \rho\left.{\bf V}\cdot \nabla\phi\right\vert _{r=a},$">|; 

$key = q/{bfr}equiv(x,,y,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6282.png"
 ALT="$ {\bf r}\equiv (x, y, z)$">|; 

$key = q/displaystylefrac{x_1^{,2}}{a_1^{,2}}+frac{x_2^{,2}}{a_2^{,2}}+frac{x_3^{,2}}{a_3^{,2}}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img865.png"
 ALT="$\displaystyle \frac{x_1^{ 2}}{a_1^{ 2}}+ \frac{x_2^{ 2}}{a_2^{ 2}}+\frac{x_3^{ 2}}{a_3^{ 2}}=1.$">|; 

$key = q/displaystyle=x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4241.png"
 ALT="$\displaystyle = x,$">|; 

$key = q/displaystyle{bfF}=frac{rho}{4pi}int_Sint_{S'}frac{[mbox{boldmathOmega}_S({bfr})tesmbox{boldmathOmega}_S({bfr}')}{|{bfr}'-{bfr}|^{,3}},dS',dS.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3458.png"
 ALT="$\displaystyle {\bf F} = \frac{\rho}{4\pi}\int_S\int_{S'}\frac{[\mbox{\boldmath$...
...box{\boldmath$\Omega$}_S({\bf r}')}{\vert{\bf r}'-{\bf r}\vert^{ 3}} dS' dS.$">|; 

$key = q/m(r)slashM_ast=y(xi)slashy_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5253.png"
 ALT="$ m(r)/M_\ast = y(\xi)/y_1$">|; 

$key = q/displaystyleL_0=pi,rho,V^{,2},S.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3515.png"
 ALT="$\displaystyle L_0 = \pi \rho V^{ 2} S.$">|; 

$key = q/displaystylep_1,S_1,v_1,deltat-p_2,S_2,v_2,deltat=mleft(frac{p_1}{rho_1}-frac{p_2}{rho_2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1371.png"
 ALT="$\displaystyle p_1 S_1 v_1 \delta t - p_2 S_2 v_2 \delta t = m\left(\frac{p_1}{\rho_1}-\frac{p_2}{\rho_2}\right).$">|; 

$key = q/{mitDelta}varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$ {\mit \Delta }\varphi $">|; 

$key = q/gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="$ \gamma$">|; 

$key = q/displaystyle{rmMa}_s=left[1+left(frac{gamma+1}{2,gamma}right)betaright]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5536.png"
 ALT="$\displaystyle {\rm Ma}_s=\left[1+\left(\frac{\gamma+1}{2 \gamma}\right)\beta\right]^{1/2}.$">|; 

$key = q/displaystyleP(r,theta)=p_0+frac{3,mu,V,a,costheta}{2,r^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3790.png"
 ALT="$\displaystyle P(r,\theta) = p_0 + \frac{3 \mu V a \cos\theta}{2 r^{ 2}},$">|; 

$key = q/F_3(z)=F_1(z)+F_2(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2013.png"
 ALT="$ F_3(z)=F_1(z)+F_2(z)$">|; 

$key = q/beta_1'-beta_1>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6170.png"
 ALT="$ \beta_1'-\beta_1&gt;0$">|; 

$key = q/betarightarrow2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2961.png"
 ALT="$ \beta\rightarrow 2$">|; 

$key = q/a_{ik};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6859.png"
 ALT="$ a_{ik}$">|; 

$key = q/k_2=3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4647.png"
 ALT="$ k_2=3/2$">|; 

$key = q/j=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$ j=1$">|; 

$key = q/displaystyleI_n(phi)=lim_{epsilonrightarrow0}left[int_0^{phi-epsilon}frac{cos(n,phi+epsilon}^pifrac{cos(n,phi'),dphi'}{cosphi'-cosphi}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="459" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3539.png"
 ALT="$\displaystyle I_n(\phi)=\lim_{\epsilon\rightarrow 0}\left[ \int_0^{\phi-\epsilo...
..._{\phi+\epsilon}^\pi \frac{\cos(n \phi') d\phi'}{\cos\phi'-\cos\phi}\right].
$">|; 

$key = q/displaystylep_c=frac{5}{4pi},frac{G,M_ast^{,2}}{R},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5286.png"
 ALT="$\displaystyle p_c = \frac{5}{4\pi} \frac{G M_\ast^{ 2}}{R},
$">|; 

$key = q/u_2=theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7106.png"
 ALT="$ u_2=\theta$">|; 

$key = q/displaystyle|zeta_b|,,|zeta|lldlla.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4469.png"
 ALT="$\displaystyle \vert\zeta_b\vert,  \vert\zeta\vert\ll d\ll a.$">|; 

$key = q/displaystyle=int_0^inftyfrac{du}{mitDelta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img859.png"
 ALT="$\displaystyle =\int_0^\infty \frac{du}{\mit\Delta},$">|; 

$key = q/displaystylebeta_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6101.png"
 ALT="$\displaystyle \beta_s$">|; 

$key = q/d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7230.png"
 ALT="$ d{\mit\Omega}$">|; 

$key = q/-K'=pislash(2,skew{3}bar{psi}_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2204.png"
 ALT="$ -K'=\pi/(2 \skew{3}\bar{\psi}_1)$">|; 

$key = q/displaystylephantom{=}-frac{G,M}{5,r^{,5}},a_2^{,2}left[x_2^{,2}-frac{1}{2},(x_1},a_3^{,2}left[x_3^{,2}-frac{1}{2},(x_1^{,2}+x_2^{,2})right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="528" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7292.png"
 ALT="$\displaystyle \phantom{=}- \frac{G M}{5 r^{ 5}} a_2^{ 2}\left[x_2^{ 2}-\f...
... r^{ 5}} a_3^{ 2}\left[x_3^{ 2}-\frac{1}{2} (x_1^{ 2}+x_2^{ 2})\right].$">|; 

$key = q/overline{p}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3843.png"
 ALT="$ \overline{p}_0$">|; 

$key = q/S_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6348.png"
 ALT="$ S_x$">|; 

$key = q/displaystyle{rmMa}_2^{,2},sin^2(beta-theta)=frac{2+(gamma-1),{rmMa}_1^{,2},sin^2beta}{1-gamma+2,gamma,{rmMa}_1^{,2},sin^2beta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5701.png"
 ALT="$\displaystyle {\rm Ma}_2^{ 2} \sin^2(\beta-\theta)= \frac{2+(\gamma-1) {\rm Ma}_1^{ 2} \sin^2\beta}{1-\gamma +2 \gamma {\rm Ma}_1^{ 2} \sin^2\beta}.$">|; 

$key = q/displaystyletanvarphi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4292.png"
 ALT="$\displaystyle \tan\varphi'$">|; 

$key = q/displaystyleX^{,3}+b,X^{,2}+c,X+d=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6087.png"
 ALT="$\displaystyle X^{ 3} + b X^{ 2}+ c X+ d = 0,$">|; 

$key = q/displaystylesimeq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5911.png"
 ALT="$\displaystyle \simeq 0.$">|; 

$key = q/{mitGamma}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2437.png"
 ALT="$ {\mit\Gamma}'$">|; 

$key = q/p_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4925.png"
 ALT="$ p_r$">|; 

$key = q/displaystylefrac{p-p_{rmatm}}{skew{3}bar{rho}}+{mitPhi}+chi=P(theta,phi,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4484.png"
 ALT="$\displaystyle \frac{p-p_{\rm atm}}{\skew{3}\bar{\rho}}+{\mit\Phi}+\chi = P(\theta,\phi,t),$">|; 

$key = q/displaystyle=left(frac{gamma+1}{4}right)left(frac{{rmMa}_1^{,2}}{{rmMa}_1^{,2}-1}right)theta_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6183.png"
 ALT="$\displaystyle = \left(\frac{\gamma+1}{4}\right)\left(\frac{{\rm Ma}_1^{ 2}}{{\rm Ma}_1^{ 2}-1}\right)\theta_2,$">|; 

$key = q/g=9.8,{rmm,s^{-2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5128.png"
 ALT="$ g=9.8 {\rm m s^{-2}}$">|; 

$key = q/displaystylec(x)=c_1left(frac{rho}{rho}_1right)^{,(gamma-1)slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5555.png"
 ALT="$\displaystyle c(x) = c_1\left(\frac{\rho}{\rho}_1\right)^{ (\gamma-1)/2}.$">|; 

$key = q/2,omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4530.png"
 ALT="$ 2 \omega_1$">|; 

$key = q/displaystyletimes({bfr}-{bfr}_0)+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6659.png"
 ALT="$\displaystyle \times ({\bf r}-{\bf r}_0) +$">|; 

$key = q/9.45times10^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4629.png"
 ALT="$ 9.45\times 10^{-1}$">|; 

$key = q/-(zeta_{mitOmega}+zeta_M+zeta_E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4544.png"
 ALT="$ -(\zeta_{\mit\Omega} + \zeta_M+\zeta_E)$">|; 

$key = q/D{bfv}slashDt={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3578.png"
 ALT="$ D{\bf v}/Dt={\bf0}$">|; 

$key = q/displaystylefrac{p_2'-p_1}{p_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6196.png"
 ALT="$\displaystyle \frac{p_2'-p_1}{p_1}$">|; 

$key = q/displaystylefrac{1}{r^{,2}}frac{d}{dr}!left(frac{r^{,2}}{rho},frac{dp}{dr}right)=-4pi,G,rho.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5178.png"
 ALT="$\displaystyle \frac{1}{r^{ 2}}\frac{d}{dr}\!\left(\frac{r^{ 2}}{\rho} \frac{dp}{dr}\right) = -4\pi G \rho.$">|; 

$key = q/displaystyle=cot^{-1}left[({rmMa}_1^{,2}-1)^{1slash2},frac{tanz}{tanz_1}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6130.png"
 ALT="$\displaystyle = \cot^{-1}\left[({\rm Ma}_1^{ 2}-1)^{1/2} \frac{\tan z}{\tan z_1}\right],$">|; 

$key = q/displaystylegleft.frac{partialphi}{partialz}right|_{z=0}+left.frac{partial^{,2}pft.frac{partial^{,3}phi}{partialz,partial^{,2}x}right|_{z=0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4146.png"
 ALT="$\displaystyle g\left.\frac{\partial\phi}{\partial z}\right\vert _{z=0} + \left....
...left.\frac{\partial^{ 3}\phi}{\partial z \partial^{ 2} x}\right\vert _{z=0}.$">|; 

$key = q/displaystylemu_0,{bfj}=nablatimes{bfB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3385.png"
 ALT="$\displaystyle \mu_0 {\bf j} = \nabla\times {\bf B}$">|; 

$key = q/F_3(z)=phi_3(x,y)+{rmi},psi_3(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2014.png"
 ALT="$ F_3(z)=\phi_3(x,y)+{\rm i} \psi_3(x,y)$">|; 

$key = q/displaystylea_{ij}=u_{ij}+v_{ij}+s_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6977.png"
 ALT="$\displaystyle a_{ij} = u_{ij} + v_{ij} + s_{ij},
$">|; 

$key = q/displaystyleleft.frac{partial}{partialt}right|_{x,y}=left.frac{partial}{partialt}right|_{r,theta}-{bfV}cdotnabla.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1810.png"
 ALT="$\displaystyle \left.\frac{\partial}{\partial t}\right\vert _{x,y} =\left. \frac{\partial}{\partial t}\right\vert _{r,\theta} -{\bf V}\cdot\nabla.$">|; 

$key = q/displaystylefrac{dzeta}{dz}=K,(zeta-a)^{1-alphaslashpi},(zeta-b)^{1-betaslashpi},(zeta-c)^{1-gammaslashpi}cdots,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2145.png"
 ALT="$\displaystyle \frac{d\zeta}{dz} = K (\zeta-a)^{1-\alpha/\pi} (\zeta-b)^{1-\beta/\pi} (\zeta-c)^{1-\gamma/\pi}\cdots,$">|; 

$key = q/S,(p-p_0+rho,v^{,2})slashR;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1595.png"
 ALT="$ S (p - p_0 + \rho v^{ 2})/R$">|; 

$key = q/displaystyle=x,'sintheta+y',costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6568.png"
 ALT="$\displaystyle = x '\sin\theta +y' \cos\theta,$">|; 

$key = q/displaystyle=a_2,b_3-a_3,b_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6781.png"
 ALT="$\displaystyle = a_2 b_3-a_3 b_2,$">|; 

$key = q/displaystyledot{Q}=-oint_Skappa,frac{partialT}{partialx_i},dS_i=-int_Vfrac{partil}{partialx_i}!left(kappa,frac{partialT}{partialx_i}right)dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="$\displaystyle \dot{Q} = -\oint_S\kappa \frac{\partial T}{\partial x_i} dS_i =...
...artial}{\partial x_i}\!\left(\kappa \frac{\partial T}{\partial x_i}\right) dV,$">|; 

$key = q/G(u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1271.png"
 ALT="$ G(u)$">|; 

$key = q/t>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5672.png"
 ALT="$ t&gt;0$">|; 

$key = q/displaystyle{calL}^{,2}(psi)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3744.png"
 ALT="$\displaystyle {\cal L}^{ 2}(\psi) = 0,$">|; 

$key = q/nabla^{,2}f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7220.png"
 ALT="$ \nabla^{ 2} f$">|; 

$key = q/p_0simeq10^5,{rmN,m^{-2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5123.png"
 ALT="$ p_0\simeq 10^5 {\rm N m^{-2}}$">|; 

$key = q/displaystylemu,nabla^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4392.png"
 ALT="$\displaystyle \mu \nabla^{ 2}$">|; 

$key = q/displaystyle=nablatimes{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3388.png"
 ALT="$\displaystyle = \nabla\times {\bf v},$">|; 

$key = q/displaystyleh(x)=frac{y_U(x)-y_L(x)}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5825.png"
 ALT="$\displaystyle h(x)= \frac{y_U(x)-y_L(x)}{2},$">|; 

$key = q/u_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1512.png"
 ALT="$ u_2$">|; 

$key = q/displaystyle2,a_{ijk}+a_{jik}+a_{kji};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6915.png"
 ALT="$\displaystyle 2 a_{ijk} + a_{jik} + a_{kji}$">|; 

$key = q/displaystyle{calS}_2-{calS}_1lesseqgtr0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5498.png"
 ALT="$\displaystyle {\cal S}_2-{\cal S}_1 \lesseqgtr 0$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialx^{,2}}+frac{1}{m^{,2}},frac{partial^{,2}phi}{partialy^{,2}}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5958.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial x^{ 2}} + \frac{1}{m^{ 2}} \frac{\partial^{ 2}\phi}{\partial y^{ 2}}=0$">|; 

$key = q/displaystyleu^{,2},frac{partialu}{partialx}+u,v,frac{partialu}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5885.png"
 ALT="$\displaystyle u^{ 2} \frac{\partial u}{\partial x} + u v \frac{\partial u}{\partial y}$">|; 

$key = q/displaystylesum_{n'=1,infty}sum_{m'=0,n'}left(left[-f_j^{,2},skew{3}hat{delta}_{f_j,skew{3}hat{beta}_{-n,,n'}^{,-m,,m'},p_{n'}^{,m'}right)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="560" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4995.png"
 ALT="$\displaystyle \sum_{n'=1,\infty}\sum_{m'=0,n'}\left(\left[-f_j^{ 2} \skew{3}\...
...-m'} + f_j \skew{3}\hat{\beta}_{-n, n'}^{ -m, m'} p_{n'}^{ m'}\right)= 0,$">|; 

$key = q/displaystyle=frac{2}{pi+2}left[tanh^{-1}(coszeta)-coszeta)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2231.png"
 ALT="$\displaystyle =\frac{2}{\pi+2}\left[\tanh^{-1}(\cos\zeta)-\cos\zeta)\right],$">|; 

$key = q/d(l)=d_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3692.png"
 ALT="$ d(l)=d_2$">|; 

$key = q/beta_{r,s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4886.png"
 ALT="$ \beta_{r,s}$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter10slashstokes2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3795.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter10/stokes2.eps}}
\end{figure}">|; 

$key = q/B'={rmi},pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2199.png"
 ALT="$ B'={\rm i} \pi/2$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}+g,r,sintheta-frac{partialphi}{partialt}=frac{p_0}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1833.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} +g r \sin\theta-\frac{\partial\phi}{\partial t} = \frac{p_0}{\rho},$">|; 

$key = q/Q=u-2,cslash(gamma-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5661.png"
 ALT="$ Q=u-2 c/(\gamma-1)$">|; 

$key = q/displaystyle({bfv}cdotnabla),{bfv}=-frac{nablap}{rho}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5864.png"
 ALT="$\displaystyle ({\bf v}\cdot\nabla) {\bf v} = -\frac{\nabla p}{\rho}.$">|; 

$key = q/displaystyleu_1(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5360.png"
 ALT="$\displaystyle u_1(x,t)$">|; 

$key = q/displaystylefrac{partial}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3016.png"
 ALT="$\displaystyle \frac{\partial}{\partial y}$">|; 

$key = q/displaystylefrac{Doverline{rho}}{Doverline{t}}ll1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="$\displaystyle \frac{D\overline{\rho}}{D\overline{t}} \ll 1.$">|; 

$key = q/displaystyleC-frac{f^{,2}}{2,l^{,2}}geq1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1164.png"
 ALT="$\displaystyle C - \frac{f^{ 2}}{2 l^{ 2}}\geq 1,$">|; 

$key = q/({bfa}times{bfb})times({bfc}times{bfb})=[{bfb}cdot({bfc}times{bfa})],{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6719.png"
 ALT="$ ({\bf a}\times {\bf b})\times ({\bf c}\times {\bf b}) = [{\bf b}\cdot({\bf c}\times {\bf a})] {\bf b}$">|; 

$key = q/8times10^{-3},{rmmslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3820.png"
 ALT="$ 8\times 10^{-3} {\rm m/s}$">|; 

$key = q/{bfF}_{rmext};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1819.png"
 ALT="$ {\bf F}_{\rm ext}$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter03slashfig6.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="371" BORDER="0"
 SRC="|."$dir".q|img1188.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter03/fig6.05.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashmerge.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="284" BORDER="0"
 SRC="|."$dir".q|img6177.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/merge.eps}}
\end{figure}">|; 

$key = q/displaystylenabla^{,2}{bfA}=nabla(nablacdot{bfA})-nablatimes(nablatimes{bfA}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7097.png"
 ALT="$\displaystyle \nabla^{ 2}{\bf A} = \nabla(\nabla\cdot{\bf A}) - \nabla\times(\nabla\times {\bf A}).$">|; 

$key = q/theta=-alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2240.png"
 ALT="$ \theta=-\alpha$">|; 

$key = q/displaystyle=-frac{3}{10},frac{(cosgamma)^{1slash3}}{sin^2gamma}left[(1-4,cos^2gamma),frac{gamma}{singamma}+3,cosgammaright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img900.png"
 ALT="$\displaystyle = -\frac{3}{10} \frac{(\cos\gamma)^{1/3}}{\sin^2\gamma}\left[(1-4 \cos^2\gamma) \frac{\gamma}{\sin\gamma}+3 \cos\gamma\right].$">|; 

$key = q/displaystylefrac{h_2}{h_1}gtreqless1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1485.png"
 ALT="$\displaystyle \frac{h_2}{h_1} \gtreqless 1$">|; 

$key = q/2slashpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2784.png"
 ALT="$ 2/\pi$">|; 

$key = q/{calL}(f,f_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1250.png"
 ALT="$ {\cal L}(f,f_z)$">|; 

$key = q/e=e_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4583.png"
 ALT="$ e=e_M$">|; 

$key = q/displaystylerho,v_i,frac{Dv_i}{Dt}=rho,frac{D}{Dt}!left(frac{1}{2},v_i,v_iright)=v_i,F_i+v_i,frac{partialsigma_{ij}}{partialx_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="354" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="$\displaystyle \rho v_i \frac{D v_i}{Dt} = \rho \frac{D}{Dt}\!\left(\frac{1}{2} v_i v_i\right) = v_i F_i + v_i \frac{\partial \sigma_{ij}}{\partial x_j}.$">|; 

$key = q/partiala_{ij}slashpartialx_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6881.png"
 ALT="$ \partial a_{ij}/\partial x_j$">|; 

$key = q/displaystylefrac{1}{{rmRe}}left(V_x,frac{partialV_y}{partialX}+V_y,frac{partialVy}{partialX^{,2}}+frac{partial^{,2}V_y}{partialY^{,2}}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="486" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2893.png"
 ALT="$\displaystyle \frac{1}{{\rm Re}}\left(V_x \frac{\partial V_y}{\partial X} + V_...
... 2} V_y}{\partial X^{ 2}}+\frac{\partial^{ 2} V_y}{\partial Y^{ 2}}\right].$">|; 

$key = q/z=pmd;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1262.png"
 ALT="$ z=\pm d$">|; 

$key = q/vslashu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3099.png"
 ALT="$ v/u$">|; 

$key = q/displaystyle=-frac{1}{2},rhooint_Cleft|frac{dF}{dz}right|^{,2}dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2395.png"
 ALT="$\displaystyle =-\frac{1}{2} \rho\oint_C \left\vert\frac{dF}{dz}\right\vert^{ 2}dx,$">|; 

$key = q/displaystyle{bfB}=W_0,{bfe}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img590.png"
 ALT="$\displaystyle {\bf B} = W_0 {\bf e}_z,$">|; 

$key = q/displaystylep=p_infty-rhoint_r^inftyfrac{v_theta^{,2}}{r},dr,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1722.png"
 ALT="$\displaystyle p = p_\infty-\rho\int_r^\infty \frac{v_\theta^{ 2}}{r} dr,$">|; 

$key = q/Qslash(U_0tmspace+thinmuskip{.1667em}delta)=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$ Q/(U_0\tmspace +\thinmuskip {.1667em}\delta )=0.5$">|; 

$key = q/{bfe}_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7050.png"
 ALT="$ {\bf e}_3$">|; 

$key = q/varpi=(x^{,2}+y^{,2})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2525.png"
 ALT="$ \varpi=(x^{ 2}+y^{ 2})^{1/2}$">|; 

$key = q/widehat{mitGamma}={mitGamma}slash[pi,V,(c+delta)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3324.png"
 ALT="$ \widehat{\mit\Gamma}= {\mit\Gamma}/[\pi V (c+\delta)]$">|; 

$key = q/displaystyle=-frac{partial{mitPhi}}{partialtheta}-frac{1}{sintheta},frac{partial{mitPsi}}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4664.png"
 ALT="$\displaystyle = -\frac{\partial{\mit\Phi}}{\partial \theta} - \frac{1}{\sin\theta} \frac{\partial{\mit\Psi}}{\partial\phi},$">|; 

$key = q/displaystyleU=-int_0^{M_ast}frac{G,m}{r},dm=frac{1}{2}int_0^{M_ast}{mitPsi},dm=-3int_0^Rp,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5267.png"
 ALT="$\displaystyle U= - \int_0^{M_\ast}\frac{G m}{r} dm = \frac{1}{2}\int_0^{M_\ast}{\mit\Psi} dm=-3\int_0^R p dV
$">|; 

$key = q/displaystylefrac{D{calE}}{Dt}=-frac{p}{rho},nablacdot{bfv}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5438.png"
 ALT="$\displaystyle \frac{D{\cal E}}{Dt} =-\frac{p}{\rho} \nabla\cdot{\bf v}.$">|; 

$key = q/displaystylep(a,theta,t)=p_0-left.frac{1}{2},rho,v^{,2}right|_{r=a}+rholeft.frac{partialt}right|_{r=a}-rholeft.{bfV}cdotnablaphiright|_{r=a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="448" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1809.png"
 ALT="$\displaystyle p(a,\theta,t) = p_0 -\left.\frac{1}{2} \rho v^{ 2}\right\vert ...
...ial t}\right\vert _{r=a} - \rho\left.{\bf V}\cdot \nabla\phi\right\vert _{r=a}.$">|; 

$key = q/displaystyle{rmMa}_{,e}=sqrt{left(frac{2}{gamma-1}right)left[left(frac{p_0}{p_{rmatm}}right)^{(gamma-1)slashgamma}-1right]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="91" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5425.png"
 ALT="$\displaystyle {\rm Ma}_{ e} =\sqrt{ \left(\frac{2}{\gamma-1}\right) \left[\left(\frac{p_0}{p_{\rm atm}}\right)^{(\gamma-1)/\gamma}-1\right]}.$">|; 

$key = q/displaystyleg=frac{G,m}{a^{,2}}=frac{4pi}{3},G,rho,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4370.png"
 ALT="$\displaystyle g = \frac{G m}{a^{ 2}}= \frac{4\pi}{3} G \rho a$">|; 

$key = q/displaystyleu';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5943.png"
 ALT="$\displaystyle u'$">|; 

$key = q/{bfp};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6389.png"
 ALT="$ {\bf p}$">|; 

$key = q/beta>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img954.png"
 ALT="$ \beta&gt;0$">|; 

$key = q/{mitPsi}=g,z=g,r,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3792.png"
 ALT="$ {\mit\Psi} = g z=g r \cos\theta$">|; 

$key = q/displaystyleint_{A_0}x,delta(x,y),dx,dy=int_{A_0}y,delta(x,y),dx,dy=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img632.png"
 ALT="$\displaystyle \int_{A_0} x  \delta(x,y) dx dy= \int_{A_0}y \delta(x,y) dx dy = 0.$">|; 

$key = q/{rmMa}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5622.png"
 ALT="$ {\rm Ma}(x)$">|; 

$key = q/displaystyle{bfv}times;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5866.png"
 ALT="$\displaystyle {\bf v}\times$">|; 

$key = q/displaystyle{rmFr}_2lesseqgtr1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1496.png"
 ALT="$\displaystyle {\rm Fr}_2\lesseqgtr 1.$">|; 

$key = q/displaystyle{calS}=frac{c_V}{M}lnleft(frac{p}{rho^{,gamma}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$\displaystyle {\cal S} = \frac{c_V}{M}\ln\left(\frac{p}{\rho^{ \gamma}}\right).$">|; 

$key = q/displaystyle=|{bfa}|^{,2},|{bfb}|^{,2}-|{bfa}|^{,2},|{bfb}|^{,2},cos^2theta=|{bfa}|^{,2},|{bfb}|^{,2},sin^2theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6376.png"
 ALT="$\displaystyle = \vert{\bf a}\vert^{ 2}  \vert{\bf b}\vert^{ 2} - \vert{\bf a...
...\cos^2\theta = \vert{\bf a}\vert^{ 2} \vert{\bf b}\vert^{ 2}  \sin^2\theta.$">|; 

$key = q/(2slash5),M,a^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7202.png"
 ALT="$ (2/5) M a^{ 2}$">|; 

$key = q/displaystyle=2,c,x,y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1902.png"
 ALT="$\displaystyle = 2 c x y,$">|; 

$key = q/displaystyleomega^{,2}=frac{G,(M+M')}{R^{,3}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img973.png"
 ALT="$\displaystyle \omega^{ 2} = \frac{G (M+M')}{R^{ 3}}.$">|; 

$key = q/displaystyle=-frac{nablap}{rho}-nabla{mitPsi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="$\displaystyle =- \frac{\nabla p}{\rho} - \nabla{\mit\Psi}.$">|; 

$key = q/displaystylep(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5150.png"
 ALT="$\displaystyle p(z)$">|; 

$key = q/{mitPi}_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4607.png"
 ALT="$ {\mit\Pi}_E$">|; 

$key = q/DHslashDt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img552.png"
 ALT="$ DH/Dt = 0$">|; 

$key = q/displaystyle=frac{4}{sqrt{{rmMa}_1^{,2}-1}}left[,overline{left(frac{dh}{dx}right)^2}+alpha_0^{,2}+overline{alpha_C^{,2}}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5860.png"
 ALT="$\displaystyle =\frac{4}{\sqrt{{\rm Ma}_1^{ 2}-1}}\left[  \overline{\left(\frac{dh}{dx}\right)^2}+\alpha_0^{ 2}+\overline{\alpha_C^{ 2}}\right],$">|; 

$key = q/{bfa}times{bfa}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6378.png"
 ALT="$ {\bf a}\times{\bf a} = {\bf0}$">|; 

$key = q/{bfa}={bfq}+lambda,{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6730.png"
 ALT="$ {\bf a}={\bf q} + \lambda {\bf r}$">|; 

$key = q/|nablavarphi|=1slash(r,sintheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3737.png"
 ALT="$ \vert\nabla\varphi\vert=1/(r \sin\theta)$">|; 

$key = q/displaystyledS_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7125.png"
 ALT="$\displaystyle dS_z$">|; 

$key = q/zeta(theta,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4519.png"
 ALT="$ \zeta(\theta,\phi,t)$">|; 

$key = q/displaystyle{rmMa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="$\displaystyle {\rm Ma}$">|; 

$key = q/x=a,theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3218.png"
 ALT="$ x=a \theta$">|; 

$key = q/displaystylep(a,theta,varphi,t)=p_0-frac{1}{16},rho,V_z^{,2}+frac{9}{16},rho,V_za)-rho,g,a,costheta+frac{1}{2},rho,a,frac{dV_z}{dt},costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="620" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2668.png"
 ALT="$\displaystyle p(a,\theta,\varphi,t) = p_0-\frac{1}{16} \rho V_z^{ 2} + \frac...
...ta) -\rho g a \cos\theta+ \frac{1}{2} \rho a \frac{dV_z}{dt} \cos\theta.$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialx^{,2}}+frac{1}{m^{,2}},frac{partial^{,2}phi}{partialy^{,2}}simeq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6004.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial x^{ 2}}+\frac{1}{m^{ 2}} \frac{\partial^{ 2}\phi}{\partial y^{ 2}}\simeq 0,$">|; 

$key = q/displaystylep(r,theta,phi)=p_0(r)+sum_{n=1,infty}left[p_n(r,theta,phi)-rho,{mitPhi}_n(r,theta,phi)right]-rho,mitPhi_{rmext}(r,theta,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="546" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4388.png"
 ALT="$\displaystyle p(r,\theta,\phi) = p_0(r) + \sum_{n=1,\infty}\left[p_n(r,\theta,\...
...ho {\mit\Phi}_n(r,\theta,\phi)\right]-\rho \mit\Phi_{\rm ext}(r,\theta,\phi),$">|; 

$key = q/displaystyle=frac{4,overline{alpha}}{sqrt{{rmMa}_1^{,2}-1}}=frac{4,alpha_0}{sqrt{{rmMa}_1^{,2}-1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5858.png"
 ALT="$\displaystyle = \frac{4 \overline{\alpha}}{\sqrt{{\rm Ma}_1^{ 2}-1}}=\frac{4 \alpha_0}{\sqrt{{\rm Ma}_1^{ 2}-1}},$">|; 

$key = q/displaystyle=3,omega,z,a.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img781.png"
 ALT="$\displaystyle = 3 \omega z a.$">|; 

$key = q/HH';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img642.png"
 ALT="$ HH'$">|; 

$key = q/displaystyle(rho-rho'),g+gamma,k^{,2}=frac{rho,k,(V-c)^2}{tanh(k,d)}+frac{rho',k,(V'-c)^2}{tanh(k,d')},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="396" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4171.png"
 ALT="$\displaystyle (\rho-\rho') g + \gamma k^{ 2} = \frac{\rho k (V-c)^2}{\tanh(k d)}+ \frac{\rho' k (V'-c)^2}{\tanh(k d')},$">|; 

$key = q/displaystyleA_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4896.png"
 ALT="$\displaystyle A_r$">|; 

$key = q/x_2leqxleqx_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5569.png"
 ALT="$ x_2\leq x\leq x_1$">|; 

$key = q/a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$ a)$">|; 

$key = q/displaystylev_psimeqleft(frac{k,gamma}{rho}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4164.png"
 ALT="$\displaystyle v_p \simeq \left(\frac{k \gamma}{\rho}\right)^{1/2},$">|; 

$key = q/-x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5549.png"
 ALT="$ -x$">|; 

$key = q/displaystylefrac{d{mitGamma}_C}{dt}=oint_Cfrac{d{bfv}}{dt}cdotd{bfr}+oint_C{bfv}cdotd{bfv}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1555.png"
 ALT="$\displaystyle \frac{d{\mit\Gamma}_C}{dt} = \oint_C \frac{d{\bf v}}{dt}\cdot d{\bf r} + \oint_C {\bf v}\cdot d{\bf v}.$">|; 

$key = q/bfr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6325.png"
 ALT="$ \bf r$">|; 

$key = q/r,dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2545.png"
 ALT="$ r d\theta$">|; 

$key = q/displaystylefrac{dz}{dx}=-h^{,-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7351.png"
 ALT="$\displaystyle \frac{dz}{dx} = -h^{ -1}.$">|; 

$key = q/displaystyle=-frac{c^{,2}}{rho},u,frac{partialrho}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5886.png"
 ALT="$\displaystyle = -\frac{c^{ 2}}{\rho} u \frac{\partial \rho}{\partial x},$">|; 

$key = q/eta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2107.png"
 ALT="$ \eta=0$">|; 

$key = q/displaystyleR;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6095.png"
 ALT="$\displaystyle R$">|; 

$key = q/displaystylefrac{dzeta''}{dzeta'}=frac{dz''}{dz'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2074.png"
 ALT="$\displaystyle \frac{d\zeta''}{d\zeta'} = \frac{dz''}{dz'}.$">|; 

$key = q/dT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6548.png"
 ALT="$ dT$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashcylinder1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1776.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/cylinder1.eps}}
\end{figure}">|; 

$key = q/displaystyleomega_varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2566.png"
 ALT="$\displaystyle \omega_\varphi$">|; 

$key = q/displaystylenablacdot{bfv}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="$\displaystyle \nabla\cdot{\bf v} = 0.$">|; 

$key = q/partialTslashpartialx_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="$ \partial T/\partial x_i$">|; 

$key = q/displaystyle_S,dS={bfL}+{bfD}+{bfF},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3437.png"
 ALT="$\displaystyle _S dS = {\bf L}+{\bf D}+{\bf F},$">|; 

$key = q/displaystyleoverline{nabla}cdotoverline{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="$\displaystyle \overline{\nabla}\cdot\overline{\bf v}$">|; 

$key = q/bar{F}=-F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2182.png"
 ALT="$ \bar{F}=-F$">|; 

$key = q/displaystyle=frac{4,q_1}{sqrt{{rmMa}_1^{,2}-1}}int_{x_L}^{x_R}left[left(frac{dh}{dx}right)^2+alpha^{,2}right]dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5849.png"
 ALT="$\displaystyle = \frac{4 q_1}{\sqrt{{\rm Ma}_1^{ 2}-1}}\int_{x_L}^{x_R}\left[\left(\frac{dh}{dx}\right)^2+\alpha^{ 2}\right]dx.$">|; 

$key = q/displaystyleepsilon_{ijk},sigma_{kj}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="$\displaystyle \epsilon_{ijk} \sigma_{kj} = 0$">|; 

$key = q/displaystyleDequivfrac{1}{sintheta}left(frac{partial}{partialtheta},sintheta,fra+frac{1}{sintheta},frac{partial^{,2}}{partialphi^{,2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4667.png"
 ALT="$\displaystyle D\equiv \frac{1}{\sin\theta}\left(\frac{\partial}{\partial\theta}...
...theta}+ \frac{1}{\sin\theta} \frac{\partial^{ 2}}{\partial\phi^{ 2}}\right).$">|; 

$key = q/dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2543.png"
 ALT="$ dz$">|; 

$key = q/displaystylew(phi)=sum_{nu=1,3,5,cdots}frac{nu,{mitGamma}_nu}{2,b},frac{sin(nu,phi)}{sinphi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3548.png"
 ALT="$\displaystyle w(\phi)= \sum_{\nu=1,3,5,\cdots} \frac{\nu {\mit\Gamma}_\nu}{2 b} \frac{\sin(\nu \phi)}{\sin\phi}.
$">|; 

$key = q/displaystyle(nabla{bfA})_{ij}=[({bfe}_jcdotnabla),{bfA}]_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7093.png"
 ALT="$\displaystyle (\nabla{\bf A})_{ij} = [({\bf e}_j\cdot\nabla) {\bf A}]_i.$">|; 

$key = q/dy);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2351.png"
 ALT="$ dy)$">|; 

$key = q/p_4slashp_1rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5652.png"
 ALT="$ p_4/p_1\rightarrow\infty$">|; 

$key = q/displaystylephantom{=}+left[frac{1}{r},frac{partial}{partialr},(r,A_theta)-frac{1}{r},frac{partialA_r}{partialtheta}right]{bfe}_phi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7182.png"
 ALT="$\displaystyle \phantom{=}+ \left[\frac{1}{r} \frac{\partial}{\partial r} (r A_\theta) - \frac{1}{r} \frac{\partial A_r}{\partial\theta}\right]{\bf e}_\phi, $">|; 

$key = q/phi=phi(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2612.png"
 ALT="$ \phi=\phi(r)$">|; 

$key = q/|{bfa}|=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6318.png"
 ALT="$ \vert{\bf a}\vert=0$">|; 

$key = q/displaystyleepsilonsimeqfrac{5}{4},frac{1}{291}simeqfrac{1}{233}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img916.png"
 ALT="$\displaystyle \epsilon \simeq \frac{5}{4} \frac{1}{291}\simeq \frac{1}{233}.$">|; 

$key = q/displaystyle=muleft(frac{1}{r},frac{partialv_r}{partialtheta}+frac{partialv_theta}{partialr}-frac{v_theta}{r}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="$\displaystyle =\mu\left(\frac{1}{r} \frac{\partial v_r}{\partial\theta} + \frac{\partial v_\theta}{\partial r}-\frac{v_\theta}{r}\right),$">|; 

$key = q/displaystylepsi(x,y)=-V_0left(r-frac{c^{,2}}{r}right)sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2136.png"
 ALT="$\displaystyle \psi(x,y) = - V_0\left(r -\frac{c^{ 2}}{r}\right)\sin\theta,$">|; 

$key = q/displaystylefrac{{mitDelta}p}{p_1}simeqleft(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right)theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5728.png"
 ALT="$\displaystyle \frac{{\mit\Delta}p}{p_1}\simeq \left(\frac{\gamma {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right)\theta,$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter02slashfig3.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img728.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter02/fig3.03.eps}}
\end{figure}">|; 

$key = q/delta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3532.png"
 ALT="$ \delta=0$">|; 

$key = q/displaystyle=y_C(x)+h(x)=y_C(x_L)-alpha_0,(x-x_L)-alpha_C(x)+h(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5834.png"
 ALT="$\displaystyle = y_C(x)+ h(x) = y_C(x_L) -\alpha_0 (x-x_L) -\alpha_C(x)+h(x),$">|; 

$key = q/CC'=v_2,deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1367.png"
 ALT="$ CC'=v_2 \delta t$">|; 

$key = q/overline{mu}ggmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3865.png"
 ALT="$ \overline{\mu}\gg \mu$">|; 

$key = q/displaystyle{bfa}cdot{bfa}times{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6368.png"
 ALT="$\displaystyle {\bf a}\cdot {\bf a}\times{\bf b}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter14slashnozzle.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5409.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter14/nozzle.eps}}
\end{figure}">|; 

$key = q/displaystyleI_1=pi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3541.png"
 ALT="$\displaystyle I_1=\pi,
$">|; 

$key = q/displaystylefrac{d}{dxi}left(frac{1}{coshxi},frac{dF}{dxi}right)-frac{2}{coshxi},F=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2760.png"
 ALT="$\displaystyle \frac{d}{d\xi}\left(\frac{1}{\cosh\xi} \frac{dF}{d\xi}\right) - \frac{2}{\cosh\xi} F=0,$">|; 

$key = q/displaystyle=p_0-rho,g,zeta+p_1(x,0_-,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4099.png"
 ALT="$\displaystyle = p_0-\rho g \zeta + p_1(x,0_-,t)$">|; 

$key = q/displaystyleh_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1890.png"
 ALT="$\displaystyle h_1$">|; 

$key = q/x=-l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3035.png"
 ALT="$ x=-l$">|; 

$key = q/theta=90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5780.png"
 ALT="$ \theta=90^\circ$">|; 

$key = q/displaystyle{mitDelta}=(a_1^{,2}+u)^{1slash2},(a_2^{,2}+u)^{1slash2},(a_3^{,2}+u)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7266.png"
 ALT="$\displaystyle {\mit\Delta} = (a_1^{ 2}+u)^{1/2} (a_2^{ 2}+u)^{1/2} (a_3^{ 2}+u)^{1/2}.$">|; 

$key = q/displaystyle=zeta_0,{rme}^{,{rmi},(sigma,t-{bfk}cdot{bfr})},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5079.png"
 ALT="$\displaystyle = \zeta_0 {\rm e}^{ {\rm i} (\sigma t-{\bf k}\cdot{\bf r})},$">|; 

$key = q/displaystylephi(r,theta)=-V,aleft[frac{r}{a}+frac{1}{2}left(frac{a}{r}right)^2right]costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2645.png"
 ALT="$\displaystyle \phi(r,\theta) = - V a\left[\frac{r}{a}+\frac{1}{2}\left(\frac{a}{r}\right)^2\right]\cos\theta,$">|; 

$key = q/(1-lambda^{,2})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2209.png"
 ALT="$ (1-\lambda^{ 2})^{1/2}$">|; 

$key = q/displaystyle=a_x,(a_y,b_z-a_z,b_y)+a_y,(a_z,b_x-a_x,b_z)+a_z,(a_x,b_y-a_y,b_x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="469" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6369.png"
 ALT="$\displaystyle = a_x (a_y  b_z-a_z  b_y) + a_y  (a_z  b_x- a_x  b_z) +a_z (a_x  b_y - a_y  b_x)$">|; 

$key = q/|dz'|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2068.png"
 ALT="$ \vert dz'\vert$">|; 

$key = q/displaystyleP_2(mu);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2588.png"
 ALT="$\displaystyle P_2(\mu)$">|; 

$key = q/displaystyle=rho_1left(frac{cosz}{cosz_1}right)^{2slash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6139.png"
 ALT="$\displaystyle =\rho_1\left(\frac{\cos z}{\cos z_1}\right)^{2/(\gamma-1)},$">|; 

$key = q/0.62;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1424.png"
 ALT="$ 0.62$">|; 

$key = q/displaystyleoverline{p}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="$\displaystyle \overline{p}_0$">|; 

$key = q/+(1slash2),zeta_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4568.png"
 ALT="$ +(1/2) \zeta_E$">|; 

$key = q/displaystyleD=frac{1}{4},rho,g,a^{,2},w.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3996.png"
 ALT="$\displaystyle D = \frac{1}{4} \rho g a^{ 2} w.$">|; 

$key = q/{bfA}={bfgrad},V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6552.png"
 ALT="$ {\bf A} = {\bf grad} V$">|; 

$key = q/beta_{-n,,n'}^{,-m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5020.png"
 ALT="$ \beta_{-n, n'}^{ -m, m'}$">|; 

$key = q/displaystylesigma_{ij}=-p,delta_{ij}+d_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="$\displaystyle \sigma_{ij} = - p \delta_{ij} + d_{ij},$">|; 

$key = q/displaystylebeta_{-r,,-s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4881.png"
 ALT="$\displaystyle \beta_{-r, -s}$">|; 

$key = q/displaystyle=-frac{2,n}{2,(2,n+3),(n+1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4429.png"
 ALT="$\displaystyle = -\frac{2 n}{2 (2 n+3) (n+1)}.$">|; 

$key = q/intnablatimes{bfA}cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6664.png"
 ALT="$ \int \nabla\times {\bf A}\cdot d{\bf S}$">|; 

$key = q/displaystylefrac{partialv_x}{partialx}+frac{partialv_y}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2860.png"
 ALT="$\displaystyle \frac{\partial v_x}{\partial x} +\frac{\partial v_y}{\partial y}$">|; 

$key = q/T(x,,y,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6538.png"
 ALT="$ T(x, y, z)$">|; 

$key = q/displaystyle={mitPsi}_r'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4839.png"
 ALT="$\displaystyle = {\mit\Psi}_r''$">|; 

$key = q/({bfA}cdotnabla),{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7134.png"
 ALT="$ ({\bf A}\cdot\nabla) {\bf A}$">|; 

$key = q/displaystyle{mitGamma}(z)={mitGamma}(z+dz)+I(z),dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3468.png"
 ALT="$\displaystyle {\mit\Gamma}(z) = {\mit\Gamma}(z+dz) + I(z) dz,$">|; 

$key = q/z_1<z_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1241.png"
 ALT="$ z_1&lt;z_2$">|; 

$key = q/displaystyleJ=4pi,a_1,a_2,a_3^{,2}int_0^{pislash2}frac{sintheta,sec^2theta,dthet^2theta)^{1slash2},(a_2^{,2}+a_3^{,2},tan^2theta)^{1slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="489" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7263.png"
 ALT="$\displaystyle J = 4\pi a_1 a_2 a_3^{ 2}\int_0^{\pi/2}\frac{\sin\theta \sec...
...2}+a_3^{ 2} \tan^2\theta)^{1/2}  (a_2^{ 2}+a_3^{ 2} \tan^2\theta)^{1/2}}.$">|; 

$key = q/displaystylefrac{p}{p_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5403.png"
 ALT="$\displaystyle \frac{p}{p_1}$">|; 

$key = q/displaystylefrac{dlambda}{(lambda-cosalpha)^3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2308.png"
 ALT="$\displaystyle \frac{d\lambda}{(\lambda-\cos\alpha)^3}$">|; 

$key = q/displaystyle=-frac{g}{a,sintheta},frac{partial}{partialphi}left[zeta-(1+k_2-h_2)ar{zeta}_2-sum_{n=1,infty}(1+k_n'-h_n'),alpha_n,zeta_nright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="536" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4510.png"
 ALT="$\displaystyle = - \frac{g}{a \sin\theta} \frac{\partial}{\partial\phi}\left[\...
...skew{5}\bar{\zeta}_2 -\sum_{n=1,\infty}(1+k_n'-h_n') \alpha_n \zeta_n\right].$">|; 

$key = q/displaystyleskew{3}bar{C}_V=m,k.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5303.png"
 ALT="$\displaystyle \skew{3}\bar{C}_V = m k.$">|; 

$key = q/v_g=v_pslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4029.png"
 ALT="$ v_g=v_p/2$">|; 

$key = q/F(z)=-V_0,z,{rme}^{-{rmi},theta_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1975.png"
 ALT="$ F(z)=-V_0 z {\rm e}^{-{\rm i} \theta_0}$">|; 

$key = q/displaystylev_x,frac{partialv_y}{partialx}+v_y,frac{partialv_y}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2863.png"
 ALT="$\displaystyle v_x \frac{\partial v_y}{\partial x} + v_y \frac{\partial v_y}{\partial y}$">|; 

$key = q/displaystylea_{ij}'={calR}_{ip},{calR}_{jq},a_{pq}=a_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6899.png"
 ALT="$\displaystyle a_{ij}' = {\cal R}_{ip} {\cal R}_{jq} a_{pq} = a_{ij}$">|; 

$key = q/{calO}({rmRe}^{-1slash2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3096.png"
 ALT="$ {\cal O}({\rm Re}^{-1/2})$">|; 

$key = q/omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4229.png"
 ALT="$ \omega_1$">|; 

$key = q/{bfv}({bfr},t)cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="$ {\bf v}({\bf r}, t)\cdot d{\bf S}$">|; 

$key = q/displaystyle{bfA}=rhoint_S({bfV}+{bfv}_{mitSigma}+{bfv}_S)times;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3436.png"
 ALT="$\displaystyle {\bf A} = \rho\int_S({\bf V}+{\bf v}_{\mit\Sigma}+{\bf v}_S)\times$">|; 

$key = q/u_c=1.1997;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1281.png"
 ALT="$ u_c=1.1997$">|; 

$key = q/displaystylecos^2beta=1-sin^2betasimeqfrac{{rmMa}_1^{,2}-1}{{rmMa}_1^{,2}}left(1-frac{2,epsilon}{sqrt{{rmMa}_1^{,2}-1}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5740.png"
 ALT="$\displaystyle \cos^2\beta =1-\sin^2\beta\simeq \frac{{\rm Ma}_1^{ 2}-1}{{\rm Ma}_1^{ 2}}\left(1-\frac{2 \epsilon}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right),$">|; 

$key = q/V_0=s,V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img602.png"
 ALT="$ V_0=s V$">|; 

$key = q/x=coshxi,coseta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7221.png"
 ALT="$ x=\cosh\xi \cos\eta$">|; 

$key = q/displaystyle{bfa}cdot{bfb}=a_x^{,2}+a_y^{,2}+a_z^{,2}=|{bfa}|^{,2}=a^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6311.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b} = a_x^{ 2}+a_y^{ 2} + a_z^{ 2} = \vert{\bf a}\vert^{ 2}=a^{ 2}.$">|; 

$key = q/{rmPr}ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="$ {\rm Pr}\ll 1$">|; 

$key = q/V_p=[2slash(gamma-1)],c_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5564.png"
 ALT="$ V_p= [2/(\gamma-1)] c_1$">|; 

$key = q/displaystylenablatimes(nablaphi)equiv{bf0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6669.png"
 ALT="$\displaystyle \nabla\times(\nabla\phi)\equiv {\bf0}.$">|; 

$key = q/2pi,beta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1751.png"
 ALT="$ 2\pi \beta_0$">|; 

$key = q/(2tmspace+thinmuskip{.1667em}a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$ (2\tmspace +\thinmuskip {.1667em}a$">|; 

$key = q/displaystylefrac{bar{F}}{k,V}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2268.png"
 ALT="$\displaystyle \frac{\bar{F}}{k V}= 1.$">|; 

$key = q/displaystylev_t(theta)=v_theta(a,theta)=left.frac{partialpsi}{partialr}right|_{r=a}=-V_0,(gamma+2,sintheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1772.png"
 ALT="$\displaystyle v_t(\theta) = v_\theta(a,\theta) = \left.\frac{\partial\psi}{\partial r}\right\vert _{r=a} = -V_0 (\gamma+2 \sin\theta).$">|; 

$key = q/displaystyle,dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1543.png"
 ALT="$\displaystyle  dV,$">|; 

$key = q/e_{13}=0.8759;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1035.png"
 ALT="$ e_{13}= 0.8759$">|; 

$key = q/d(0)=d_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3691.png"
 ALT="$ d(0)=d_1$">|; 

$key = q/b=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2158.png"
 ALT="$ b=0$">|; 

$key = q/displaystyle=alpha,(1-k^{,2},sin^2phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1319.png"
 ALT="$\displaystyle =\alpha (1-k^{ 2} \sin^2\phi),$">|; 

$key = q/v_-<v_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1436.png"
 ALT="$ v_-&lt;v_c$">|; 

$key = q/displaystyleP_l(mu)=frac{(-1)^{,l}}{2^{,l},l!},frac{d^{,l}}{dmu^{,l}}left(1-mu^{,2}right)^l.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2583.png"
 ALT="$\displaystyle P_l(\mu)= \frac{(-1)^{ l}}{2^{ l} l!} \frac{d^{ l}}{d\mu^{ l}}\left(1-\mu^{ 2}\right)^l.$">|; 

$key = q/displaystylep(a,theta,varphi)=p_0-frac{1}{16},rho,V^{,2}+frac{9}{16},rho,V^{,2},cos(2,theta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="365" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2653.png"
 ALT="$\displaystyle p(a,\theta,\varphi) = p_0-\frac{1}{16} \rho V^{ 2} + \frac{9}{16} \rho V^{ 2} \cos(2 \theta).$">|; 

$key = q/{bfa}equiv(0,,1,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6294.png"
 ALT="$ {\bf a} \equiv (0, 1, 0)$">|; 

$key = q/displaystyle={calR},rho_{1,2},T_{1,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5470.png"
 ALT="$\displaystyle = {\cal R} \rho_{1,2} T_{1,2},$">|; 

$key = q/displaystyleC_p=frac{C_{p_0}}{sqrt{1-{rmMa}_infty^{,2}}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6035.png"
 ALT="$\displaystyle C_p = \frac{C_{p_0}}{\sqrt{1-{\rm Ma}_\infty^{ 2}}}.$">|; 

$key = q/displaystyleG(u_0)=u_0-frac{1}{tanhu_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1272.png"
 ALT="$\displaystyle G(u_0) = u_0-\frac{1}{\tanh u_0},$">|; 

$key = q/displaystylew(z)=-frac{1}{4pi}int_{-bslash2}^{bslash2}frac{I(z'),dz'}{z-z'}=frac{1}{4pi}intfrac{d{mitGamma}(z')}{z-z'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3480.png"
 ALT="$\displaystyle w(z) =- \frac{1}{4\pi}\int_{-b/2}^{b/2} \frac{I(z') dz'}{z-z'}= \frac{1}{4\pi}\int\frac{d{\mit\Gamma}(z')}{z-z'},$">|; 

$key = q/displaystyleI_{n,,n'}^{,m,,m'}=int_{-1}^1P_n^{,m}(mu),P_{n'}^{,m'}(mu),dmu.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4966.png"
 ALT="$\displaystyle I_{n, n'}^{ m, m'} = \int_{-1}^1 P_n^{ m}(\mu) P_{n'}^{ m'}(\mu) d\mu.$">|; 

$key = q/displaystyleD,{mitPsi}_r''&=-frac{1}{sintheta}left[frac{partial}{partialtheta}leeft(costheta,frac{partial{mitPsi}_r}{partialphi}right)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="524" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4854.png"
 ALT="$\displaystyle D {\mit\Psi}_r''&amp;= - \frac{1}{\sin\theta}\left[\frac{\partial}{\...
...hi}\left( \cos\theta \frac{\partial{\mit\Psi}_r}{\partial \phi}\right)\right],$">|; 

$key = q/displaystylefrac{d^{,2}Y}{dt^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1848.png"
 ALT="$\displaystyle \frac{d^{ 2}Y}{dt^{ 2}}$">|; 

$key = q/y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2055.png"
 ALT="$ y)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter07slashdipole1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img2633.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter07/dipole1.eps}}
\end{figure}">|; 

$key = q/displaystylemu=sin^{-1}left(frac{1}{{rmMa}_1}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5696.png"
 ALT="$\displaystyle \mu = \sin^{-1}\left(\frac{1}{{\rm Ma}_1}\right)$">|; 

$key = q/gammaslash(gamma-1)simeq3.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5154.png"
 ALT="$ \gamma/(\gamma-1)\simeq 3.5$">|; 

$key = q/displaystyle=frac{partialv_x}{partialx}+frac{partialv_y}{partialy}+frac{partialv_z}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="$\displaystyle =\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z},$">|; 

$key = q/displaystyledT={bfgrad},Tcdotd{bfr}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6546.png"
 ALT="$\displaystyle dT = {\bf grad} T \cdot d{\bf r} = 0.$">|; 

$key = q/sigma_{12}=-(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$ \sigma_{12} = -($">|; 

$key = q/displaystyledM=x,dY-y,dX=P,(x,dx+y,dy).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2387.png"
 ALT="$\displaystyle dM = x dY-y dX = P (x dx+y dy).$">|; 

$key = q/v_x(eta)proptof'(eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2967.png"
 ALT="$ v_x(\eta)\propto f'(\eta)$">|; 

$key = q/displaystyleomega=sqrt{frac{g}{delta_0}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img618.png"
 ALT="$\displaystyle \omega = \sqrt{\frac{g}{\delta_0}}.$">|; 

$key = q/displaystyleM,(skew{3}ddot{r}-omega^{,2},r_G)=-M_0,omega^{,2},r_H,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img827.png"
 ALT="$\displaystyle M (\skew{3}\ddot{r} - \omega^{ 2} r_G) = -M_0 \omega^{ 2} r_H,$">|; 

$key = q/nrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5758.png"
 ALT="$ n\rightarrow\infty$">|; 

$key = q/delta(x,-y)=delta(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img630.png"
 ALT="$ \delta(x,-y)=\delta(x,y)$">|; 

$key = q/y_U(x_R)=y_C(x_R)=y_L(x_R);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5822.png"
 ALT="$ y_U(x_R)=y_C(x_R)=y_L(x_R)$">|; 

$key = q/{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2185.png"
 ALT="$ {\mit\Omega}$">|; 

$key = q/displaystylephantom{simeqfrac{G,m',a^{2}}{R^{,3}}}-frac{1}{4},{calR}_2^{,(-2)}left(r,theta,varphi-omega_1,tright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4307.png"
 ALT="$\displaystyle \phantom{\simeq \frac{G m' a^{2}}{R^{ 3}}} - \frac{1}{4} {\cal R}_2^{ (-2)}\left(r,\theta,\varphi-\omega_1 t\right)$">|; 

$key = q/theta=theta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1693.png"
 ALT="$ \theta=\theta_0$">|; 

$key = q/displaystyleleft(frac{partialrho}{partialt}+v_j,frac{partialrho}{partialx_j}+rhoi}{partialx_j}right)=F_i+frac{partialsigma_{ij}}{partialx_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="493" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="$\displaystyle \left(\frac{\partial\rho}{\partial t} + v_j \frac{\partial\rho}{...
...tial v_i}{\partial x_j}\right)= F_i + \frac{\partial\sigma_{ij}}{\partial x_j}.$">|; 

$key = q/displaystyleH=frac{1}{2},v^{,2}+frac{gamma}{gamma-1},frac{p}{rho}+{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img551.png"
 ALT="$\displaystyle H = \frac{1}{2} v^{ 2} + \frac{\gamma}{\gamma-1} \frac{p}{\rho} + {\mit\Psi}
$">|; 

$key = q/gamma=2.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$ \gamma =2.5$">|; 

$key = q/partiala_{ij}slashpartialx_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6880.png"
 ALT="$ \partial a_{ij}/\partial x_k$">|; 

$key = q/displaystyle=left(frac{gamma+1}{gamma-1}right)^{1slash2}tan^{-1}left(left[left(fright)-tan^{-1}left(left[{rmMa}^{,2}-1right]^{1slash2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="612" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5773.png"
 ALT="$\displaystyle = \left(\frac{\gamma+1}{\gamma-1}\right)^{1/2}\tan^{-1}\left(\lef...
...\right]^{1/2}\right)-\tan^{-1}\left(\left[{\rm Ma}^{ 2}-1\right]^{1/2}\right).$">|; 

$key = q/displaystyleX-{rmi},Y=frac{1}{2},{rmi},rhooint_Cleft|frac{dF}{dz}right|^{,2}dskew{2}bar{z}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2401.png"
 ALT="$\displaystyle X - {\rm i} Y = \frac{1}{2} {\rm i} \rho\oint_C \left\vert\frac{dF}{dz}\right\vert^{ 2}d\skew{2}\bar{z}.$">|; 

$key = q/u_3(x_1,x_2,x_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7028.png"
 ALT="$ u_3(x_1, x_2, x_3)$">|; 

$key = q/uslashc={rmMa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5412.png"
 ALT="$ u/c={\rm Ma}$">|; 

$key = q/displaystylec=c_0left(frac{rho}{rho_0}right)^{,(gamma-1)slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5366.png"
 ALT="$\displaystyle c = c_0\left(\frac{\rho}{\rho_0}\right)^{ (\gamma-1)/2}.$">|; 

$key = q/{bfa}times{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6363.png"
 ALT="$ {\bf a}\times{\bf b}$">|; 

$key = q/gammasimeq2times10^{-2},{rmN,m^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1084.png"
 ALT="$ \gamma\simeq 2\times 10^{-2} {\rm N m^{-1}}$">|; 

$key = q/displaystyleI=int_a^bF(y,y',x),dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7307.png"
 ALT="$\displaystyle I = \int_a^b F(y, y', x) dx,$">|; 

$key = q/displaystyle=epsilon_{lmn}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6839.png"
 ALT="$\displaystyle =\epsilon_{lmn}.$">|; 

$key = q/displaystyle=-frac{1}{rho},frac{partialp}{partialx}-frac{partial{mitPsi}}{partiaa^{,2}v_x+frac{1}{3},frac{partial{mitDelta}}{partialx}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="$\displaystyle = - \frac{1}{\rho} \frac{\partial p}{\partial x} - \frac{\partia...
...\nabla^{ 2} v_x + \frac{1}{3} \frac{\partial{\mit\Delta}}{\partial x}\right),$">|; 

$key = q/{mitPhi}_{mitTheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="$ {\mit\Phi}_{\mit\Theta}$">|; 

$key = q/z_0+dz';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2064.png"
 ALT="$ z_0+dz'$">|; 

$key = q/nablatimes{bfv}=(0,,0,,2,omega);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6657.png"
 ALT="$ \nabla\times{\bf v} = (0 ,0 ,2 \omega)$">|; 

$key = q/displaystylefrac{p-p_{rmatm}}{skew{3}bar{rho}}+{mitPhi}+chi=P(theta,phi,t)+{calO}(d,{mitOmega},v),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4494.png"
 ALT="$\displaystyle \frac{p-p_{\rm atm}}{\skew{3}\bar{\rho}}+{\mit\Phi}+\chi = P(\theta,\phi,t) + {\cal O}(d {\mit\Omega} v),$">|; 

$key = q/displaystylefrac{d}{dt}left({bfa}times{bfb}right)=frac{d{bfa}}{dt}times{bfb}+{bfa}timesfrac{d{bfb}}{dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6440.png"
 ALT="$\displaystyle \frac{d}{dt}\left({\bf a}\times{\bf b}\right) = \frac{d{\bf a}}{dt}\times{\bf b} + {\bf a}\times \frac{d{\bf b}}{dt}.$">|; 

$key = q/MllM';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1021.png"
 ALT="$ M\ll M'$">|; 

$key = q/{bfV}=-V,cosalpha,{bfe}_x+V,sinalpha,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3275.png"
 ALT="$ {\bf V} = -V \cos\alpha {\bf e}_x+V \sin\alpha {\bf e}_y$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter13slashfig4.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img5255.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter13/fig4.05.eps}}
\end{figure}">|; 

$key = q/displaystylenabla_{x'}=costheta,nabla_x+sintheta,nabla_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6572.png"
 ALT="$\displaystyle \nabla_{x'} = \cos\theta  \nabla_x + \sin\theta  \nabla_y.$">|; 

$key = q/displaystylembox{hspace{1cm}as{rslasharightarrowinfty{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1765.png"
 ALT="$\displaystyle \mbox{\hspace{1cm}as $r/a\rightarrow\infty$}$">|; 

$key = q/displaystyleomega=(g,k)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3926.png"
 ALT="$\displaystyle \omega = (g k)^{1/2},$">|; 

$key = q/displaystyle{bfA}=frac{1}{2},rhoint_S{bfU}times({bfn}times{bfU}),dS.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3435.png"
 ALT="$\displaystyle {\bf A} = \frac{1}{2} \rho\int_S {\bf U}\times({\bf n}\times {\bf U}) dS.$">|; 

$key = q/V=v_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3990.png"
 ALT="$ V=v_p$">|; 

$key = q/displaystylem,frac{dV_x}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1842.png"
 ALT="$\displaystyle m \frac{dV_x}{dt}$">|; 

$key = q/displaystyle=frac{1}{2},(3,x^{,2}-1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4243.png"
 ALT="$\displaystyle = \frac{1}{2} (3 x^{ 2}-1).$">|; 

$key = q/displaystylesigma_{varphivarphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img496.png"
 ALT="$\displaystyle \sigma_{\varphi\varphi}$">|; 

$key = q/displaystyled_{ij}=alpha,frac{partialv_k}{partialx_k},delta_{ij}+beta,frac{partialv_i}{partialx_j}+gamma,frac{partialv_j}{partialx_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="$\displaystyle d_{ij} = \alpha \frac{\partial v_k}{\partial x_k} \delta_{ij} +...
... \frac{\partial v_i}{\partial x_j} +\gamma \frac{\partial v_j}{\partial x_i}.$">|; 

$key = q/displaystyle=-int_{mitOmega}costhetaleft(frac{partial{mitPsi}_r}{partialtheta},ftialphi},frac{partial{mitPhi}_s}{partialphi}right)d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4873.png"
 ALT="$\displaystyle = -\int_{\mit\Omega}\cos\theta\left(\frac{\partial{\mit\Psi}_r}{\...
...r}{\partial\phi} \frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}$">|; 

$key = q/displaystyle)=-nabla^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3742.png"
 ALT="$\displaystyle ) = -\nabla^{ 2}$">|; 

$key = q/displaystylez=d+zeta_b(theta,phi,t)+zeta(theta,phi,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4468.png"
 ALT="$\displaystyle z = d + \zeta_b(\theta,\phi,t)+\zeta(\theta,\phi,t),$">|; 

$key = q/e=0.92995;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1078.png"
 ALT="$ e=0.92995$">|; 

$key = q/displaystylez+{rmi},varpi=c,cosh(xi+{rmi},eta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2795.png"
 ALT="$\displaystyle z+{\rm i} \varpi = c \cosh(\xi+{\rm i} \eta),$">|; 

$key = q/-|theta_2|<eta<-|theta_{rmmax}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6155.png"
 ALT="$ -\vert\theta_2\vert&lt; \eta&lt;-\vert\theta_{\rm max}\vert$">|; 

$key = q/displaystyleu=frac{t^{,2}}{36,s^{,2}}+left[(1-s)-frac{t^{,2}}{12,s}right]^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img753.png"
 ALT="$\displaystyle u = \frac{t^{ 2}}{36 s^{ 2}} + \left[(1-s)-\frac{t^{ 2}}{12 s}\right]^{ 2},$">|; 

$key = q/displaystyle(1-{rmMa}_infty^{,2}),frac{partial^{,2}phi}{partialx^{,2}}+frac{partial^{,2}phi}{partialy^{,2}}simeq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5954.png"
 ALT="$\displaystyle (1-{\rm Ma}_\infty^{ 2}) \frac{\partial^{ 2} \phi}{\partial x^{ 2}} + \frac{\partial^{ 2}\phi}{\partial y^{ 2}}\simeq 0,$">|; 

$key = q/displaystyle{mitPsi}=-frac{1}{2},G,rholeft(J-sum_{i=1,3},A_i,x_i^{,2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7255.png"
 ALT="$\displaystyle {\mit\Psi} = - \frac{1}{2} G \rho\left(J - \sum_{i=1,3} A_i x_i^{ 2}\right),$">|; 

$key = q/13.5,{rmmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1148.png"
 ALT="$ 13.5 {\rm mm}$">|; 

$key = q/v=V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2239.png"
 ALT="$ v=V$">|; 

$key = q/AP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2829.png"
 ALT="$ AP$">|; 

$key = q/displaystyleb=frac{a}{cosh(k,d)-(gslashk,V^{,2}),sinh(k,d)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4127.png"
 ALT="$\displaystyle b = \frac{a}{\cosh(k d) - (g/k V^{ 2}) \sinh(k d)},$">|; 

$key = q/rho_c=rho(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5199.png"
 ALT="$ \rho_c= \rho(0)$">|; 

$key = q/l>gslashomega^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1058.png"
 ALT="$ l&gt;g/\omega^{ 2}$">|; 

$key = q/displaystyle=mu_sint_{mitOmega}{mitPsi}_r',{mitPsi}_s,d{mitOmega}=-int_{mitOmega}{mitPsi}_r',D,{mitPsi}_s,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4882.png"
 ALT="$\displaystyle =\mu_s\int_{\mit\Omega} {\mit\Psi}_r' {\mit\Psi}_s d{\mit\Omega} = -\int_{\mit\Omega} {\mit\Psi}_r' D {\mit\Psi}_s d{\mit\Omega}$">|; 

$key = q/displaystyle=-int_{mitOmega}costhetaleft(xi,frac{partial{mitPsi}_r}{partialthetaintheta},frac{partial{mitPsi}_r}{partialphi}right)d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4906.png"
 ALT="$\displaystyle = -\int_{\mit\Omega}\cos\theta\left(\xi \frac{\partial{\mit\Psi}...
...eta}{\sin\theta} \frac{\partial{\mit\Psi}_r}{\partial\phi}\right)d{\mit\Omega}$">|; 

$key = q/displaystyle=n,m,{bfa}=m,(n,{bfa}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6247.png"
 ALT="$\displaystyle = n m {\bf a} = m (n {\bf a}),$">|; 

$key = q/displaystyleint_{OAP}{bfv}cdotd{bfr}=int_{OBP}{bfv}cdotd{bfr}=-phi_P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1576.png"
 ALT="$\displaystyle \int_{OAP} {\bf v}\cdot d{\bf r} =\int_{OBP} {\bf v}\cdot d{\bf r} = -\phi_P$">|; 

$key = q/displaystyle{rmPr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="$\displaystyle {\rm Pr}$">|; 

$key = q/displaystylefrac{z}{k}=int_0^{bar{F}slashk,V}frac{du}{-{rmi},u^{,-1slash2}+(1-1shk,V}left[,{rmi},u^{,-1slash2}+(1-1slashu)^{1slash2}right]du,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="602" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2265.png"
 ALT="$\displaystyle \frac{z}{k}=\int_0^{\bar{F}/k V}\frac{du}{-{\rm i} u^{ -1/2}+(...
...{1/2}}=\int_0^{\bar{F}/k V}\left[ {\rm i} u^{ -1/2}+(1-1/u)^{1/2}\right]du,$">|; 

$key = q/bggc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3447.png"
 ALT="$ b\gg c$">|; 

$key = q/k=0.95;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$ k=0.95$">|; 

$key = q/e=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2783.png"
 ALT="$ e=0$">|; 

$key = q/displaystyleK=frac{1}{2},I_{33},omega^{,2}=frac{1}{10},M,(a_1^{,2}+a_2^{,2}),omega^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img854.png"
 ALT="$\displaystyle K =\frac{1}{2} I_{33} \omega^{ 2}= \frac{1}{10} M (a_1^{ 2}+a_2^{ 2}) \omega^{ 2}.$">|; 

$key = q/displaystyleq_i=-kappa,frac{partialT}{partialx_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="$\displaystyle q_i = -\kappa \frac{\partial T}{\partial x_i},$">|; 

$key = q/displaystyle=-nablap-rho,g,{bfe}_z+mu,nabla^{,2}{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3896.png"
 ALT="$\displaystyle = - \nabla p -\rho g {\bf e}_z + \mu \nabla^{ 2}{\bf v},$">|; 

$key = q/displaystyle=costheta,frac{partial{mitPsi}_r}{partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4850.png"
 ALT="$\displaystyle = \cos\theta \frac{\partial{\mit\Psi}_r}{\partial \theta},$">|; 

$key = q/displaystyleint_P^Qx,y^{,2},dl=int_0^1x^{,3},sqrt{2},dx=frac{sqrt{2}}{4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6448.png"
 ALT="$\displaystyle \int_P^Q x y^{ 2} dl = \int_0^1 x^{ 3} \sqrt{2}  dx = \frac{\sqrt{2}}{4}.$">|; 

$key = q/{calR}_n^{,(0)}(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4325.png"
 ALT="$ {\cal R}_n^{ (0)}(r,\theta)$">|; 

$key = q/{figure}{epsfysize{=1.25in{centerline{epsffile{AppendixAslashfigA.24.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="139" BORDER="0"
 SRC="|."$dir".q|img6599.png"
 ALT="\begin{figure}
\epsfysize =1.25in
\centerline{\epsffile{AppendixA/figA.24.eps}}
\end{figure}">|; 

$key = q/alpha=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1875.png"
 ALT="$ \alpha=1/2$">|; 

$key = q/f_xslashl^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1162.png"
 ALT="$ f_x/l^{ 2}$">|; 

$key = q/-g,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2665.png"
 ALT="$ -g {\bf e}_z$">|; 

$key = q/4,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2820.png"
 ALT="$ 4 a$">|; 

$key = q/m=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1743.png"
 ALT="$ m=0$">|; 

$key = q/displaystyle{calE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="$\displaystyle {\cal E}$">|; 

$key = q/partial^{,2}phislashpartialx_k,partialx_j=partial^{,2}phislashpartialx_j,partialx_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6887.png"
 ALT="$ \partial^{ 2}\phi/\partial x_k \partial x_j
= \partial^{ 2}\phi/\partial x_j \partial x_k$">|; 

$key = q/p_0=p(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3702.png"
 ALT="$ p_0=p(0)$">|; 

$key = q/displaystyleleft[int_{x_1}^{x_2}D(x,y),dxright]dy.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6488.png"
 ALT="$\displaystyle \left[\int_{x_1}^{x_2} D(x,y) dx\right]dy.$">|; 

$key = q/displaystylel=int_{-aslash2}^{aslash2}(1+y'^{,2})^{1slash2},dx=int_{-aslash2}^{aslash2}cosh(xslashh),dx=2,h,sinh(aslash2,h).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="519" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7360.png"
 ALT="$\displaystyle l= \int_{-a/2}^{a/2}(1+y'^{ 2})^{1/2} dx = \int_{-a/2}^{a/2} \cosh(x/h)  dx = 2 h \sinh(a/2 h).$">|; 

$key = q/displaystyle(rho-rho'),g+gamma,k^{,2}=(rho+rho'),k,c^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4172.png"
 ALT="$\displaystyle (\rho-\rho') g + \gamma k^{ 2} = (\rho+\rho') k c^{ 2}.$">|; 

$key = q/u_4=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5639.png"
 ALT="$ u_4=0$">|; 

$key = q/AQ_1P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2531.png"
 ALT="$ AQ_1 P$">|; 

$key = q/displaystyle=-frac{c^{,2}}{rho},v,frac{partialrho}{partialy}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5888.png"
 ALT="$\displaystyle = -\frac{c^{ 2}}{\rho} v \frac{\partial \rho}{\partial y}.$">|; 

$key = q/partialuslashpartialy=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3139.png"
 ALT="$ \partial u/\partial y=0$">|; 

$key = q/displaystyle=eta_1-left(frac{gamma+1}{gamma-1}right)^{1slash2}(z_2-z_1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6146.png"
 ALT="$\displaystyle = \eta_1-\left(\frac{\gamma+1}{\gamma-1}\right)^{1/2}(z_2-z_1).$">|; 

$key = q/displaystyle{calS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5586.png"
 ALT="$\displaystyle {\cal S}$">|; 

$key = q/T=2pislashsigma_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5031.png"
 ALT="$ T=2\pi/\sigma_j$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{Chapter08slashwedge.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="169" BORDER="0"
 SRC="|."$dir".q|img3224.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{Chapter08/wedge.eps}}
\end{figure}">|; 

$key = q/h_v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7216.png"
 ALT="$ h_v$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter03slashfig6.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="371" BORDER="0"
 SRC="|."$dir".q|img1166.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter03/fig6.04.eps}}
\end{figure}">|; 

$key = q/displaystylea_z=-left(frac{s-1}{s+1slash2}right)g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2675.png"
 ALT="$\displaystyle a_z =- \left(\frac{s-1}{s+1/2}\right)g,$">|; 

$key = q/displaystyle=a_{ijss},delta_{kl}+a_{isks},delta_{jl}+a_{sjks},delta_{il},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6945.png"
 ALT="$\displaystyle =a_{ijss} \delta_{kl}+a_{isks} \delta_{jl}+a_{sjks} \delta_{il},$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialx^{,2}}+frac{partial^{,2}phi}{partialy^{,2}}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1660.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial x^{ 2}}+\frac{\partial^{ 2}\phi}{\partial y^{ 2}} = 0.$">|; 

$key = q/y=pm(1slash27)^{1slash2},X_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4057.png"
 ALT="$ y=\pm (1/27)^{1/2} X_0$">|; 

$key = q/[(gamma-1)slashgamma],(T_0slashH);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5147.png"
 ALT="$ [(\gamma-1)/\gamma] (T_0/H)$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter04slashkelvin.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="254" BORDER="0"
 SRC="|."$dir".q|img1561.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter04/kelvin.eps}}
\end{figure}">|; 

$key = q/F_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$ F_i$">|; 

$key = q/displaystyle=|alpha|,(1-k^{,2},sin^2phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1334.png"
 ALT="$\displaystyle =\vert\alpha\vert (1-k^{ 2} \sin^2\phi),$">|; 

$key = q/displaystyleh=h_1+h_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1887.png"
 ALT="$\displaystyle h= h_1+h_2.$">|; 

$key = q/displaystyle={rmMa}_infty^{,2}left[(gamma+1),frac{u'}{U}+left(frac{gamma+1}{2}riright)left(frac{v'}{U}right)^2right]frac{partialu'}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="518" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5904.png"
 ALT="$\displaystyle = {\rm Ma}_\infty^{ 2}\left[(\gamma+1) \frac{u'}{U} +\left(\fra...
...ma-1}{2}\right)\left(\frac{v'}{U}\right)^2\right]\frac{\partial u'}{\partial x}$">|; 

$key = q/displaystyleint_{OAP}{bfv}cdotd{bfr}+int_{PBO}{bfv}cdotd{bfr}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1575.png"
 ALT="$\displaystyle \int_{OAP} {\bf v}\cdot d{\bf r} + \int_{PBO} {\bf v}\cdot d{\bf r} = 0,$">|; 

$key = q/({bfa}times{bfb})times({bfc}times{bfd})=({bfa}times{bfb}cdot{bfd}),{bfc}-({bfa}times{bfb}cdot{bfc}),{bfd};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6986.png"
 ALT="$ ({\bf a}\times {\bf b})\times ({\bf c}\times {\bf d})= ({\bf a}\times
{\bf b}\cdot{\bf d}) {\bf c} - ({\bf a}\times {\bf b}\cdot{\bf c}) {\bf d}$">|; 

$key = q/displaystyled=frac{M}{L,cosalpha}=frac{1}{4},(c-delta),frac{sinalpha}{widehat{mitGamma}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3323.png"
 ALT="$\displaystyle d = \frac{M}{L \cos\alpha} = \frac{1}{4} (c-\delta) \frac{\sin\alpha}{\widehat{\mit\Gamma}}$">|; 

$key = q/v_r(a,theta)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3844.png"
 ALT="$ v_r(a,\theta)=0$">|; 

$key = q/(nabla^{,2}{bfA})_i=nabla^{,2}A_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7103.png"
 ALT="$ (\nabla^{ 2} {\bf A})_i=\nabla^{ 2} A_i$">|; 

$key = q/F(Q);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6465.png"
 ALT="$ F(Q)$">|; 

$key = q/displaystylef=2,{mitOmega},costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5067.png"
 ALT="$\displaystyle f= 2 {\mit\Omega} \cos\theta
$">|; 

$key = q/displaystyle=-frac{f_j,n,(n+1),(n-m_j-1),(n-m_j)}{(2,n-3),(2,n-1),[m_j-n,(n-1),f_j]},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4701.png"
 ALT="$\displaystyle = - \frac{f_j n (n+1) (n-m_j-1) (n-m_j)}{(2 n-3) (2 n-1) [m_j-n (n-1) f_j]},$">|; 

$key = q/rho(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5173.png"
 ALT="$ \rho(r)$">|; 

$key = q/u_{ii}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6980.png"
 ALT="$ u_{ii}=0$">|; 

$key = q/(AB);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6315.png"
 ALT="$ (AB)$">|; 

$key = q/displaystyle=frac{c}{2},costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3309.png"
 ALT="$\displaystyle = \frac{c}{2} \cos\theta,$">|; 

$key = q/nabla^{,2}phi=nablacdotnablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7078.png"
 ALT="$ \nabla^{ 2}\phi = \nabla\cdot\nabla\phi$">|; 

$key = q/displaystyler,xi_{r,n}equivx_i,xi_{i,n}=(n,A_n+B_n),r^{,2},p_n+n,phi_n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4415.png"
 ALT="$\displaystyle r \xi_{r n}\equiv x_i \xi_{i n} = (n A_n+B_n) r^{ 2} p_n + n \phi_n,$">|; 

$key = q/displaystylefrac{Doverline{bfv}}{Doverline{t}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="$\displaystyle \frac{D\overline{\bf v}}{D\overline{t}}$">|; 

$key = q/displaystyleleft.frac{du_0}{dy}right|_{y=0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3137.png"
 ALT="$\displaystyle \left.\frac{du_0}{dy}\right\vert _{y=0}$">|; 

$key = q/displaystylefrac{partialu_1}{partialt}+frac{c_0^{,2}}{rho_0},frac{partialrho_1}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5356.png"
 ALT="$\displaystyle \frac{\partial u_1}{\partial t} + \frac{c_0^{ 2}}{\rho_0} \frac{\partial \rho_1}{\partial x}$">|; 

$key = q/displaystylefrac{T_2}{T_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5487.png"
 ALT="$\displaystyle \frac{T_2}{T_1}$">|; 

$key = q/v(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1521.png"
 ALT="$ v(x)$">|; 

$key = q/displaystylenabla^2phi=sum_{i=1,3}frac{1}{h},frac{partial}{partialu_i}!left(frac{h}{h_i^{,2}},frac{partialphi}{partialu_i}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7079.png"
 ALT="$\displaystyle \nabla^2\phi =\sum_{i=1,3}\frac{1}{h} \frac{\partial}{\partial u_i}\!\left(\frac{h}{h_i^{ 2}} \frac{\partial\phi}{\partial u_i}\right).$">|; 

$key = q/displaystylepi,a,h^{,2},rho,gleft(1+frac{omega^{,2},a^{,2}}{4,g,h}right)^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1062.png"
 ALT="$\displaystyle \pi a h^{ 2} \rho g\left(1+ \frac{\omega^{ 2} a^{ 2}}{4 g h}\right)^2.
$">|; 

$key = q/displaystyleint!int!intdV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6509.png"
 ALT="$\displaystyle \int\!\int\!\int dV$">|; 

$key = q/displaystyleint_P^QdT=int_P^Q{bfgrad},Tcdotd{bfr}=T(Q)-T(P).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6549.png"
 ALT="$\displaystyle \int_P^Q dT = \int_P^Q {\bf grad} T\cdot d{\bf r} = T(Q)-T(P).$">|; 

$key = q/|{bfr}-{bfk}|^{n},({bfr}-{bfk});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6756.png"
 ALT="$ \vert{\bf r} - {\bf k}\vert^{n} ({\bf r}-{\bf k})$">|; 

$key = q/displaystyle=p_0+rho,g,(h_2-z),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1473.png"
 ALT="$\displaystyle =p_0+\rho g (h_2-z),$">|; 

$key = q/x=(8slash27)^{1slash2},X_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4056.png"
 ALT="$ x=(8/27)^{1/2} X_0$">|; 

$key = q/displaystyle{bfa}ast{bfb}equiv(a_x,b_x,,a_y,b_y,,a_z,b_z).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6350.png"
 ALT="$\displaystyle {\bf a} \ast{\bf b} \equiv (a_x  b_x,  a_y  b_y,  a_z  b_z).$">|; 

$key = q/y(-aslash2)=y(+aslash2)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7341.png"
 ALT="$ y(-a/2) = y(+a/2) = 0$">|; 

$key = q/-2,lleqhleq2,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1231.png"
 ALT="$ -2 l\leq h\leq 2 l$">|; 

$key = q/displaystyle=a_1,cosbeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img927.png"
 ALT="$\displaystyle =a_1 \cos\beta,$">|; 

$key = q/displaystyleF(z)=alpha,tanh(alpha,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3257.png"
 ALT="$\displaystyle F(z)=\alpha \tanh(\alpha z)
$">|; 

$key = q/vartheta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5035.png"
 ALT="$ \vartheta=0$">|; 

$key = q/displaystyledS=mleft({calC}_p,frac{dv}{v}+{calC}_V,frac{dp}{p}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5323.png"
 ALT="$\displaystyle dS =m \left({\cal C}_p \frac{dv}{v}+ {\cal C}_V \frac{dp}{p}\right),$">|; 

$key = q/displaystylegammaoint_C{bft}times{bfn},dr=gammaint_Sleft[(nabla{bfn})cdot{bfn}-(nablacdot{bfn}),{bfn}right]dS.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1103.png"
 ALT="$\displaystyle \gamma\oint_C {\bf t}\times {\bf n} dr= \gamma\int_S \left[(\nabla {\bf n})\cdot {\bf n} - (\nabla\cdot {\bf n}) {\bf n}\right]dS.$">|; 

$key = q/displaystyle(2,n,A_n+B_n),a^{,2},p_n({bfr})+2,(n-1),phi_n({bfr})=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4420.png"
 ALT="$\displaystyle (2 n A_n+B_n) a^{ 2} p_n({\bf r}) + 2 (n-1) \phi_n({\bf r}) = 0.$">|; 

$key = q/displaystylefrac{dF}{dz}simeqV,{rme}^{,{rmi},alpha}+{rmi},frac{{mitGamma}}{2pi,z},{rme}^{,{rmi},alpha}-a^{,2},{rme}^{-{rmi},alpha})}{z^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3315.png"
 ALT="$\displaystyle \frac{dF}{dz}\simeq V {\rm e}^{ {\rm i} \alpha} + {\rm i} \fr...
...2} {\rm e}^{ {\rm i} \alpha}-a^{ 2} {\rm e}^{-{\rm i} \alpha})}{z^{ 2}}.$">|; 

$key = q/displaystyledelta_{ii};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6793.png"
 ALT="$\displaystyle \delta_{ii}$">|; 

$key = q/dz';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2077.png"
 ALT="$ dz'$">|; 

$key = q/{bfv}=u,{bfe}_x+v,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5072.png"
 ALT="$ {\bf v}= u {\bf e}_x+v {\bf e}_y$">|; 

$key = q/displaystylep=p_0-frac{1}{2},rholeft[left(frac{3}{2},G,M,alpha_1-3,lambdaright)x^{,2}+left(frac{3}{2},G,M,alpha_3+lambdaright)x_3^{,2}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="619" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img992.png"
 ALT="$\displaystyle p = p_0 -\frac{1}{2} \rho\left[\left(\frac{3}{2} G M \alpha_1...
..._2 x_2^{ 2}+\left(\frac{3}{2} G M \alpha_3+\lambda\right)x_3^{ 2}\right],$">|; 

$key = q/displaystyle{bfv}=nablavarphitimesnablapsi+{mitOmega}_varphi,nablavarphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2553.png"
 ALT="$\displaystyle {\bf v} = \nabla\varphi\times\nabla\psi + {\mit\Omega}_\varphi \nabla\varphi,$">|; 

$key = q/displaystylenabla^{,2}left(frac{partialpsi}{partialt}right)+beta,frac{partialpsi}{partialy}simeq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5109.png"
 ALT="$\displaystyle \nabla^{ 2}\left(\frac{\partial\psi}{\partial t}\right)+\beta \frac{\partial\psi}{\partial y} \simeq 0.
$">|; 

$key = q/displaystyleint_P^Qleft[f(x,y),dx+g(x,y),dyright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6455.png"
 ALT="$\displaystyle \int_P^Q \left[ f(x,y) dx + g(x,y) dy\right].$">|; 

$key = q/displaystylefrac{dp}{dz}=-frac{g,M}{R,T},p.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5121.png"
 ALT="$\displaystyle \frac{dp}{dz} = -\frac{g M}{R T} p.$">|; 

$key = q/cosh^{-1}(z)equivln[z+(z^{,2}-1)^{1slash2}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1230.png"
 ALT="$ \cosh^{-1}(z)\equiv \ln[z+(z^{ 2}-1)^{1/2}]$">|; 

$key = q/s<0.2113;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img802.png"
 ALT="$ s&lt;0.2113$">|; 

$key = q/displaystyle=frac{1}{r^{,2}},frac{partial}{partialr}!left(r^{,2},frac{partial}{p1}{r^{,2},sin^2theta},frac{partial^{,2}}{partialvarphi^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="493" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img513.png"
 ALT="$\displaystyle =\frac{1}{r^{ 2}} \frac{\partial}{\partial r}\!\left(r^{ 2} \...
...\frac{1}{r^{ 2} \sin^2\theta} \frac{\partial^{ 2}}{\partial \varphi^{ 2}},$">|; 

$key = q/displaystyle=frac{4,a^{,2},{mitOmega}^{,2}}{g,d}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4688.png"
 ALT="$\displaystyle = \frac{4 a^{ 2} {\mit\Omega}^{ 2}}{g d}.$">|; 

$key = q/d=3.8,{rmkm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4714.png"
 ALT="$ d=3.8 {\rm km}$">|; 

$key = q/a_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img836.png"
 ALT="$ a_1$">|; 

$key = q/displaystylenabla(phi,psi)=phi,nablapsi+psi,nablaphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6565.png"
 ALT="$\displaystyle \nabla(\phi  \psi) = \phi  \nabla\psi + \psi  \nabla \phi.$">|; 

$key = q/displaystyleY';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4743.png"
 ALT="$\displaystyle Y'$">|; 

$key = q/alpha=sin^{-1}(sinbetaslashsingamma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img998.png"
 ALT="$ \alpha=\sin^{-1}(\sin\beta/\sin\gamma)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter09slashcirc.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3325.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/circ.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{Chapter10slashplate.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="167" BORDER="0"
 SRC="|."$dir".q|img3575.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{Chapter10/plate.eps}}
\end{figure}">|; 

$key = q/c^{,2}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4180.png"
 ALT="$ c^{ 2}&gt;0$">|; 

$key = q/{bfgrad},hequiv(partialhslashpartialx,,partialhslashpartialy);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6536.png"
 ALT="$ {\bf grad} h \equiv (\partial h/\partial x,  \partial h/\partial y)$">|; 

$key = q/{mitPsi}={mitPsi}(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img563.png"
 ALT="$ {\mit\Psi}={\mit\Psi}(z)$">|; 

$key = q/usimeqpartialpsislashpartialy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5107.png"
 ALT="$ u\simeq \partial\psi/\partial y$">|; 

$key = q/displaystyleomega=0.5298,(G,Mslasha_0^{,3})^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1074.png"
 ALT="$\displaystyle \omega = 0.5298 (G M/a_0^{ 3})^{1/2},
$">|; 

$key = q/widehat{omega}=omegaslash(2pi,G,rho)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img892.png"
 ALT="$ \widehat{\omega}=\omega/(2\pi G \rho)^{1/2}$">|; 

$key = q/displaystyleD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2275.png"
 ALT="$\displaystyle D$">|; 

$key = q/displaystylee_{13};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img904.png"
 ALT="$\displaystyle e_{13}$">|; 

$key = q/oint{bfv}cdotd{bfS}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6594.png"
 ALT="$ \oint {\bf v}\cdot d{\bf S} = 0$">|; 

$key = q/displaystylenablacdot{bfn}=frac{1}{f,f_z},frac{d}{dz}!left[frac{f}{(1+f_z^{,2})^{1slash2}}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1235.png"
 ALT="$\displaystyle \nabla\cdot {\bf n} = \frac{1}{f f_z} \frac{d}{dz}\!\left[\frac{f}{(1+f_z^{ 2})^{1/2}}\right].$">|; 

$key = q/p'_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4973.png"
 ALT="$ p'_{\rm max}$">|; 

$key = q/displaystyle(nabla^{,2}{bfv})_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3656.png"
 ALT="$\displaystyle (\nabla^{ 2} {\bf v})_x$">|; 

$key = q/h_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7217.png"
 ALT="$ h_z$">|; 

$key = q/gamma=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$ \gamma =1$">|; 

$key = q/displaystylex_i,frac{partial{calR}_n}{partialx_i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4330.png"
 ALT="$\displaystyle x_i \frac{\partial {\cal R}_n}{\partial x_i}$">|; 

$key = q/{bfv}_S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3415.png"
 ALT="$ {\bf v}_S$">|; 

$key = q/displaystyleQ=frac{pi,a^{,4}}{8,mu}left(G+rho,g,sinalpharight),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3874.png"
 ALT="$\displaystyle Q = \frac{\pi a^{ 4}}{8 \mu}\left(G + \rho g \sin\alpha\right),
$">|; 

$key = q/displaystylefrac{h^{,2}}{2,l^{,2}}=1-sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1223.png"
 ALT="$\displaystyle \frac{h^{ 2}}{2 l^{ 2}} = 1-\sin\theta,$">|; 

$key = q/displaystyle|theta|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6131.png"
 ALT="$\displaystyle \vert\theta\vert$">|; 

$key = q/_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1540.png"
 ALT="$ _2$">|; 

$key = q/displaystylefrac{x}{l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1184.png"
 ALT="$\displaystyle \frac{x}{l}$">|; 

$key = q/displaystylefrac{D{calE}}{Dt}=-frac{p}{rho},nablacdot{bfv}+frac{chi}{rho}+frac{nabla!cdot(kappa,nablaT)}{rho}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="$\displaystyle \frac{D{\cal E}}{Dt} = - \frac{p}{\rho} \nabla\cdot{\bf v} + \frac{\chi}{\rho} +\frac{\nabla\!\cdot( \kappa \nabla T)}{\rho}.$">|; 

$key = q/hat{z}=zslashH;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5144.png"
 ALT="$ \hat{z}=z/H$">|; 

$key = q/displaystyle=left[frac{1+[(gamma-1)slash2],{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right]theta_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6187.png"
 ALT="$\displaystyle = \left[\frac{1+[(\gamma-1)/2] {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right]\theta_1,$">|; 

$key = q/displaystyle=-frac{1}{rho},frac{partialp}{partialr}-frac{partial{mitPsi}}{partia,2}}-frac{2}{r^{,2}},frac{partialv_theta}{partialtheta}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img503.png"
 ALT="$\displaystyle = - \frac{1}{\rho} \frac{\partial p}{\partial r} - \frac{\partia...
..._r}{r^{ 2}}-\frac{2}{r^{ 2}} \frac{\partial v_\theta}{\partial\theta}\right.$">|; 

$key = q/displaystylefrac{d^{,2}v_x}{dy^{,2}}=-frac{G}{mu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3591.png"
 ALT="$\displaystyle \frac{d^{ 2} v_x}{dy^{ 2}} = - \frac{G}{\mu},$">|; 

$key = q/displaystyleoint_Sa_i,dS_i=int_Vfrac{partiala_i}{partialx_i},dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6892.png"
 ALT="$\displaystyle \oint_S a_i dS_i = \int_V \frac{\partial a_i}{\partial x_i} dV,$">|; 

$key = q/displaystylefrac{r}{R},frac{d}{dr}!left(r,frac{dR}{dr}right)=-frac{1}{{mitTheta}},frac{d^{,2}{mitTheta}}{dtheta^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1731.png"
 ALT="$\displaystyle \frac{r}{R} \frac{d}{dr}\!\left(r \frac{dR}{dr}\right)= - \frac{1}{{\mit\Theta}} \frac{d^{ 2}{\mit\Theta}}{d\theta^{ 2}},$">|; 

$key = q/displaystyle{bfD}=rho,int_{-bslash2}^{bslash2}w(z),{mitGamma}(z),dz,,{bfe}_parallel.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3456.png"
 ALT="$\displaystyle {\bf D} = \rho \int_{-b/2}^{b/2} w(z) {\mit\Gamma}(z) dz  {\bf e}_\parallel.$">|; 

$key = q/m'=7.35times10^{22},{rmkg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4571.png"
 ALT="$ m'=7.35\times 10^{22} {\rm kg}$">|; 

$key = q/displaystyle{bfe}_1cdot{bfe}_2times{bfe}_3=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7038.png"
 ALT="$\displaystyle {\bf e}_1\cdot {\bf e}_2\times {\bf e}_3 = 1$">|; 

$key = q/displaystylep+frac{1}{2},rho,v^{,2}=p_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2006.png"
 ALT="$\displaystyle p +\frac{1}{2} \rho v^{ 2} = p_0,$">|; 

$key = q/lambda=omega_2,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4297.png"
 ALT="$ \lambda=\omega_2 t$">|; 

$key = q/lnz=lnr+{rmi},theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1977.png"
 ALT="$ \ln z = \ln r + {\rm i} \theta$">|; 

$key = q/{calT};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1359.png"
 ALT="$ {\cal T}$">|; 

$key = q/displaystyleB_i=-int_Vfrac{partialp}{partialx_i},dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img579.png"
 ALT="$\displaystyle B_i =- \int_V\frac{\partial p}{\partial x_i} dV,$">|; 

$key = q/{bfa}cdot{bfb}times{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6408.png"
 ALT="$ {\bf a}\cdot {\bf b}\times {\bf c}$">|; 

$key = q/displaystylephantom{=}+(3+cos^2gamma)left[E(gamma,alpha)+[F(gamma,alpha),cos^2alpha-2,E(gamma,alpha)],frac{cos^2beta}{cos^2alpha}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="511" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1011.png"
 ALT="$\displaystyle \phantom{=}+ (3+\cos^2\gamma)\left[E(\gamma,\alpha)+[F(\gamma,\alpha) \cos^2\alpha-2 E(\gamma,\alpha)] \frac{\cos^2\beta}{\cos^2\alpha}\right.$">|; 

$key = q/lslashdgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3009.png"
 ALT="$ l/d\gg 1$">|; 

$key = q/U(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2859.png"
 ALT="$ U(x)$">|; 

$key = q/displaystyle={bfA}times(nablatimes{bfB})+{bfB}times(nablatimes{bfA})+({bfA}cdotnabla),{bfB}+({bfB}cdotnabla),{bfA},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="474" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6684.png"
 ALT="$\displaystyle = {\bf A}\times(\nabla\times{\bf B}) + {\bf B}\times(\nabla\times{\bf A}) +({\bf A}\cdot \nabla) {\bf B} + ({\bf B} \cdot \nabla) {\bf A},$">|; 

$key = q/5^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4594.png"
 ALT="$ 5^\circ$">|; 

$key = q/nusimeq1.0times10^{-4},{rmm^2,s^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="$ \nu\simeq 1.0\times 10^{-4} {\rm m^2 s^{-1}}$">|; 

$key = q/+(7,e_Mslash4),zeta_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4561.png"
 ALT="$ +(7 e_M/4) \zeta_M$">|; 

$key = q/u_2=-V_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5551.png"
 ALT="$ u_2=-V_p$">|; 

$key = q/h_3=|nablau_3|^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7032.png"
 ALT="$ h_3=\vert\nabla u_3\vert^{-1}$">|; 

$key = q/beta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4053.png"
 ALT="$ \beta=0$">|; 

$key = q/{mitDelta}v_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3408.png"
 ALT="$ {\mit\Delta} v_z$">|; 

$key = q/displaystylembox{boldmath{{pi{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$\displaystyle \mbox{\boldmath$\pi$}$">|; 

$key = q/h_1=|nablau_1|^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7030.png"
 ALT="$ h_1=\vert\nabla u_1\vert^{-1}$">|; 

$key = q/xsimV^{,1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$ x\sim V^{ 1/3}$">|; 

$key = q/displaystyle[({bfA}cdotnabla),{bfB}]_i=sum_{j=1,3}left(frac{A_j}{h_j},frac{partii}+frac{A_i,B_j}{h_i,h_j},frac{partialh_i}{partialu_j}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7088.png"
 ALT="$\displaystyle [({\bf A}\cdot\nabla) {\bf B}]_i= \sum_{j=1,3}\left(\frac{A_j}{h...
...ial u_i} + \frac{A_i B_j}{h_i h_j} \frac{\partial h_i}{\partial u_j}\right).$">|; 

$key = q/langlecos^2(omega,t-k,x)rangle=langlesin^2(omega,t-k,x)rangle=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3975.png"
 ALT="$ \langle \cos^2(\omega t-k x)\rangle =\langle \sin^2(\omega t-k x)\rangle=1/2$">|; 

$key = q/displaystyle{calS}_n-{calS}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5755.png"
 ALT="$\displaystyle {\cal S}_n-{\cal S}_1$">|; 

$key = q/z=z_0+{rmi},pislashK';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2157.png"
 ALT="$ z=z_0+{\rm i} \pi/K'$">|; 

$key = q/i=1,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img999.png"
 ALT="$ i=1,3$">|; 

$key = q/displaystylenu=frac{mu}{rho};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="$\displaystyle \nu = \frac{\mu}{\rho}$">|; 

$key = q/displaystylefrac{d}{dmu}left[(1-mu^{,2}),frac{d{mitTheta}}{dmu}right]+l,(l+1),{mitTheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2576.png"
 ALT="$\displaystyle \frac{d}{d\mu}\left[(1-\mu^{ 2}) \frac{d{\mit\Theta}}{d\mu}\right]+l (l+1) {\mit\Theta}$">|; 

$key = q/rho_2slashrho_1>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5505.png"
 ALT="$ \rho_2/\rho_1&gt;1$">|; 

$key = q/displaystyle(widetilde{nabla{bfA}})_{thetaz}=(widetilde{nabla{bfA}})_{ztheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7157.png"
 ALT="$\displaystyle (\widetilde{\nabla {\bf A}})_{\theta z}=(\widetilde{\nabla {\bf A}})_{z\theta}$">|; 

$key = q/displaystyleint_Srho,v_x,{bfv}cdotd{bfS}=F_x+int_Ssigma_{xj},dS_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3053.png"
 ALT="$\displaystyle \int_S \rho v_x {\bf v}\cdot d{\bf S} = F_x + \int_S \sigma_{xj} dS_j,$">|; 

$key = q/j=k=l=m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6812.png"
 ALT="$ j=k=l=m$">|; 

$key = q/displaystyle=p_0+rho,g,(h_1-z),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1471.png"
 ALT="$\displaystyle = p_0+\rho g (h_1-z),$">|; 

$key = q/displaystyle=|alpha|,(1-k^{,2},sin^2phi)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1304.png"
 ALT="$\displaystyle = \vert\alpha\vert (1-k^{ 2} \sin^2\phi)^{1/2},$">|; 

$key = q/oint{bfA}cdotd{bfr}=I_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6626.png"
 ALT="$ \oint {\bf A}\cdot d{\bf r}
= I_{\rm max}$">|; 

$key = q/partialuslashpartialy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3138.png"
 ALT="$ \partial u/\partial y$">|; 

$key = q/D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1570.png"
 ALT="$ D$">|; 

$key = q/X'=Y'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4754.png"
 ALT="$ X'=Y'=0$">|; 

$key = q/T(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5439.png"
 ALT="$ T(x)$">|; 

$key = q/displaystyleepsilon_{ijk},epsilon_{ilm}=delta_{jl},delta_{km}-delta_{jm},delta_{kl}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6803.png"
 ALT="$\displaystyle \epsilon_{ijk} \epsilon_{ilm} = \delta_{jl} \delta_{km}-\delta_{jm} \delta_{kl}.$">|; 

$key = q/displaystyle=-pi,a^{,2},rho,frac{dV_x}{dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1816.png"
 ALT="$\displaystyle = -\pi a^{ 2} \rho \frac{dV_x}{dt},$">|; 

$key = q/displaystyley(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7380.png"
 ALT="$\displaystyle y(\theta)$">|; 

$key = q/displaystylerho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6138.png"
 ALT="$\displaystyle \rho$">|; 

$key = q/=(0,,0,,omega);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6649.png"
 ALT="$ = (0, 0, \omega)$">|; 

$key = q/displaystyleV_0'=int_{A_0}[delta(x,y)-theta,y],dx,dy=V_0+thetaint_{A_0}y,dx,dy=V_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img637.png"
 ALT="$\displaystyle V_0'=\int_{A_0} [\delta(x,y) -\theta y] dx dy = V_0 + \theta\int_{A_0} y dx dy = V_0,$">|; 

$key = q/displaystyleomega_1^{,2}=frac{G,(m+m')}{r'^{,3}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4230.png"
 ALT="$\displaystyle \omega_1^{ 2} = \frac{G (m+m')}{r'^{ 3}}.$">|; 

$key = q/u_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5086.png"
 ALT="$ u_0$">|; 

$key = q/V^{,-1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="$ V^{ -1/3}$">|; 

$key = q/displaystyle(delta_{ii},delta_{ks}-delta_{is},delta_{ki}),a_{sj}+(delta_{ij},delta_{ks}-delta_{is},delta_{kj}),a_{is}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="374" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6903.png"
 ALT="$\displaystyle (\delta_{ii} \delta_{ks}-\delta_{is} \delta_{ki}) a_{sj} + (\delta_{ij} \delta_{ks} -\delta_{is} \delta_{kj}) a_{is} = 0,$">|; 

$key = q/displaystyleddot{p}_{-r}+2,{mitOmega}sum_{s=1,infty}(beta_{-r,,s},dot{p}_s+beta_{-r,,-s},dot{p}_{-s});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4914.png"
 ALT="$\displaystyle \ddot{p}_{-r} + 2 {\mit\Omega}\sum_{s=1,\infty} (\beta_{-r, s} \dot{p}_s+\beta_{-r, -s} \dot{p}_{-s})$">|; 

$key = q/displaystylephi(r,theta,t)=frac{1}{2},V_z(t),frac{a^{,3}}{r^{,2}},costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2661.png"
 ALT="$\displaystyle \phi(r,\theta,t) = \frac{1}{2} V_z(t) \frac{a^{ 3}}{r^{ 2}} \cos\theta.$">|; 

$key = q/q=|{bfv}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1932.png"
 ALT="$ q=\vert{\bf v}\vert$">|; 

$key = q/a_0=6.37times10^6,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img913.png"
 ALT="$ a_0=6.37\times 10^6 {\rm m}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter12slashampm2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5050.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter12/ampm2.eps}}
\end{figure}">|; 

$key = q/v_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1437.png"
 ALT="$ v_+$">|; 

$key = q/displaystyle=int_{-b}^b(p-p_0)_{x=0},dy=frac{1}{2},rho,V^{,2}int_{-b}^bleft(1-frac{v_y^{,2}}{V^{,2}}right)_{x=0}dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="416" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2331.png"
 ALT="$\displaystyle =\int_{-b}^b(p-p_0)_{x=0} dy=\frac{1}{2} \rho V^{ 2}\int_{-b}^b\left(1-\frac{v_y^{ 2}}{V^{ 2}}\right)_{x=0}dy$">|; 

$key = q/c_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="$ c_V$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter11slashoblique.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="278" BORDER="0"
 SRC="|."$dir".q|img4016.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter11/oblique.eps}}
\end{figure}">|; 

$key = q/A'P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4032.png"
 ALT="$ A'P$">|; 

$key = q/3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="$ 3$">|; 

$key = q/displaystyle=frac{1}{gamma,{rmMa}^{,2},(1+1slash{rmFr}^{,2}+1slash{rmRe})},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="$\displaystyle = \frac{1}{\gamma {\rm Ma}^{ 2} (1+1/{\rm Fr}^{ 2}+1/{\rm Re})},$">|; 

$key = q/d{bfv}=d(d{bfr}slashdt)=d(d{bfr})slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1554.png"
 ALT="$ d{\bf v} = d(d{\bf r}/dt)= d(d{\bf r})/dt$">|; 

$key = q/oint{bfA}cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6578.png"
 ALT="$ \oint {\bf A}\cdot d{\bf S}$">|; 

$key = q/nabla{calS}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5872.png"
 ALT="$ \nabla {\cal S}=0$">|; 

$key = q/L_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3531.png"
 ALT="$ L_0$">|; 

$key = q/a_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img845.png"
 ALT="$ a_3$">|; 

$key = q/displaystyleg,b=k,V,A.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4126.png"
 ALT="$\displaystyle g b = k V A.$">|; 

$key = q/displaystylefrac{partialphi}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6018.png"
 ALT="$\displaystyle \frac{\partial\phi}{\partial y}$">|; 

$key = q/displaystylepsi(r,theta)=-frac{D}{2pi},frac{sin(theta-theta_0)}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1691.png"
 ALT="$\displaystyle \psi(r,\theta) = -\frac{D}{2\pi} \frac{\sin(\theta-\theta_0)}{r},$">|; 

$key = q/epsilon_{mkt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6911.png"
 ALT="$ \epsilon_{mkt}$">|; 

$key = q/muggmu';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1017.png"
 ALT="$ \mu\gg \mu'$">|; 

$key = q/alpha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$ \alpha _0$">|; 

$key = q/displaystylefrac{dy}{dlambda}=frac{partialy}{partialskew{3}bar{phi}},frac{dskew{(lambda-cosalpha)^3},left(lambdapmsqrt{lambda^{,2}-1},right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2303.png"
 ALT="$\displaystyle \frac{d y}{d\lambda} = \frac{\partial y}{\partial\skew{3}\bar{\ph...
...2 k}{(\lambda-\cos\alpha)^3} \left(\lambda\pm\sqrt{\lambda^{ 2}-1} \right).$">|; 

$key = q/displaystylebeta_{n,,-n'}^{,m,,-m'}=-beta_{-n',,n}^{,-m',,m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4964.png"
 ALT="$\displaystyle \beta_{n, -n'}^{ m, -m'}=-\beta_{-n', n}^{ -m', m},$">|; 

$key = q/displaystyle2,a_{ijkl}+(a_{ijlk}+a_{ikjl}+a_{ilkj});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6965.png"
 ALT="$\displaystyle 2 a_{ijkl}+(a_{ijlk}+a_{ikjl}+a_{ilkj})$">|; 

$key = q/mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="$ \mu$">|; 

$key = q/displaystyle=-V_0left(r-frac{a^{,2}}{r}right)sintheta+frac{mitGamma}{2pi},lnleft(frac{r}{a}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1990.png"
 ALT="$\displaystyle =-V_0\left(r-\frac{a^{ 2}}{r}\right)\sin\theta + \frac{\mit\Gamma}{2\pi} \ln\left(\frac{r}{a}\right).$">|; 

$key = q/displaystyle2pi,(psi_{R'}-psi_R)=2pi,varpi,deltas,v_perp,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2539.png"
 ALT="$\displaystyle 2\pi (\psi_{R'}- \psi_R) = 2\pi \varpi \delta s v_\perp,$">|; 

$key = q/widehat{L};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img885.png"
 ALT="$ \widehat{L}$">|; 

$key = q/displaystylephantom{=}-3,mu,frac{V}{a}left[frac{mu+(3slash2),overline{mu}}{mu+overline{mu}}right]costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3861.png"
 ALT="$\displaystyle \phantom{=}- 3 \mu \frac{V}{a}\left[\frac{\mu+(3/2) \overline{\mu}}{\mu+\overline{\mu}}\right]\cos\theta.$">|; 

$key = q/{rmMa}_2=w_2slashc_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5697.png"
 ALT="$ {\rm Ma}_2 = w_2/c_2$">|; 

$key = q/overline{rho};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3810.png"
 ALT="$ \overline{\rho}$">|; 

$key = q/psi_3(x,y)=psi_1(x,y)+psi_2(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2015.png"
 ALT="$ \psi_3(x,y)=\psi_1(x,y)+\psi_2(x,y)$">|; 

$key = q/rho=MslashV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img598.png"
 ALT="$ \rho = M/V$">|; 

$key = q/H_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img691.png"
 ALT="$ H_2$">|; 

$key = q/displaystyley=epsilon,sin(alpha,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5950.png"
 ALT="$\displaystyle y = \epsilon \sin(\alpha x).$">|; 

$key = q/displaystyle=-frac{2}{3},pi,rho,a,b^{,2},V^{,2}left{frac{1+(1slash2,e)lnleft[(1-e^{,2})^{,-1}+(1slash2,e)lnleft[(1-e)slash(1+e)right]}right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="503" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2811.png"
 ALT="$\displaystyle = -\frac{2}{3} \pi \rho a b^{ 2} V^{ 2}\left\{\frac{1 +(1/...
...(1+e)\right]} {(1-e^{ 2})^{ -1}+(1/2 e)\ln\left[(1-e)/(1+e)\right]}\right\}.$">|; 

$key = q/displaystylep({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4068.png"
 ALT="$\displaystyle p({\bf r},t)$">|; 

$key = q/displaystylefrac{rho_1}{rho_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5399.png"
 ALT="$\displaystyle \frac{\rho_1}{\rho_0}$">|; 

$key = q/{bfe}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7048.png"
 ALT="$ {\bf e}_1$">|; 

$key = q/{mitDelta}theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="$ {\mit \Delta }\theta $">|; 

$key = q/{mitPhi}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4809.png"
 ALT="$ {\mit\Phi}_r$">|; 

$key = q/displaystyle=int_{mitOmega}left(frac{partial{mitPsi}'}{partialtheta},frac{partiaialphi},frac{partial{mitPsi}_r}{partialphi}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4905.png"
 ALT="$\displaystyle =\int_{\mit\Omega} \left(\frac{\partial {\mit\Psi}'}{\partial\the...
...}{\partial\phi} \frac{\partial{\mit\Psi}_r}{\partial\phi}\right)d{\mit\Omega},$">|; 

$key = q/xill1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5221.png"
 ALT="$ \xi\ll 1$">|; 

$key = q/displaystyle_{mitSigma}=-I(z),{bfe}_parallel,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3406.png"
 ALT="$\displaystyle _{\mit\Sigma}= -I(z) {\bf e}_\parallel,$">|; 

$key = q/displaystylef_y';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3714.png"
 ALT="$\displaystyle f_y'$">|; 

$key = q/x=(u^{,2}-v^{,2})slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7213.png"
 ALT="$ x=(u^{ 2}-v^{ 2})/2$">|; 

$key = q/displaystyle{calR}_n^{,(0)}(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4274.png"
 ALT="$\displaystyle {\cal R}_n^{ (0)}(r,\theta)$">|; 

$key = q/p_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="$ p_0$">|; 

$key = q/displaystyle{rmFr}_1gtreqless1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1495.png"
 ALT="$\displaystyle {\rm Fr}_1 \gtreqless 1$">|; 

$key = q/displaystylesimeqleft(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right)theta_1',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6197.png"
 ALT="$\displaystyle \simeq \left(\frac{\gamma {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right)\theta_1',$">|; 

$key = q/stackrel{displaystylerightarrow}{PS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="30" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6233.png"
 ALT="$ \stackrel{\displaystyle \rightarrow}{PS}$">|; 

$key = q/a_{ijk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6767.png"
 ALT="$ a_{ijk}$">|; 

$key = q/displaystyle{calR}_n^{,(+m)}(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4276.png"
 ALT="$\displaystyle {\cal R}_n^{ (+m)}(r,\theta,\varphi)$">|; 

$key = q/displaystylezeta_n^{,m}=-lambda_n^{,m},p_n^{,m}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5027.png"
 ALT="$\displaystyle \zeta_n^{ m} = -\lambda_n^{ m} p_n^{ m}.$">|; 

$key = q/pslashrho+{calT};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1373.png"
 ALT="$ p/\rho+{\cal T}$">|; 

$key = q/varpi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2817.png"
 ALT="$ \varpi=0$">|; 

$key = q/deltaslashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3369.png"
 ALT="$ \delta/c$">|; 

$key = q/displaystylephantom{simeqfrac{G,m',a^{2}}{R^{,3}}}+frac{1}{2},sindelta,{calR}_2^{,(-1)}(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4304.png"
 ALT="$\displaystyle \phantom{\simeq \frac{G m' a^{2}}{R^{ 3}}} + \frac{1}{2} \sin\delta {\cal R}_2^{ (-1)}(r,\theta,\varphi)$">|; 

$key = q/displaystyle{bfa}=({bfB}times{bfC})slash({bfA}cdot{bfB}times{bfC}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6736.png"
 ALT="$\displaystyle {\bf a} = ({\bf B}\times {\bf C})/({\bf A}\cdot{\bf B}\times {\bf C}),
$">|; 

$key = q/T_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5631.png"
 ALT="$ T_4$">|; 

$key = q/displaystyle=-frac{G,M'}{R}left(1-2,frac{x_1}{R}+frac{x_1^{,2}+x_2^{,2}+x_3^{,2}}{R^{,2}}right)^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img984.png"
 ALT="$\displaystyle =-\frac{G M'}{R}\left(1-2 \frac{x_1}{R} + \frac{x_1^{ 2}+x_2^{ 2}+x_3^{ 2}}{R^{ 2}}\right)^{-1/2}$">|; 

$key = q/displaystylep(r)=left{{array}{lll}p_infty-(rhoslash2),({mitGamma}slash2pi,a)^2,(ash2),({mitGamma}slash2pi,a)^2,(aslashr)^2&&r>a{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="472" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1724.png"
 ALT="$\displaystyle p(r)=\left\{ \begin{array}{lll} p_\infty - (\rho/2) ({\mit\Gamma...
... p_\infty-(\rho/2) ({\mit\Gamma}/2\pi a)^2 (a/r)^2&amp;&amp;r&gt;a \end{array} \right..$">|; 

$key = q/displaystyle=nablavarphitimesnabla[-{calL}(psi)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3740.png"
 ALT="$\displaystyle = \nabla\varphi\times \nabla[-{\cal L}(\psi)].$">|; 

$key = q/displaystylenabla^{,2}T=frac{partial^{,2}T}{partialx^{,2}}+frac{partial^{,2}T}{partialy^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6618.png"
 ALT="$\displaystyle \nabla^{ 2} T = \frac{\partial^{ 2} T}{\partial x^{ 2}}+ \frac{\partial^{ 2} T}{\partial y^{ 2}}.$">|; 

$key = q/0<ell1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4285.png"
 ALT="$ 0&lt;e\ll 1$">|; 

$key = q/displaystylenablacdot{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="$\displaystyle \nabla\cdot{\bf v}$">|; 

$key = q/displaystyle={rmi}^{,n+n'+1},epsilon_{n,,n'}^{,m,,m'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5007.png"
 ALT="$\displaystyle = {\rm i}^{ n+n'+1}  \epsilon_{n, n'}^{ m, m'},$">|; 

$key = q/displaystyleK=frac{pi,G,M_ast^{,2slash3}}{(4pi,y_1)^{2slash3}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5230.png"
 ALT="$\displaystyle K = \frac{\pi G M_\ast^{ 2/3}}{(4\pi y_1)^{2/3}}.$">|; 

$key = q/displaystyleF_z=-6pi,a,mu,V+frac{4pi}{3},a^{,3},rho,g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3808.png"
 ALT="$\displaystyle F_z = -6\pi a \mu V + \frac{4\pi}{3} a^{ 3} \rho g.$">|; 

$key = q/displaystylefrac{1}{sqrt{1-n^{,2}}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6215.png"
 ALT="$\displaystyle \frac{1}{\sqrt{1-n^{ 2}}},
$">|; 

$key = q/displaystyle{calH}={calE}+p,v={calE}+{calR},T={calC}_p,T=frac{c_p}{M},T,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5318.png"
 ALT="$\displaystyle {\cal H} = {\cal E} + p v={\cal E}+{\cal R} T = {\cal C}_p T =\frac{c_p}{M} T,$">|; 

$key = q/eta=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2496.png"
 ALT="$ \eta = a$">|; 

$key = q/{rmMa}<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5417.png"
 ALT="$ {\rm Ma}&lt;1$">|; 

$key = q/displaystyle=int_0^gammafrac{dtheta}{(1-sin^2alpha,sin^2theta)^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img946.png"
 ALT="$\displaystyle = \int_0^\gamma \frac{d\theta}{(1-\sin^2\alpha  \sin^2\theta)^{1/2}},$">|; 

$key = q/displaystyle(nabla^{,2}{bfA})_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7098.png"
 ALT="$\displaystyle (\nabla^{ 2}{\bf A})_i$">|; 

$key = q/{bfe}_1=h_1,nablau_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7033.png"
 ALT="$ {\bf e}_1=h_1 \nabla u_1$">|; 

$key = q/displaystylenabla^{,2}{mitPhi}_{n>0}=4pi,Gleft(rho,zeta_{b,n}+skew{3}bar{rho},zeta_nright)delta(r-a).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4356.png"
 ALT="$\displaystyle \nabla^{ 2}{\mit\Phi}_{n&gt;0} = 4\pi G\left(\rho \zeta_{b n}+ \skew{3}\bar{\rho} \zeta_n\right)\delta(r-a).$">|; 

$key = q/displaystyle-sum_{n=1,infty}nablap_n=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4393.png"
 ALT="$\displaystyle - \sum_{n=1,\infty}\nabla p_n=0.$">|; 

$key = q/c_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5893.png"
 ALT="$ c_\infty$">|; 

$key = q/1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2106.png"
 ALT="$ 1)$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{Chapter14slashshock.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="424" HEIGHT="198" BORDER="0"
 SRC="|."$dir".q|img5627.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{Chapter14/shock.eps}}
\end{figure}">|; 

$key = q/gamma=skew{3}bar{phi}slash(k,V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2281.png"
 ALT="$ \gamma=\skew{3}\bar{\phi}/(k V)$">|; 

$key = q/displaystyleC_p=frac{p-p_1}{q_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5787.png"
 ALT="$\displaystyle C_p =\frac{p-p_1}{q_1},$">|; 

$key = q/displaystylefrac{D{bfv}}{Dt}=-frac{nablap}{rho}-nabla{mitPsi}+nu,nabla^{,2}{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="$\displaystyle \frac{D{\bf v}}{Dt}= - \frac{\nabla p}{\rho}-\nabla{\mit\Psi} + \nu \nabla^{ 2} {\bf v},$">|; 

$key = q/omega_z=-nabla^{,2}psi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1676.png"
 ALT="$ \omega_z=-\nabla^{ 2}\psi=0$">|; 

$key = q/displaystylesimeqfrac{3}{2},V,costheta,frac{a}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3889.png"
 ALT="$\displaystyle \simeq\frac{3}{2} V \cos\theta \frac{a}{r},$">|; 

$key = q/displaystyle=frac{nu}{kappa_H},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="$\displaystyle = \frac{\nu}{\kappa_H},$">|; 

$key = q/displaystylefrac{delta(x)}{x}simeqfrac{1}{[{rmRe}(x)]^{,1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2951.png"
 ALT="$\displaystyle \frac{\delta(x)}{x} \simeq \frac{1}{[{\rm Re}(x)]^{ 1/2}},$">|; 

$key = q/({bfa}times{bfb})cdot({bfc}times{bfd})=({bfa}cdot{bfc}),({bfb}cdot{bfd})-({bfa}cdot{bfd}),({bfb}cdot{bfc});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6985.png"
 ALT="$ ({\bf a}\times {\bf b})\cdot({\bf c}\times {\bf d})=
({\bf a}\cdot{\bf c}) ({\bf b}\cdot{\bf d}) - ({\bf a}\cdot{\bf d}) ({\bf b}\cdot{\bf c})$">|; 

$key = q/s>0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img806.png"
 ALT="$ s&gt;0.5$">|; 

$key = q/m>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2954.png"
 ALT="$ m&gt;0$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashpres.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="396" HEIGHT="342" BORDER="0"
 SRC="|."$dir".q|img3095.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/pres.eps}}
\end{figure}">|; 

$key = q/dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6453.png"
 ALT="$ dx$">|; 

$key = q/{bfk}times{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6754.png"
 ALT="$ {\bf k}\times {\bf r}$">|; 

$key = q/-3,e_M,zeta_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4549.png"
 ALT="$ -3 e_M \zeta_M$">|; 

$key = q/P_1P_2=deltas;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1629.png"
 ALT="$ P_1 P_2=\delta s$">|; 

$key = q/OC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1896.png"
 ALT="$ OC$">|; 

$key = q/displaystylefrac{1}{r},frac{d}{partialr}!left(r,frac{dpsi}{dr}right)=left{{arraypi,a^{,2})&mbox{hspace{1cm}}&rleqa[0.5ex]0&&r>a{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1712.png"
 ALT="$\displaystyle \frac{1}{r} \frac{d}{\partial r}\!\left(r \frac{d\psi}{d r}\rig...
...(\pi a^{ 2})&amp;\mbox{\hspace{1cm}}&amp;r\leq a\ [0.5ex] 0&amp;&amp;r&gt;a \end{array}\right..$">|; 

$key = q/displaystyleF_1(lambda);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3191.png"
 ALT="$\displaystyle F_1(\lambda)$">|; 

$key = q/hslasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1042.png"
 ALT="$ h/a$">|; 

$key = q/displaystylezeta(theta,phi,t)=zeta_c(theta,phi),cos(sigma_j,t)+zeta_s(theta,phi),sin(sigma_j,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5022.png"
 ALT="$\displaystyle \zeta(\theta,\phi,t) = \zeta_c(\theta,\phi) \cos(\sigma_j t) + \zeta_s(\theta,\phi) \sin(\sigma_j t),$">|; 

$key = q/displaystyleoint{bfA}cdotd{bfr}=left(frac{partialA_x}{partialz}-frac{partialA_z}{partialx}right)dS_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6639.png"
 ALT="$\displaystyle \oint {\bf A} \cdot d{\bf r} = \left(\frac{\partial A_x}{\partial z}- \frac{\partial A_z}{\partial x}\right) dS_y.$">|; 

$key = q/displaystyle.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1376.png"
 ALT="$\displaystyle .$">|; 

$key = q/14.77;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4722.png"
 ALT="$ 14.77$">|; 

$key = q/_S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3411.png"
 ALT="$ _S$">|; 

$key = q/displaystyleV=frac{dQ}{dh}=frac{G,h^{,2}}{mu}=left(frac{9,Q^{,2},g,sinalpha}{nu}right)^{1slash3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3607.png"
 ALT="$\displaystyle V = \frac{dQ}{dh} = \frac{G h^{ 2}}{\mu} = \left(\frac{9 Q^{ 2} g \sin\alpha}{\nu}\right)^{1/3}.$">|; 

$key = q/v=v_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1432.png"
 ALT="$ v=v_c$">|; 

$key = q/displaystyle2pi,gammaleft[frac{f}{(1+f_z^{,2})^{1slash2}}-frac{f^{,2}}{2,a}right]=C',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1252.png"
 ALT="$\displaystyle 2\pi \gamma\left[\frac{f}{(1+f_z^{ 2})^{1/2}}-\frac{f^{ 2}}{2 a}\right] = C',$">|; 

$key = q/displaystylerho(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5152.png"
 ALT="$\displaystyle \rho(z)$">|; 

$key = q/displaystyle=-frac{1}{sintheta},frac{partial{mitPhi}}{partialphi}+frac{partial{mitPsi}}{partialtheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4665.png"
 ALT="$\displaystyle = -\frac{1}{\sin\theta} \frac{\partial{\mit\Phi}}{\partial\phi} +\frac{\partial{\mit\Psi}}{\partial\theta}.$">|; 

$key = q/displaystyler(xi)=c,{rme}^{,xi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2729.png"
 ALT="$\displaystyle r(\xi)=c {\rm e}^{ \xi}.$">|; 

$key = q/displaystyleint!int_Sx,y^{,2},dx,dy,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6492.png"
 ALT="$\displaystyle \int\!\int_S x  y^{ 2} dx dy,$">|; 

$key = q/displaystyle=frac{delta_0}{2},sinphi,sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3482.png"
 ALT="$\displaystyle =\frac{\delta_0}{2} \sin\phi \sin\theta,$">|; 

$key = q/displaystyle=frac{v}{c_0}left[1-frac{1}{2},(gamma-1)left(frac{v}{c_0}right)^2right]^{,-1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5615.png"
 ALT="$\displaystyle = \frac{v}{c_0}\left[1-\frac{1}{2} (\gamma-1)\left(\frac{v}{c_0}\right)^2\right]^{ -1/2},$">|; 

$key = q/k,l=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4167.png"
 ALT="$ k l=1$">|; 

$key = q/displaystyle=frac{M}{R},frac{p}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="$\displaystyle =\frac{M}{R} \frac{p}{\rho},$">|; 

$key = q/displaystyle=muleft(frac{partialv_x}{partialz}+frac{partialv_z}{partialx}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="$\displaystyle = \mu\left(\frac{\partial v_x}{\partial z}+\frac{\partial v_z}{\partial x}\right),$">|; 

$key = q/displaystyle{rmdet}(A)=epsilon_{ijk},A_{1i},A_{2j},A_{3k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6998.png"
 ALT="$\displaystyle {\rm det}(A) = \epsilon_{ijk} A_{1i} A_{2j} A_{3k}
$">|; 

$key = q/displaystyle=frac{2}{a_1},frac{sin^{-1}e_{13}}{e_{13}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img883.png"
 ALT="$\displaystyle = \frac{2}{a_1} \frac{\sin^{-1} e_{13}}{e_{13}}.$">|; 

$key = q/{mitLambda}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3214.png"
 ALT="$ {\mit\Lambda}=0$">|; 

$key = q/_S({bfr})simeq{bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3463.png"
 ALT="$ _S({\bf r})\simeq {\bf0}$">|; 

$key = q/displaystyle{bfa}+(-{bfa})={bf0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6239.png"
 ALT="$\displaystyle {\bf a} + (-{\bf a}) = {\bf0}.$">|; 

$key = q/displaystyle{calF}(theta,phi,t)=(1+k_2-h_2),skew{5}bar{zeta}_2(theta,phi,t)+ointG_F(theta,phi;theta',phi'),zeta(theta',phi',t),d{mitOmega}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="564" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4659.png"
 ALT="$\displaystyle {\cal F}(\theta,\phi,t)=(1+k_2-h_2) \skew{5}\bar{\zeta}_2(\theta...
...)+\oint G_F(\theta,\phi;\theta',\phi') \zeta(\theta',\phi',t) d{\mit\Omega}',$">|; 

$key = q/_x({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$ _x({\bf r}, t)$">|; 

$key = q/v_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1353.png"
 ALT="$ v_1$">|; 

$key = q/displaystylefrac{d}{dr}!left(r^{,4},frac{d{mitOmega}}{dr}right)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3884.png"
 ALT="$\displaystyle \frac{d}{dr}\!\left(r^{ 4} \frac{d{\mit\Omega}}{dr}\right) = 0.
$">|; 

$key = q/delta_{kl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6948.png"
 ALT="$ \delta_{kl}$">|; 

$key = q/displaystylenablatimes(nablatimes;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3741.png"
 ALT="$\displaystyle \nabla\times (\nabla\times$">|; 

$key = q/C_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2509.png"
 ALT="$ C_3$">|; 

$key = q/displaystyle=frac{M}{5},(a_1^{,2}+a_3^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7282.png"
 ALT="$\displaystyle = \frac{M}{5}  (a_1^{ 2}+a_3^{ 2}),$">|; 

$key = q/displaystylerho,frac{partial{bfv}}{partialt}simeq-nablap_1+mu,nabla^{,2}{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3899.png"
 ALT="$\displaystyle \rho \frac{\partial {\bf v}}{\partial t} \simeq - \nabla p_1 + \mu \nabla^{ 2}{\bf v},$">|; 

$key = q/displaystyle=F_y-m,g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1845.png"
 ALT="$\displaystyle = F_y-m g,$">|; 

$key = q/displaystyledX;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2383.png"
 ALT="$\displaystyle dX$">|; 

$key = q/A_0=a,b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img665.png"
 ALT="$ A_0=a b$">|; 

$key = q/displaystylesigma,k^{,2}+beta,k_y=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5111.png"
 ALT="$\displaystyle \sigma k^{ 2}+\beta k_y=0.
$">|; 

$key = q/displaystylesigma_{xz}=sigma_{zx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="$\displaystyle \sigma_{xz}=\sigma_{zx}$">|; 

$key = q/displaystyle-rho,S_1,V^{,2}+rho,S_2,V^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1400.png"
 ALT="$\displaystyle -\rho S_1 V^{ 2}+\rho S_2 V^{ 2},$">|; 

$key = q/displaystylev_r=frac{Q}{4pi,r^{,2}}=-frac{partialphi}{partialr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2613.png"
 ALT="$\displaystyle v_r = \frac{Q}{4\pi r^{ 2}} =-\frac{\partial\phi}{\partial r}.$">|; 

$key = q/displaystylelambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3189.png"
 ALT="$\displaystyle \lambda$">|; 

$key = q/alpha_C(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5831.png"
 ALT="$ \alpha_C(x)$">|; 

$key = q/displaystyleleft(frac{partial^{,2}}{partialx^{,2}}+frac{partial^{,2}}{partialy^{,2}}right)v_z=-frac{G}{mu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3880.png"
 ALT="$\displaystyle \left(\frac{\partial^{ 2}}{\partial x^{ 2}} + \frac{\partial^{ 2}}{\partial y^{ 2}}\right)v_z = - \frac{G}{\mu},
$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialx^{,2}}+frac{partial^{,2}phi}{partialy^alx}+frac{partialphi}{partialy},frac{partialpsi}{partialy}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1967.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial x^{ 2}}+\frac{\partial^{ 2}\...
...tial x} + \frac{\partial\phi}{\partial y} \frac{\partial\psi}{\partial y} = 0.$">|; 

$key = q/displaystyleJ';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2364.png"
 ALT="$\displaystyle J'$">|; 

$key = q/displaystylefrac{l}{2,h}=sinhleft(frac{a}{2,h}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7361.png"
 ALT="$\displaystyle \frac{l}{2 h} = \sinh\left(\frac{a}{2 h}\right).$">|; 

$key = q/1slash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img678.png"
 ALT="$ 1/4$">|; 

$key = q/displaystyleG_{rmc}=frac{3}{2}left(frac{Q^{,2}}{g}right)^{1slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1448.png"
 ALT="$\displaystyle G_{\rm c} = \frac{3}{2}\left(\frac{Q^{ 2}}{g}\right)^{1/3},$">|; 

$key = q/displaystyle=frac{kappa^{,2},A}{V_2}-H_2G=frac{1}{V_2}left[A,kappa^{,2}-left(frac{V_1,V_2}{V_1+V_2}right)H_1H_2right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img697.png"
 ALT="$\displaystyle = \frac{\kappa^{ 2} A}{V_2} - H_2G =\frac{1}{V_2}\left[A \kappa^{ 2}- \left(\frac{V_1 V_2}{V_1+V_2}\right)H_1H_2\right],$">|; 

$key = q/j=i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6924.png"
 ALT="$ j=i$">|; 

$key = q/displaystyleoverline{alpha^{,2}}=overline{(alpha_0+alpha_C)^2}=overline{alpha_0^}+overline{alpha_C^{,2}}=alpha_0^{,2}+overline{alpha_C^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5855.png"
 ALT="$\displaystyle \overline{\alpha^{ 2}} = \overline{(\alpha_0+\alpha_C)^2}=\overl...
...lpha_C}+\overline{\alpha_C^{ 2}} = \alpha_0^{ 2} + \overline{\alpha_C^{ 2}}.$">|; 

$key = q/lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img625.png"
 ALT="$ \lambda$">|; 

$key = q/displaystyle={bfA}cdotnablaA_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7140.png"
 ALT="$\displaystyle = {\bf A}\cdot\nabla A_z.$">|; 

$key = q/v=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3106.png"
 ALT="$ v=0$">|; 

$key = q/SllA_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1602.png"
 ALT="$ S\ll A_1$">|; 

$key = q/displaystylefrac{T_1}{T_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5395.png"
 ALT="$\displaystyle \frac{T_1}{T_0}$">|; 

$key = q/{bfA}cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6587.png"
 ALT="$ {\bf A}\cdot d{\bf S}$">|; 

$key = q/displaystyle={rmi}^{,n+n'},beta_{n,,-n'}^{,m,,-m'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5001.png"
 ALT="$\displaystyle = {\rm i}^{ n+n'} \beta_{n, -n'}^{ m, -m'},$">|; 

$key = q/{mitPsi}(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5175.png"
 ALT="$ {\mit\Psi}(r)$">|; 

$key = q/|phi_+-phi_-|<2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4725.png"
 ALT="$ \vert\phi_+-\phi_-\vert&lt; 2\pi$">|; 

$key = q/W=rho_0,V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img705.png"
 ALT="$ W=\rho_0 V_0$">|; 

$key = q/V_0sim1,{rmm,s^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="$ V_0\sim 1 {\rm m s^{-1}}$">|; 

$key = q/z_0=-{rmi},pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2196.png"
 ALT="$ z_0=-{\rm i} \pi/2$">|; 

$key = q/x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$ x_2$">|; 

$key = q/displaystyle=-f_y=frac{mu,U,l}{d_0},frac{1}{k}left[2lnleft(frac{1+k}{1-k}right)-3,kright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3715.png"
 ALT="$\displaystyle =-f_y = \frac{\mu U l}{d_0} \frac{1}{k}\left[2\ln\left(\frac{1+k}{1-k}\right)-3 k\right],$">|; 

$key = q/deltaslashLll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2876.png"
 ALT="$ \delta/L\ll 1$">|; 

$key = q/k_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4446.png"
 ALT="$ k_n$">|; 

$key = q/displaystyleoint{bfA}cdotd{bfS}=left(frac{partialA_x}{partialx}+frac{partialA_y}{partialy}+frac{partialA_z}{partialz}right)dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6580.png"
 ALT="$\displaystyle \oint {\bf A}\cdot d{\bf S} = \left(\frac{\partial A_x}{\partial x}+ \frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\right)dV.$">|; 

$key = q/I_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6630.png"
 ALT="$ I_{\rm max}$">|; 

$key = q/alpharightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img951.png"
 ALT="$ \alpha\rightarrow0$">|; 

$key = q/displaystyleF_x=F_y=F_z=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2658.png"
 ALT="$\displaystyle F_x=F_y=F_z=0.$">|; 

$key = q/tan^{-1}(1slashsinhxi_0)=sin^{-1}e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2772.png"
 ALT="$ \tan^{-1}(1/\sinh\xi_0)=\sin^{-1}e$">|; 

$key = q/displaystylefrac{rho_2}{rho_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5485.png"
 ALT="$\displaystyle \frac{\rho_2}{\rho_1}$">|; 

$key = q/displaystyle=(p_2'-p_2),c,cosalpha_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5783.png"
 ALT="$\displaystyle =(p_2'-p_2) c \cos\alpha_0,$">|; 

$key = q/displaystyles=frac{rho}{rho_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img600.png"
 ALT="$\displaystyle s = \frac{\rho}{\rho_0}$">|; 

$key = q/|w|ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3565.png"
 ALT="$ \vert w\vert\ll 1$">|; 

$key = q/p(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1908.png"
 ALT="$ p(r)$">|; 

$key = q/displaystyle(widetilde{nabla{bfA}})_{thetaphi}=(widetilde{nabla{bfA}})_{phitheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7193.png"
 ALT="$\displaystyle (\widetilde{\nabla {\bf A}})_{\theta \phi}=(\widetilde{\nabla {\bf A}})_{\phi\theta}$">|; 

$key = q/|zeta|slash{muskipz@5mudividemuskipz@tw@mkernmuskipz@mathaccentV{tilde}07E{mkern-muskipz@{zeta}mkernmuskipz@}mkern-muskipz@}{}_8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$ \vert\zeta \vert/{\muskip \z@ 5mu\divide \muskip \z@ \tw@ \mkern \muskip \z@ \...
...e}07E{\mkern -\muskip \z@ {\zeta }\mkern \muskip \z@ }\mkern -\muskip \z@ }{}_8$">|; 

$key = q/|costheta|leq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1130.png"
 ALT="$ \vert\cos\theta\vert\leq 1$">|; 

$key = q/{mitDelta}J={mitDelta}J_r+{rmi},{mitDelta}J_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2367.png"
 ALT="$ {\mit\Delta} J={\mit\Delta} J_r+{\rm i} {\mit\Delta} J_i$">|; 

$key = q/displaystylep_2(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1472.png"
 ALT="$\displaystyle p_2(z)$">|; 

$key = q/{mitPhi}_{rmext};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4390.png"
 ALT="$ {\mit\Phi}_{\rm ext}$">|; 

$key = q/v_x(x,y)=U(x),F[yslashdelta(x)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2914.png"
 ALT="$ v_x(x,y)= U(x) F[y/\delta(x)]$">|; 

$key = q/displaystylep(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2867.png"
 ALT="$\displaystyle p(x,y)$">|; 

$key = q/displaystyle=tan^{-1}left[left(frac{gamma-1}{gamma+1}right)^{1slash2}({rmMa}_1^{,2}-1)^{1slash2}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6129.png"
 ALT="$\displaystyle = \tan^{-1}\left[\left(\frac{\gamma-1}{\gamma+1}\right)^{1/2}({\rm Ma}_1^{ 2}-1)^{1/2}\right],$">|; 

$key = q/times{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4454.png"
 ALT="$ \times {\bf v}$">|; 

$key = q/{bfa},%,{bfb}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6296.png"
 ALT="$ {\bf a} \% {\bf b}= 1$">|; 

$key = q/partialTslashpartialxequiv(partialTslashpartialx)_{y,z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6542.png"
 ALT="$ \partial T/\partial x\equiv (\partial T/\partial x)_{y, z}$">|; 

$key = q/epsilon_{ijk},epsilon_{ilm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6814.png"
 ALT="$ \epsilon_{ijk} \epsilon_{ilm}$">|; 

$key = q/displaystyledeltasimeqfrac{T}{W}-frac{2}{A},frac{W}{L_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3529.png"
 ALT="$\displaystyle \delta\simeq \frac{T}{W} - \frac{2}{A} \frac{W}{L_0}.$">|; 

$key = q/displaystyle=(,[a_x,c_x+a_y,c_y],b_x,,0,,0)-(a_x,b_x,c_x,,a_x,b_x,c_y,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6430.png"
 ALT="$\displaystyle = ( [a_x  c_x + a_y  c_y]  b_x,  0,  0) - (a_x  b_x  c_x,  a_x  b_x  c_y,  0)$">|; 

$key = q/displaystylea'=a.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6843.png"
 ALT="$\displaystyle a' = a.$">|; 

$key = q/displaystylea_{sjks};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6953.png"
 ALT="$\displaystyle a_{sjks}$">|; 

$key = q/|{bfh}|^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6741.png"
 ALT="$ \vert{\bf h}\vert^{-1}$">|; 

$key = q/skew{5}bar{zeta}'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5077.png"
 ALT="$ \skew{5}\bar{\zeta}'=0$">|; 

$key = q/displaystylev(x_0,y)=infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3105.png"
 ALT="$\displaystyle v(x_0,y) = \infty$">|; 

$key = q/y_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1981.png"
 ALT="$ y_0)$">|; 

$key = q/displaystyle=frac{V_0}{(g,L)^{1slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="$\displaystyle = \frac{V_0}{(g L)^{1/2}}.$">|; 

$key = q/mu_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4829.png"
 ALT="$ \mu_r$">|; 

$key = q/displaystyleint_{mitOmega}g,D,f,d{mitOmega}=-int_{mitOmega}left(frac{partialg}{ppartialphi}right)d{mitOmega}=int_{mitOmega}f,D,g,d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="536" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4803.png"
 ALT="$\displaystyle \int_{\mit\Omega}g D f d{\mit\Omega} =-\int_{\mit\Omega}\left(...
...f}{\partial\phi}\right)d{\mit\Omega}= \int_{\mit\Omega} f D g d{\mit\Omega}.$">|; 

$key = q/displaystylefrac{partial{bfv}}{partialt}+f,{rme}_ztimes{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5070.png"
 ALT="$\displaystyle \frac{\partial{\bf v}}{\partial t} + f {\rm e}_z\times {\bf v}$">|; 

$key = q/displaystylefrac{p_2}{p_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5483.png"
 ALT="$\displaystyle \frac{p_2}{p_1}$">|; 

$key = q/(1slash2),v^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1374.png"
 ALT="$ (1/2) v^{ 2}$">|; 

$key = q/displaystyle{bfa}times{bfb}=-{bfb}times{bfa},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6364.png"
 ALT="$\displaystyle {\bf a}\times{\bf b} = - {\bf b} \times{\bf a},$">|; 

$key = q/displaystyle=-frac{mitGamma}{2pi},theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1985.png"
 ALT="$\displaystyle =-\frac{\mit\Gamma}{2\pi} \theta,$">|; 

$key = q/displaystyleleft(frac{partialT}{partialv}right)_{calS}=-left(frac{partialp}{partial{calS}}right)_v.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5580.png"
 ALT="$\displaystyle \left(\frac{\partial T}{\partial v}\right)_{\cal S} = -\left(\frac{\partial p}{\partial {\cal S}}\right)_v.
$">|; 

$key = q/displaystyle=a,omega,(k,d)^{-1},sin(omega,t-k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3959.png"
 ALT="$\displaystyle = a \omega (k d)^{-1} \sin(\omega t-k x)$">|; 

$key = q/displaystyle2,a_{ijk}+a_{jik}+a_{ikj};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6917.png"
 ALT="$\displaystyle 2 a_{ijk} + a_{jik} + a_{ikj}$">|; 

$key = q/displaystyle=nu,delta_{jk}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6954.png"
 ALT="$\displaystyle = \nu \delta_{jk}.$">|; 

$key = q/displaystylenablatimes{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2560.png"
 ALT="$\displaystyle \nabla\times {\bf v}$">|; 

$key = q/displaystyle=frac{2,gamma_1+(gamma_1+1),(p_2slashp_1-1)}{2,gamma_1+(gamma_1-1),(p_2slashp_1-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5647.png"
 ALT="$\displaystyle = \frac{2 \gamma_1+(\gamma_1+1) (p_2/p_1-1)}{2 \gamma_1+(\gamma_1-1) (p_2/p_1-1)},$">|; 

$key = q/oint;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6560.png"
 ALT="$ \oint$">|; 

$key = q/nablatimesnablaphiequiv0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1565.png"
 ALT="$ \nabla\times \nabla\phi\equiv 0$">|; 

$key = q/displaystyle=V,{bfe}_parallel,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3318.png"
 ALT="$\displaystyle = V {\bf e}_\parallel,$">|; 

$key = q/displaystylesimeqleft(frac{gamma-1}{gamma}right)frac{{mitDelta}p}{p_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5513.png"
 ALT="$\displaystyle \simeq \left(\frac{\gamma-1}{\gamma}\right)\frac{{\mit\Delta}p}{p_1},$">|; 

$key = q/u_3+du_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7044.png"
 ALT="$ u_3+d u_3$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter05slashwedge.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="307" BORDER="0"
 SRC="|."$dir".q|img1868.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter05/wedge.eps}}
\end{figure}">|; 

$key = q/x>V_s,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5525.png"
 ALT="$ x&gt;V_s t$">|; 

$key = q/dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$ d V$">|; 

$key = q/displaystyleW_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img580.png"
 ALT="$\displaystyle W_x$">|; 

$key = q/x=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3005.png"
 ALT="$ x=l$">|; 

$key = q/displaystylephantom{simeqfrac{G,m',a^{2}}{R^{,3}}}+frac{1}{2},sindelta,{calR}_2^{,(-1)}(r,theta,varphi-2,omega_1,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4305.png"
 ALT="$\displaystyle \phantom{\simeq \frac{G m' a^{2}}{R^{ 3}}}+\frac{1}{2} \sin\delta {\cal R}_2^{ (-1)}(r,\theta,\varphi-2 \omega_1 t)$">|; 

$key = q/a_2=a_1,cosbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img996.png"
 ALT="$ a_2=a_1 \cos\beta$">|; 

$key = q/displaystylexi_{i,n}=A_n,r^{,2},frac{partialp_n}{partialx_i}+B_n,p_n,x_i+frac{partialphi_n}{partialx_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4411.png"
 ALT="$\displaystyle \xi_{i n} = A_n r^{ 2} \frac{\partial p_n}{\partial x_i} + B_n p_n x_i + \frac{\partial\phi_n}{\partial x_i}.$">|; 

$key = q/displaystyleintx_i^{,2},dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img848.png"
 ALT="$\displaystyle \int x_i^{ 2} dV$">|; 

$key = q/rho';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4083.png"
 ALT="$ \rho'$">|; 

$key = q/displaystyleoverline{z}=left.int!int!intz,dVrightslashint!int!intdV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6508.png"
 ALT="$\displaystyle \overline{z} = \left. \int\!\int\!\int z dV\right/ \int\!\int\!\int dV.$">|; 

$key = q/V_s=u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5529.png"
 ALT="$ V_s=u_1$">|; 

$key = q/Rggl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1119.png"
 ALT="$ R\gg l$">|; 

$key = q/y=infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2906.png"
 ALT="$ y=\infty$">|; 

$key = q/displaystyleQ=frac{pi,G,a^{,4}}{8,mu}left[n^4-1-frac{(n^2-1)^2}{lnn}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3877.png"
 ALT="$\displaystyle Q = \frac{\pi G a^{ 4}}{8 \mu}\left[n^4-1 - \frac{(n^2-1)^2}{\ln n}\right],
$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{AppendixAslashfigA.16.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="249" BORDER="0"
 SRC="|."$dir".q|img6483.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{AppendixA/figA.16.eps}}
\end{figure}">|; 

$key = q/displaystylecosgamma=costheta,costheta'+sintheta,sintheta',cos(varphi-varphi').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4256.png"
 ALT="$\displaystyle \cos\gamma = \cos\theta \cos\theta'+\sin\theta \sin\theta' \cos(\varphi-\varphi').$">|; 

$key = q/displaystyle={bfA}cdotnablaA_r-frac{A_theta^{,2}+A_phi^{,2}}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7185.png"
 ALT="$\displaystyle ={\bf A}\cdot\nabla A_r - \frac{A_\theta^{ 2}+A_\phi^{ 2}}{r},$">|; 

$key = q/u-u_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3113.png"
 ALT="$ u-u_0$">|; 

$key = q/displaystylev_r(r,theta,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1804.png"
 ALT="$\displaystyle v_r(r,\theta,t)$">|; 

$key = q/j=6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$ j=6$">|; 

$key = q/displaystyle=left(frac{gamma+1}{gamma-1}right)^{1slash2}tan^{-1}left(left[left(f}{gamma+1}right)({rmMa}^{,2}-1)right]^{1slash2}right)nonumber;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6211.png"
 ALT="$\displaystyle = \left(\frac{\gamma+1}{\gamma-1}\right)^{1/2}\tan^{-1}\left(\lef...
...\frac{\gamma-1}{\gamma+1}\right)({\rm Ma}^{ 2}-1)\right]^{1/2}\right)\nonumber$">|; 

$key = q/displaystylezeta=a,cos(omega,t-k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4204.png"
 ALT="$\displaystyle \zeta = a \cos(\omega t-k x)
$">|; 

$key = q/displaystylenablau_icdotnablau_jtimesnablau_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7061.png"
 ALT="$\displaystyle \nabla u_i\cdot\nabla u_j\times \nabla u_k$">|; 

$key = q/c_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4191.png"
 ALT="$ c_1$">|; 

$key = q/{calE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img518.png"
 ALT="$ {\cal E}$">|; 

$key = q/beta=23.1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4715.png"
 ALT="$ \beta=23.1$">|; 

$key = q/{rmMa}_inftysimeq1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6039.png"
 ALT="$ {\rm Ma}_\infty\simeq 1)$">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$ x$">|; 

$key = q/x=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6743.png"
 ALT="$ x=a$">|; 

$key = q/gamma_{23};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1125.png"
 ALT="$ \gamma_{23}$">|; 

$key = q/mu=0.06757;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1033.png"
 ALT="$ \mu=0.06757$">|; 

$key = q/overline{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="$ \overline{p}$">|; 

$key = q/urightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3048.png"
 ALT="$ u\rightarrow 0$">|; 

$key = q/displaystylephantom{=}+left[frac{1}{r,sintheta},frac{partialA_r}{partialphi}-frac{1}{r}frac{partial}{partialr},(r,A_phi)right]{bfe}_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7181.png"
 ALT="$\displaystyle \phantom{=}+\left[\frac{1}{r \sin\theta} \frac{\partial A_r}{\p...
... \phi}-\frac{1}{r}\frac{\partial}{\partial r} (r A_\phi)\right]{\bf e}_\theta$">|; 

$key = q/theta=sin^{-1}(-gammaslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1774.png"
 ALT="$ \theta = \sin^{-1}(-\gamma/2)$">|; 

$key = q/displaystyle-frac{1}{2},V,a^{,2}=F(xi_0)=frac{1}{2},Bleft[frac{(1-e^{,2})^{1slash2}}{e}-frac{sin^{-1}e}{e^{,2}}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2773.png"
 ALT="$\displaystyle -\frac{1}{2} V a^{ 2} = F(\xi_0)= \frac{1}{2} B\left[\frac{(1-e^{ 2})^{1/2}}{e}-\frac{\sin^{-1}e}{e^{ 2}}\right],$">|; 

$key = q/x=y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6446.png"
 ALT="$ x=y$">|; 

$key = q/v_y=v_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6764.png"
 ALT="$ v_y= v_2$">|; 

$key = q/displaystyle{bfc}={bfa}+{bfb}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6230.png"
 ALT="$\displaystyle {\bf c} = {\bf a} + {\bf b}.$">|; 

$key = q/displaystyle=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="$\displaystyle =0.$">|; 

$key = q/u(x,t)=u_1(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5352.png"
 ALT="$ u(x,t)=u_1(x,t)$">|; 

$key = q/u_{ji}=u_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6979.png"
 ALT="$ u_{ji}=u_{ij}$">|; 

$key = q/c=a^{,2}slashb;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2046.png"
 ALT="$ c=a^{ 2}/b$">|; 

$key = q/displaystyle+f_jleft[frac{n,(n+2),(n+m_j+1)}{2,n+3}right]{mitPsi}_{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4696.png"
 ALT="$\displaystyle +f_j\left[\frac{n (n+2) (n+m_j+1)}{2 n+3}\right]{\mit\Psi}_{n+1}$">|; 

$key = q/displaystyle=left.int_0^1x,y^{,2},dxright|_{y=0}+left.int_0^1x,y^{,2},dyright|_{x=1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6450.png"
 ALT="$\displaystyle = \left.\int_0^1 x y^{ 2} dx\right\vert _{y=0}+\left.\int_0^1 x y^{ 2}  dy \right\vert _{x=1}$">|; 

$key = q/Dsim{calO}(alpha^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3525.png"
 ALT="$ D\sim{\cal O}(\alpha^{ 2})$">|; 

$key = q/displaystyle=a,omega,(1+zslashd),cos(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3960.png"
 ALT="$\displaystyle =a \omega (1+z/d) \cos(\omega t-k x).$">|; 

$key = q/partialxslashpartialu=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3110.png"
 ALT="$ \partial x/\partial u=0$">|; 

$key = q/{bfr}equiv(x',,y',,z');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6283.png"
 ALT="$ {\bf r}\equiv (x', y', z')$">|; 

$key = q/{mitPhi}_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4359.png"
 ALT="$ {\mit\Phi}_n$">|; 

$key = q/alpha<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1329.png"
 ALT="$ \alpha&lt;0$">|; 

$key = q/displaystyle=a,(2,s-tantheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img741.png"
 ALT="$\displaystyle = a (2 s-\tan\theta),$">|; 

$key = q/displaystyleJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2362.png"
 ALT="$\displaystyle J$">|; 

$key = q/lggL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3039.png"
 ALT="$ l\gg L$">|; 

$key = q/displaystylefrac{dw}{w}=frac{d{rmMa}}{{rmMa}}+frac{dc}{c}=left(frac{1}{1+[(gamma-1)slash2],{rmMa}^{,2}}right)frac{d{rmMa}}{{rmMa}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5770.png"
 ALT="$\displaystyle \frac{dw}{w} = \frac{d{\rm Ma}}{{\rm Ma}}+\frac{dc}{c} = \left(\frac{1}{1+[(\gamma-1)/2] {\rm Ma}^{ 2}}\right)\frac{d{\rm Ma}}{{\rm Ma}}.$">|; 

$key = q/y(x)rightarrowy(x)+deltay(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7302.png"
 ALT="$ y(x)\rightarrow y(x)+\delta y(x)$">|; 

$key = q/displaystyle{bfa}=a_x,{bfe}_x+a_y,{bfe}_y+a_z,{bfe}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6259.png"
 ALT="$\displaystyle {\bf a} = a_x {\bf e}_x+ a_y {\bf e}_y+a_z {\bf e}_z,$">|; 

$key = q/e_{13}=0.92995;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img920.png"
 ALT="$ e_{13}=0.92995$">|; 

$key = q/displaystylenablacdot{bfa}=frac{partiala_i}{partialx_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6883.png"
 ALT="$\displaystyle \nabla\cdot {\bf a} = \frac{\partial a_i}{\partial x_i}.$">|; 

$key = q/beta_{pmn,,pmn'}^{,pmm,,pmm'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4957.png"
 ALT="$ \beta_{\pm n, \pm n'}^{ \pm m, \pm m'}$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialy^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6022.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial y^{ 2}}$">|; 

$key = q/displaystyle=frac{1}{36,s^{,2}}left[3,t^{,2}-12,s,(1-s)+2right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img759.png"
 ALT="$\displaystyle = \frac{1}{36 s^{ 2}}\left[3 t^{ 2}-12 s (1-s)+2\right].$">|; 

$key = q/displaystylefrac{x}{l}=cosh^{-1}left(frac{2,l}{z}right)-cosh^{-1}left(frac{2,l}{right)^{1slash2}-left(4-frac{z^{,2}}{l^{,2}}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="548" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1229.png"
 ALT="$\displaystyle \frac{x}{l} = \cosh^{-1}\left(\frac{2 l}{z}\right)-\cosh^{-1}\le...
...c{h^{ 2}}{l^{ 2}}\right)^{1/2}- \left(4-\frac{z^{ 2}}{l^{ 2}}\right)^{1/2},$">|; 

$key = q/displaystyleh_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7117.png"
 ALT="$\displaystyle h_\theta$">|; 

$key = q/displaystyle=V,t-v_g,t,sinbeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4034.png"
 ALT="$\displaystyle = V t-v_g t \sin\beta,$">|; 

$key = q/-psi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2353.png"
 ALT="$ -\psi)$">|; 

$key = q/partial{mitOmega}_varphislashpartialvarphi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2556.png"
 ALT="$ \partial{\mit\Omega}_\varphi/\partial\varphi=0$">|; 

$key = q/displaystyle=a_{ki},b_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6802.png"
 ALT="$\displaystyle = a_{ki} b_i,$">|; 

$key = q/g_0=9.81,{rmm,s^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img914.png"
 ALT="$ g_0=9.81 {\rm m s^{-1}}$">|; 

$key = q/sigma_{ij},dS_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$ \sigma_{ij} dS_j$">|; 

$key = q/displaystyleY(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1855.png"
 ALT="$\displaystyle Y(t)$">|; 

$key = q/|z|slasha>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1998.png"
 ALT="$ \vert z\vert/a&gt;1$">|; 

$key = q/displaystyle=-rho,k,V^{,2}int_{+1}^{,-1}left(frac{V}{v_y}-frac{v_y}{V}right)frac{dlambda}{(lambda-cosalpha)^3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2333.png"
 ALT="$\displaystyle =-\rho k V^{ 2}\int_{+1}^{ -1}\left(\frac{V}{v_y}-\frac{v_y}{V}\right)\frac{d\lambda}{(\lambda-\cos\alpha)^3},$">|; 

$key = q/cproptosqrt{T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5768.png"
 ALT="$ c\propto \sqrt{T}$">|; 

$key = q/displaystylefrac{dz}{dphi}=frac{1}{f_z},frac{df}{dphi}=pmleft(f+frac{alpha,beta}{f}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1307.png"
 ALT="$\displaystyle \frac{dz}{d\phi} = \frac{1}{f_z} \frac{df}{d\phi} = \pm \left(f + \frac{\alpha \beta}{f}\right),$">|; 

$key = q/b_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6771.png"
 ALT="$ b_i$">|; 

$key = q/displaystyle=frac{1}{sinbeta,tanbeta,tangamma}left[F(gamma,alpha),(1+cos^2beta)-E(gamma,alpha)left(1+frac{cos^2beta}{cos^2alpha}right)right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="534" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1008.png"
 ALT="$\displaystyle = \frac{1}{\sin\beta \tan\beta \tan\gamma}\left[F(\gamma,\alpha...
...^2\beta)-E(\gamma,\alpha)\left(1+\frac{\cos^2\beta}{\cos^2\alpha}\right)\right.$">|; 

$key = q/A';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2191.png"
 ALT="$ A'$">|; 

$key = q/displaystylefrac{partialpsi(x,0)}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2910.png"
 ALT="$\displaystyle \frac{\partial\psi(x,0)}{\partial y}$">|; 

$key = q/c,|{rmcosech}|,eta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2476.png"
 ALT="$ c \vert{\rm cosech}\vert \eta_0$">|; 

$key = q/displaystyle(rho-rho'),g=k,rho,(V-c)^2+k,rho',(V'-c)^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4111.png"
 ALT="$\displaystyle (\rho-\rho') g = k \rho (V-c)^2+k \rho' (V'-c)^2.$">|; 

$key = q/displaystylealpha_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4431.png"
 ALT="$\displaystyle \alpha_n$">|; 

$key = q/displaystyleint_a^bleft(frac{partialF}{partialy},deltay+frac{partialF}{partialy'},deltay'right),dx=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7312.png"
 ALT="$\displaystyle \int_a^b\left(\frac{\partial F}{\partial y} \delta y+ \frac{\partial F}{\partial y'} \delta y'\right) dx = 0.$">|; 

$key = q/displaystyleh=h_1,h_2,h_3.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7055.png"
 ALT="$\displaystyle h = h_1 h_2 h_3.$">|; 

$key = q/displaystyle=sum_{i=1,3}frac{n_i^{,2}}{a_i^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7242.png"
 ALT="$\displaystyle = \sum_{i=1,3}\frac{n_i^{ 2}}{a_i^{ 2}},$">|; 

$key = q/s=s_0<1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img726.png"
 ALT="$ s=s_0&lt;1/2$">|; 

$key = q/e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1070.png"
 ALT="$ e$">|; 

$key = q/displaystyle={rmi}^{,n+n'+1},beta_{n,,n'}^{,m,,m'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4999.png"
 ALT="$\displaystyle = {\rm i}^{ n+n'+1} \beta_{n, n'}^{ m, m'},$">|; 

$key = q/skew{5}tilde{zeta}_3=4.25times10^{-2},{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5037.png"
 ALT="$ \skew{5}\tilde{\zeta}_3=4.25\times 10^{-2} {\rm m}$">|; 

$key = q/displaystyleU=-frac{3}{8},G,M^{,2}left(alpha_0-frac{1}{5},sum_{i=1,3}alpha_i,a_i^{,2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7274.png"
 ALT="$\displaystyle U = -\frac{3}{8} G M^{ 2}\left(\alpha_0 - \frac{1}{5} \sum_{i=1,3}\alpha_i a_i^{ 2}\right).$">|; 

$key = q/displaystyle=frac{1}{2}+frac{1}{pi}left[sin^{-1}(lambda)+lambda,(1-lambda^{,2})^{1slash2}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2520.png"
 ALT="$\displaystyle = \frac{1}{2}+\frac{1}{\pi}\left[\sin^{-1}(\lambda)+\lambda (1-\lambda^{ 2})^{1/2}\right].$">|; 

$key = q/{calH}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5863.png"
 ALT="$ {\cal H}_0$">|; 

$key = q/displaystylenablaleft(frac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi}-frac{partialphi}{partialt}right)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1590.png"
 ALT="$\displaystyle \nabla\left(\frac{p}{\rho} + \frac{1}{2} v^{ 2} + {\mit\Psi} - \frac{\partial\phi}{\partial t}\right)=0,$">|; 

$key = q/z_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4679.png"
 ALT="$ z_n$">|; 

$key = q/g(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4797.png"
 ALT="$ g(\theta,\phi)$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{AppendixAslashfigA.22.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="284" BORDER="0"
 SRC="|."$dir".q|img6588.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{AppendixA/figA.22.eps}}
\end{figure}">|; 

$key = q/v_g,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4033.png"
 ALT="$ v_g t$">|; 

$key = q/p_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1723.png"
 ALT="$ p_\infty$">|; 

$key = q/displaystyle=frac{1}{6},(b,c-3,d)-frac{1}{27},b^{,3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6096.png"
 ALT="$\displaystyle = \frac{1}{6} (b c-3 d)-\frac{1}{27} b^{ 3},$">|; 

$key = q/h_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1466.png"
 ALT="$ h_2$">|; 

$key = q/displaystyle{bfgrad},hequivleft(frac{partialh}{partialx},,frac{partialh}{partialy}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6529.png"
 ALT="$\displaystyle {\bf grad} h \equiv \left(\frac{\partial h}{\partial x},  \frac{\partial h}{\partial y} \right).$">|; 

$key = q/displaystylefrac{{mitDelta}w}{w_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5744.png"
 ALT="$\displaystyle \frac{{\mit\Delta}w}{w_1}$">|; 

$key = q/displaystyle=frac{c_0}{2},sinphi,costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3481.png"
 ALT="$\displaystyle =\frac{c_0}{2} \sin\phi \cos\theta,$">|; 

$key = q/displaystylep_0(a)=skew{3}bar{rho},g,d+p_{rmatm}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4397.png"
 ALT="$\displaystyle p_0(a) = \skew{3}\bar{\rho} g  d+p_{\rm atm}.$">|; 

$key = q/displaystyleleft[frac{partial}{partialt}+(u+c),frac{partial}{partialx}right]left(u+frac{2,c}{gamma-1}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5658.png"
 ALT="$\displaystyle \left[\frac{\partial}{\partial t} + (u+c) \frac{\partial}{\partial x}\right]\left(u+\frac{2 c}{\gamma-1}\right)$">|; 

$key = q/x=+d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1198.png"
 ALT="$ x=+d$">|; 

$key = q/displaystylep=K,rho^{,(1+n)slashn},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5271.png"
 ALT="$\displaystyle p = K \rho^{ (1+n)/n},
$">|; 

$key = q/displaystyle=frac{sqrt{6}}{5},frac{(cosgamma)^{-1slash6}}{singamma}left[(1+2,cos^2gamma),frac{gamma}{singamma}-3,cosgammaright]^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="430" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img898.png"
 ALT="$\displaystyle = \frac{\sqrt{6}}{5} \frac{(\cos\gamma)^{-1/6}}{\sin\gamma}\left[(1+2 \cos^2\gamma) \frac{\gamma}{\sin\gamma}-3 \cos\gamma\right]^{1/2},$">|; 

$key = q/displaystyle={bfb}cdot{bfa},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6306.png"
 ALT="$\displaystyle = {\bf b} \cdot {\bf a},$">|; 

$key = q/displaystyleA_z(y+dy),dz-A_z(y),dz=frac{partialA_z}{partialy},dy,dz.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6631.png"
 ALT="$\displaystyle A_z(y+dy) dz - A_z(y) dz = \frac{\partial A_z}{\partial y}  dy dz.$">|; 

$key = q/dslashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1689.png"
 ALT="$ d/r$">|; 

$key = q/-10.36times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4625.png"
 ALT="$ -10.36\times 10^{-2}$">|; 

$key = q/displaystylefrac{1}{V},frac{DV}{Dt}=frac{DlnV}{Dt}=nablacdot{bfv}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="$\displaystyle \frac{1}{V} \frac{DV}{Dt} = \frac{D\ln V}{Dt} = \nabla\cdot{\bf v}.$">|; 

$key = q/displaystyle=-frac{1}{rho},frac{partialp}{partialy}-frac{partial{mitPsi}}{partiaa^{,2}v_y+frac{1}{3},frac{partial{mitDelta}}{partialy}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="$\displaystyle = - \frac{1}{\rho} \frac{\partial p}{\partial y} - \frac{\partia...
...\nabla^{ 2} v_y + \frac{1}{3} \frac{\partial{\mit\Delta}}{\partial y}\right),$">|; 

$key = q/AQ_2P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2532.png"
 ALT="$ A Q_2 P$">|; 

$key = q/displaystylesimeqfrac{(gamma+1)}{12,gamma^{,2}}left(frac{{mitDelta}p}{p_1}right)^3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6115.png"
 ALT="$\displaystyle \simeq \frac{(\gamma+1)}{12 \gamma^{ 2}}\left(\frac{{\mit\Delta}p}{p_1}\right)^3,$">|; 

$key = q/displaystyle{calL}=frac{partial^{,2}}{partialr^{,2}}+frac{sintheta}{r^{,2}},frac{partialtheta},frac{1}{sintheta},frac{partial}{partialtheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3736.png"
 ALT="$\displaystyle {\cal L} = \frac{\partial^{ 2}}{\partial r^{ 2}} + \frac{\sin\t...
...artial}{\partial\theta} \frac{1}{\sin\theta} \frac{\partial}{\partial\theta}.$">|; 

$key = q/displaystylef_y=-int_0^l[p(x)-p_1],dx=-frac{6,mu,U}{alpha^{,2}}left[lnleft(frac{d_1}{d_2}right)-2left(frac{d_1-d_2}{d_1+d_2}right)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="509" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3710.png"
 ALT="$\displaystyle f_y = -\int_0^l[p(x)-p_1] dx = -\frac{6 \mu U}{\alpha^{ 2}}\left[\ln\left(\frac{d_1}{d_2}\right)-2\left(\frac{d_1-d_2}{d_1+d_2}\right)\right].$">|; 

$key = q/x^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3004.png"
 ALT="$ x^{1/2}$">|; 

$key = q/costheta<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1152.png"
 ALT="$ \cos\theta&lt;0$">|; 

$key = q/displaystyleskew{5}ddot{mitPhi}-2,{mitOmega},skew{5}dot{mitPhi}'-frac{g,d}{a^{,2}},(zeta-{calF});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4791.png"
 ALT="$\displaystyle \skew{5}\ddot{\mit\Phi} -2 {\mit\Omega} \skew{5}\dot{\mit\Phi}' - \frac{g d}{a^{ 2}} (\zeta-{\cal F})$">|; 

$key = q/displaystyleI(z)=-frac{d{mitGamma}}{dz}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3469.png"
 ALT="$\displaystyle I(z) = - \frac{d{\mit\Gamma}}{dz}.$">|; 

$key = q/displaystyleint_0^{xslash({rmMa}_infty^{,2}-1)^{,1slash2}}pi^{-1},cos^{-1}left(1-2sqrt{{rmMa}_infty^{,2}-1},frac{y}{x}right)dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6071.png"
 ALT="$\displaystyle \int_0^{x/({\rm Ma}_\infty^{ 2}-1)^{ 1/2}}\pi^{-1}  \cos^{-1}\left(1-2\sqrt{{\rm Ma}_\infty^{ 2}-1} \frac{y}{x}\right)dy$">|; 

$key = q/{figure}{epsfysize{=1.25in{centerline{epsffile{Chapter09slashsheet.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="513" HEIGHT="128" BORDER="0"
 SRC="|."$dir".q|img3373.png"
 ALT="\begin{figure}
\epsfysize =1.25in
\centerline{\epsffile{Chapter09/sheet.eps}}
\end{figure}">|; 

$key = q/(0,,0,,a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6507.png"
 ALT="$ (0, 0, a)$">|; 

$key = q/MslashM'll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img981.png"
 ALT="$ M/M'\ll 1$">|; 

$key = q/H(krightarrow0)rightarrow3slash(4,k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3723.png"
 ALT="$ H(k\rightarrow 0)\rightarrow 3/(4 k)$">|; 

$key = q/displaystylefrac{d}{dx},(rho,u)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5451.png"
 ALT="$\displaystyle \frac{d}{dx} (\rho u) = 0.$">|; 

$key = q/phi({bfr})=-int_O^P{bfv}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1578.png"
 ALT="$ \phi({\bf r})=-\int_O^P {\bf v}\cdot d{\bf r}$">|; 

$key = q/h_2slashh_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1484.png"
 ALT="$ h_2/h_1$">|; 

$key = q/displaystyle={calA},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6079.png"
 ALT="$\displaystyle ={\cal A},$">|; 

$key = q/displaystyled=frac{1}{4},(c-delta),frac{sinalpha}{widehat{mitGamma}_infty}=frac{1}{4},(c-delta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3351.png"
 ALT="$\displaystyle d = \frac{1}{4} (c-\delta) \frac{\sin\alpha}{\widehat{\mit\Gamma}_\infty} = \frac{1}{4} (c-\delta)$">|; 

$key = q/displaystyle=-frac{g,d}{a^{,2}},F_r,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4913.png"
 ALT="$\displaystyle = -\frac{g d}{a^{ 2}} F_r,$">|; 

$key = q/theta(xi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$ \theta (\xi )$">|; 

$key = q/displaystylefrac{dp}{dx}=-G(x)=6,muleft[frac{U}{d^{,2}(x)}-frac{2,Q}{d^{,3}(x)}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3699.png"
 ALT="$\displaystyle \frac{dp}{dx} = -G(x) = 6 \mu\left[\frac{U}{d^{ 2}(x)}- \frac{2 Q}{d^{ 3}(x)}\right],$">|; 

$key = q/displaystyle=-frac{f_j,n,(n+1),(n+m_j+1),(n+m_j+2)}{(2,n+3),(2,n+5),[m_j-(n+1),(n+2),f_j]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="392" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4707.png"
 ALT="$\displaystyle =-\frac{f_j n (n+1) (n+m_j+1) (n+m_j+2)}{(2 n+3) (2 n+5) [m_j-(n+1) (n+2) f_j]}.$">|; 

$key = q/displaystyleF_r=P_r-sum_{s=1,infty}gamma_{r,,s},p_s,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4916.png"
 ALT="$\displaystyle F_r = P_r - \sum_{s=1,\infty}\gamma_{r, s} p_s,$">|; 

$key = q/displaystylefrac{d{mitTheta}}{dt}=S_{mitTheta}-{mitPhi}_{mitTheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="$\displaystyle \frac{d{\mit\Theta}}{dt} = S_{\mit\Theta}-{\mit\Phi}_{\mit\Theta}.$">|; 

$key = q/displaystyle{calC}_p=frac{skew{3}bar{C}_p}{m}={calC}_V+{calR}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5314.png"
 ALT="$\displaystyle {\cal C}_p = \frac{\skew{3}\bar{C}_p}{m} = {\cal C}_V+{\cal R}.$">|; 

$key = q/displaystyleF(x)=A_1,sin(k,x)+A_2,cos(k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5966.png"
 ALT="$\displaystyle F(x) = A_1 \sin(k x)+ A_2 \cos(k x),$">|; 

$key = q/displaystyle{bfv}({bfr})=v_r(r,theta),{bfe}_r+v_theta(r,theta),{bfe}_theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3730.png"
 ALT="$\displaystyle {\bf v}({\bf r}) = v_r(r,\theta) {\bf e}_r + v_\theta(r,\theta) {\bf e}_\theta.$">|; 

$key = q/ngeqm>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4278.png"
 ALT="$ n\geq m&gt;0$">|; 

$key = q/epsilonll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6121.png"
 ALT="$ \epsilon\ll 1$">|; 

$key = q/displaystyleP_1(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4240.png"
 ALT="$\displaystyle P_1(x)$">|; 

$key = q/displaystyle{mitPsi}_r(theta,phi_pm)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4828.png"
 ALT="$\displaystyle {\mit\Psi}_r(\theta,\phi_\pm)=0.$">|; 

$key = q/zeta_c=0.6627;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1282.png"
 ALT="$ \zeta_c=0.6627$">|; 

$key = q/displaystylefrac{partial^{,3}psi}{partialy^{,3}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2934.png"
 ALT="$\displaystyle \frac{\partial^{ 3}\psi}{\partial y^{ 3}}$">|; 

$key = q/F(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5362.png"
 ALT="$ F(x)$">|; 

$key = q/displaystylerho,u,frac{du}{dx}+frac{dp}{dx}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5453.png"
 ALT="$\displaystyle \rho u \frac{du}{dx} + \frac{dp}{dx} = 0,$">|; 

$key = q/x=x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3103.png"
 ALT="$ x=x_0$">|; 

$key = q/displaystyle=left[frac{gamma}{rho,g,(1-s)}right]^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4175.png"
 ALT="$\displaystyle =\left[\frac{\gamma}{\rho g (1-s)}\right]^{1/2}$">|; 

$key = q/-pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4049.png"
 ALT="$ -\pi/2$">|; 

$key = q/displaystyle=left(frac{T}{T_0}right)^{gammaslash(gamma-1)}=left[1+frac{1}{2},(gamma-1),{rmMa}^{,2}right]^{-gammaslash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5389.png"
 ALT="$\displaystyle =\left(\frac{T}{T_0}\right)^{\gamma/(\gamma-1)}=\left[1+\frac{1}{2} (\gamma-1) {\rm Ma}^{ 2}\right]^{-\gamma/(\gamma-1)} ,$">|; 

$key = q/29^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1136.png"
 ALT="$ 29^\circ$">|; 

$key = q/{mitPsi}({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="$ {\mit\Psi}({\bf r}, t)$">|; 

$key = q/displaystyleint_{mitOmega}left[left(frac{partialP'}{partialtheta}right)^2+left(fheta},frac{partialP'}{partialphi}right)^2right]d{mitOmega}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="325" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4752.png"
 ALT="$\displaystyle \int_{\mit\Omega}\left[\left(\frac{\partial P'}{\partial\theta}\r...
...}{\sin\theta} \frac{\partial P'}{\partial\phi}\right)^2\right]d{\mit\Omega}=0.$">|; 

$key = q/c=omegaslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4082.png"
 ALT="$ c=\omega/k$">|; 

$key = q/displaystylea_{ijk}'={calR}_{il},{calR}_{jm},{calR}_{kn},a_{lmn}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6842.png"
 ALT="$\displaystyle a_{ijk}' = {\cal R}_{il} {\cal R}_{jm} {\cal R}_{kn} a_{lmn}.$">|; 

$key = q/zrightarrowz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1191.png"
 ALT="$ z\rightarrow z$">|; 

$key = q/displaystyle={mitOmega}_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3629.png"
 ALT="$\displaystyle = {\mit\Omega}_1,$">|; 

$key = q/rho_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5630.png"
 ALT="$ \rho_4$">|; 

$key = q/displaystylekappa_x^{,2}=frac{int_{-aslash2}^{aslash2}int_{-bslash2}^{bslash2}y^{,2},dx,dy}{a,b}=frac{b^{,2}}{12}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img673.png"
 ALT="$\displaystyle \kappa_x^{ 2} = \frac{\int_{-a/2}^{a/2}\int_{-b/2}^{b/2} y^{ 2} dx dy}{a b} = \frac{b^{ 2}}{12}.$">|; 

$key = q/-{bfF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2427.png"
 ALT="$ -{\bf F}$">|; 

$key = q/{mitDelta}vllv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4502.png"
 ALT="$ {\mit\Delta} v\ll v$">|; 

$key = q/2pi,psi(varpi,z)=-pi,varpi^{,2},V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2599.png"
 ALT="$ 2\pi \psi(\varpi,z) = -\pi \varpi^{ 2} V$">|; 

$key = q/displaystyle=int_{mitOmega}frac{costheta}{sintheta}left(frac{partial{mitPsi}_r}{altheta},frac{partial{mitPsi}_s}{partialphi}right)d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4884.png"
 ALT="$\displaystyle = \int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\left(\frac{\part...
...{\partial\theta} \frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}$">|; 

$key = q/delta_{ik};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6922.png"
 ALT="$ \delta_{ik}$">|; 

$key = q/displaystyleY'(theta,phi_pm,t)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4745.png"
 ALT="$\displaystyle Y'(\theta,\phi_\pm,t) = 0.$">|; 

$key = q/displaystylewidehat{mitGamma}=sinalphaleft[1-exp(-hat{t})right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3342.png"
 ALT="$\displaystyle \widehat{\mit\Gamma} =\sin\alpha\left[1-\exp(-\hat{t})\right].$">|; 

$key = q/displaystyle=skew{5}ddot{mitPsi}-2,{mitOmega},skew{5}dot{mitPsi}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4783.png"
 ALT="$\displaystyle =\skew{5}\ddot{\mit\Psi} - 2 {\mit\Omega} \skew{5}\dot{\mit\Psi}'.$">|; 

$key = q/displaystyleh(x)=2,a,s+x,tantheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img751.png"
 ALT="$\displaystyle h(x) = 2 a s+x \tan\theta.$">|; 

$key = q/{bfv}=-nablaphi+delta{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1610.png"
 ALT="$ {\bf v}=-\nabla\phi + \delta{\bf v}$">|; 

$key = q/x_2-x_1=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3671.png"
 ALT="$ x_2-x_1=l$">|; 

$key = q/p_c=2.38times10^{16},{rmN,m^{-2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5249.png"
 ALT="$ p_c= 2.38\times 10^{16} {\rm
N m^{-2}}$">|; 

$key = q/+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2580.png"
 ALT="$ +1$">|; 

$key = q/0>gammaslash(p_0,|alpha|)geq-0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1333.png"
 ALT="$ 0&gt;\gamma/(p_0 \vert\alpha\vert)\geq -0.5$">|; 

$key = q/displaystyleint_0^1dxint_0^1x^{,2},y,dy=left(int_0^1x^{,2},dxright)left(int_0^1y,dyright)=frac{1}{3},frac{1}{2}=frac{1}{6},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="463" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6498.png"
 ALT="$\displaystyle \int_0^1 dx \int_0^1 x^{ 2}  y dy = \left(\int_0^1 x^{ 2} dx\right) \left(\int_0^1 y  dy\right) = \frac{1}{3} \frac{1}{2} = \frac{1}{6},$">|; 

$key = q/displaystyle=frac{1}{2}left(frac{1}{r,sintheta},frac{partialA_theta}{partialphi}ac{partialA_phi}{partialtheta}-frac{cottheta,A_phi}{r}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7194.png"
 ALT="$\displaystyle =\frac{1}{2}\left(\frac{1}{r \sin\theta} \frac{\partial A_\thet...
...{r}\frac{\partial A_\phi}{\partial \theta}-\frac{\cot\theta A_\phi}{r}\right).$">|; 

$key = q/delta_0=V_0slashA_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img610.png"
 ALT="$ \delta_0= V_0/A_0$">|; 

$key = q/d^{,2}uslashdt^{,2}<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img762.png"
 ALT="$ d^{ 2}u/dt^{ 2}&lt;0$">|; 

$key = q/displaystyleeta=left(frac{U_0}{2,nu,x}right)^{1slash2}|y|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2983.png"
 ALT="$\displaystyle \eta = \left(\frac{U_0}{2 \nu x}\right)^{1/2} \vert y\vert.$">|; 

$key = q/displaystyleZ_1(x+c,t,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5096.png"
 ALT="$\displaystyle Z_1(x+c t,y)$">|; 

$key = q/a_{ij},b_j=a_{i1},b_1+a_{i2},b_2+a_{i3},b_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6775.png"
 ALT="$ a_{ij} b_j= a_{i1} b_1+a_{i2} b_2+a_{i3} b_3$">|; 

$key = q/displaystylel=intds=int_{-aslash2}^{aslash2}(1+y'^{,2})^{1slash2},dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7344.png"
 ALT="$\displaystyle l = \int ds = \int_{-a/2}^{a/2}(1+y'^{ 2})^{1/2} dx$">|; 

$key = q/displaystylesigma_{rtheta}=sigma_{thetar};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="$\displaystyle \sigma_{r\theta}=\sigma_{\theta r}$">|; 

$key = q/zeta_1=zeta_{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4576.png"
 ALT="$ \zeta_1=\zeta_{\mit\Omega}$">|; 

$key = q/{mitGamma}=oint_C{bfv}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1788.png"
 ALT="$ {\mit\Gamma}=\oint_C{\bf v}\cdot d{\bf r}$">|; 

$key = q/theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$ \theta $">|; 

$key = q/deltal=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7306.png"
 ALT="$ \delta l =0$">|; 

$key = q/displaystylenabla^{,2}chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1926.png"
 ALT="$\displaystyle \nabla^{ 2}\chi$">|; 

$key = q/delta(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2915.png"
 ALT="$ \delta(x)$">|; 

$key = q/displaystylepsi(x,y)=psi_0,x^{,p},F(yslashx^{,q}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3248.png"
 ALT="$\displaystyle \psi(x,y)=\psi_0 x^{ p} F(y/x^{ q}).
$">|; 

$key = q/Q+deltaQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3601.png"
 ALT="$ Q+\delta Q$">|; 

$key = q/displaystyle{bfn}=frac{nabla(z-zeta)}{|nabla(z-zeta)|}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4135.png"
 ALT="$\displaystyle {\bf n} = \frac{\nabla(z-\zeta)}{\vert\nabla(z-\zeta)\vert}.$">|; 

$key = q/s>9slash32=0.28125;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img798.png"
 ALT="$ s&gt;9/32=0.28125$">|; 

$key = q/displaystyle=frac{1}{2},rho,V^{,2}int_a^inftyint_0^pileft(frac{a}{r}right)^6lefteta+frac{1}{4},sin^2thetaright)2pi,r^{,2},sintheta,dr,dtheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="498" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2678.png"
 ALT="$\displaystyle = \frac{1}{2} \rho V^{ 2}\int_a^\infty\int_0^\pi \left(\frac{...
...2\theta+\frac{1}{4} \sin^2\theta\right)2\pi r^{ 2} \sin\theta dr d\theta,$">|; 

$key = q/displaystyleb_n^{,m}=left[frac{2,n+1}{4pi},frac{(n-m)!}{(n+m)!}right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4921.png"
 ALT="$\displaystyle b_n^{ m} = \left[\frac{2 n+1}{4\pi} \frac{(n-m)!}{(n+m)!}\right]^{1/2}.$">|; 

$key = q/T_0proptop_0slashrho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5881.png"
 ALT="$ T_0\propto p_0/\rho_0$">|; 

$key = q/displaystyledelta_{ij},a_{ki},b_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6801.png"
 ALT="$\displaystyle \delta_{ij} a_{ki} b_j$">|; 

$key = q/displaystylephi(xi,eta)=frac{1}{2},V,a^{,3},frac{coseta}{(c,{rme}^{,xi})^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2742.png"
 ALT="$\displaystyle \phi(\xi,\eta)= \frac{1}{2} V a^{ 3} \frac{\cos\eta}{(c {\rm e}^{ \xi})^2}.$">|; 

$key = q/d';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4084.png"
 ALT="$ d'$">|; 

$key = q/a_i';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6821.png"
 ALT="$ a_i'$">|; 

$key = q/displaystyledelta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3264.png"
 ALT="$\displaystyle \delta_1$">|; 

$key = q/dot{phantom{eta}}equivpartialslashpartialt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4663.png"
 ALT="$ \dot{\phantom{\eta}}\equiv \partial/\partial t$">|; 

$key = q/rho_0,A,k,(h-k),g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1053.png"
 ALT="$ \rho_0 A k (h-k) g$">|; 

$key = q/displaystyleX-{rmi},Y=frac{1}{8},{rmi},rholeft(frac{Q}{pi}right)^2oint_{C'}left[rac{2}{(z+{rmi},a),(z-{rmi},a)}+frac{1}{(z+{rmi},a)}right]dz.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="593" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2416.png"
 ALT="$\displaystyle X-{\rm i} Y = \frac{1}{8} {\rm i} \rho\left(\frac{Q}{\pi}\righ...
...}+\frac{2}{(z+{\rm i} a) (z-{\rm i} a)} + \frac{1}{(z+{\rm i} a)}\right]dz.$">|; 

$key = q/displaystyle|theta|rightarrow|theta_{rmmax}|equivnu_{rmmax}-nu({rmMa}_1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6151.png"
 ALT="$\displaystyle \vert\theta\vert\rightarrow\vert\theta_{\rm max}\vert\equiv \nu_{\rm max}-\nu({\rm Ma}_1),
$">|; 

$key = q/displaystyleu,frac{drho}{dx}=-rho,frac{du}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5450.png"
 ALT="$\displaystyle u \frac{d\rho}{dx} = -\rho \frac{du}{dx},$">|; 

$key = q/(nablatimes{bfv})_z=2,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6656.png"
 ALT="$ (\nabla\times {\bf v})_z = 2 \omega$">|; 

$key = q/displaystylezeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4894.png"
 ALT="$\displaystyle \zeta$">|; 

$key = q/R=a,xi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5284.png"
 ALT="$ R=a \xi_1$">|; 

$key = q/displaystylefrac{partiala_i'}{partialx_j'}=frac{partiala_i'}{partialx_k},frac{paR}_{jk}={calR}_{il},{calR}_{jk},frac{partiala_l}{partialx_k},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6878.png"
 ALT="$\displaystyle \frac{\partial a_i'}{\partial x_j'} = \frac{\partial a_i'}{\parti...
...{\cal R}_{jk}= {\cal R}_{il} {\cal R}_{jk} \frac{\partial a_l}{\partial x_k},$">|; 

$key = q/displaystyleF(z)=phi(x,y)+{rmi},psi(x,y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1948.png"
 ALT="$\displaystyle F(z) = \phi(x, y) + {\rm i} \psi(x, y),$">|; 

$key = q/displaystyle=-left[1+frac{(2,n+1),(2,n^{,2}+4,n+3)}{(6+n^{,2})},frac{mu}{rho,g,a}right]^{,-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4443.png"
 ALT="$\displaystyle = -\left[1+\frac{(2 n+1) (2 n^{ 2}+4 n+3)}{(6+n^{ 2})} \frac{\mu}{\rho g a}\right]^{ -1}.$">|; 

$key = q/displaystyle=frac{1}{2}left(frac{1}{r,sintheta},frac{partialA_r}{partialphi}+frac{partialA_phi}{partialr}-frac{A_phi}{r}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7192.png"
 ALT="$\displaystyle =\frac{1}{2}\left(\frac{1}{r \sin\theta} \frac{\partial A_r}{\partial \phi} + \frac{\partial A_\phi}{\partial r}-\frac{A_\phi}{r}\right),$">|; 

$key = q/displaystyle{rmFr}_1gtreqless1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1486.png"
 ALT="$\displaystyle {\rm Fr}_1\gtreqless 1.$">|; 

$key = q/{calO}(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2886.png"
 ALT="$ {\cal O}(1)$">|; 

$key = q/displaystyle=-frac{3,mu,V,a,costheta}{r^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3787.png"
 ALT="$\displaystyle = -\frac{3 \mu V a \cos\theta}{r^{ 3}},$">|; 

$key = q/displaystyle{rmMa}_1^{,2},sin^{,2}betasimeq1+2,epsilonsqrt{{rmMa}_1^{,2}-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5737.png"
 ALT="$\displaystyle {\rm Ma}_1^{ 2} \sin^{ 2}\beta \simeq 1+2 \epsilon\sqrt{{\rm Ma}_1^{ 2}-1}.$">|; 

$key = q/displaystyle=frac{a}{d},frac{partialeta}{partialt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4662.png"
 ALT="$\displaystyle = \frac{a}{d} \frac{\partial\eta}{\partial t}.$">|; 

$key = q/displaystyled{bfx}cdotd{bfx}=sum_{i=1,3}h_i^{,2},du_i^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7046.png"
 ALT="$\displaystyle d{\bf x}\cdot d{\bf x}= \sum_{i=1,3} h_i^{ 2} du_i^{ 2}.$">|; 

$key = q/lambda_sneqlambda_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4815.png"
 ALT="$ \lambda_s\neq \lambda_r$">|; 

$key = q/lambda'=-cosh(aslash2,h);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7358.png"
 ALT="$ \lambda' = - \cosh (a/2 h)$">|; 

$key = q/dz={rmi},R,{rme}^{,{rmi},theta},dtheta={rmi},z,dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3280.png"
 ALT="$ dz={\rm i} R {\rm e}^{ {\rm i} \theta} d\theta={\rm i} z d\theta$">|; 

$key = q/-S,p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1414.png"
 ALT="$ -S p$">|; 

$key = q/displaystylephantom{=}left.+frac{2}{r^{,2},sin^2theta},frac{partialv_r}{partialv}{3,r,sintheta},frac{partial{mitDelta}}{partialvarphi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img510.png"
 ALT="$\displaystyle \phantom{=}\left. + \frac{2}{r^{ 2} \sin^2\theta} \frac{\parti...
...rac{1}{3 r \sin\theta} \frac{\partial{\mit\Delta}}{\partial \varphi}\right),$">|; 

$key = q/displaystyle{bfr}''=left(frac{m'}{m+m'}right){bfr}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4228.png"
 ALT="$\displaystyle {\bf r}'' = \left(\frac{m'}{m+m'}\right){\bf r}'$">|; 

$key = q/mu_r,int_{mitOmega}(cdots),{mitPsi}_r,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4911.png"
 ALT="$ \mu_r \int_{\mit\Omega} (\cdots) {\mit\Psi}_r d{\mit\Omega}$">|; 

$key = q/-sinalpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2448.png"
 ALT="$ -\sin\alpha)$">|; 

$key = q/M_0=M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img829.png"
 ALT="$ M_0=M$">|; 

$key = q/displaystyle{bfn}equivfrac{nabla(r-f)}{|nabla(r-f)|}=frac{{bfe}_r-f_z,{bfe}_z}{(1+f_z^{,2})^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1233.png"
 ALT="$\displaystyle {\bf n} \equiv \frac{\nabla(r-f)}{\vert\nabla(r-f)\vert} = \frac{{\bf e}_r- f_z {\bf e}_z}{(1+f_z^{ 2})^{1/2}},$">|; 

$key = q/displaystyleB=-V,a^{,2},{rme}^{-{rmi},alpha}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3296.png"
 ALT="$\displaystyle B = - V a^{ 2} {\rm e}^{-{\rm i} \alpha}.$">|; 

$key = q/y'={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7329.png"
 ALT="$ y' = {\rm constant}$">|; 

$key = q/psi_Q({bfr}-{bfr}_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1685.png"
 ALT="$ \psi_Q({\bf r}-{\bf r}_0)$">|; 

$key = q/displaystylefrac{1}{2}pmsqrt{frac{1}{4}-frac{a^{,2}}{h^{,2}}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1045.png"
 ALT="$\displaystyle \frac{1}{2}\pm \sqrt{\frac{1}{4}-\frac{a^{ 2}}{h^{ 2}}}.
$">|; 

$key = q/3,{bfa}+2,{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6699.png"
 ALT="$ 3 {\bf a}+2 {\bf b}$">|; 

$key = q/theta_1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5778.png"
 ALT="$ \theta_1=0$">|; 

$key = q/nabla(nablacdot{bfA});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6602.png"
 ALT="$ \nabla(\nabla\cdot{\bf A})$">|; 

$key = q/displaystylefrac{du}{dz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img786.png"
 ALT="$\displaystyle \frac{du}{dz}$">|; 

$key = q/displaystylezeta={rmi},{rme}^{,pi,zslasha}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2091.png"
 ALT="$\displaystyle \zeta = {\rm i} {\rm e}^{ \pi z/a}.$">|; 

$key = q/displaystylenuleft.frac{partialu}{partialy}right|_{y=0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3079.png"
 ALT="$\displaystyle \nu\left.\frac{\partial u}{\partial y}\right\vert _{y=0}$">|; 

$key = q/d=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1205.png"
 ALT="$ d=l$">|; 

$key = q/displaystylefrac{partial(h_eta,v_eta)}{partialxi}-frac{partial(h_xi,v_xi)}{partialeta}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2699.png"
 ALT="$\displaystyle \frac{\partial(h_\eta v_\eta)}{\partial\xi} -\frac{\partial(h_\xi v_\xi)}{\partial \eta}=0,$">|; 

$key = q/v=v(theta,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4475.png"
 ALT="$ v=v(\theta,\phi,t)$">|; 

$key = q/zeta=zeta_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4597.png"
 ALT="$ \zeta=\zeta_E$">|; 

$key = q/displaystyle{bfl}={bfr}times{bfp}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6390.png"
 ALT="$\displaystyle {\bf l}={\bf r}\times {\bf p}.$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter08slashplate.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="489" HEIGHT="279" BORDER="0"
 SRC="|."$dir".q|img2981.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter08/plate.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter12slashphasek1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5048.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter12/phasek1.eps}}
\end{figure}">|; 

$key = q/Qslash(U_0,delta)=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3030.png"
 ALT="$ Q/(U_0 \delta)=0.5$">|; 

$key = q/displaystyle=a,omega,{rme}^{,k,z},sin(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3942.png"
 ALT="$\displaystyle = a \omega {\rm e}^{ k z} \sin(\omega t-k x),$">|; 

$key = q/displaystylev_g=frac{1}{2}left(frac{g}{k}right)^{1slash2}=frac{1}{2},v_p.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3932.png"
 ALT="$\displaystyle v_g = \frac{1}{2}\left(\frac{g}{k}\right)^{1/2} = \frac{1}{2} v_p.$">|; 

$key = q/0<sll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1862.png"
 ALT="$ 0&lt;s\ll 1$">|; 

$key = q/displaystyle{mitGamma}_r=oint{bfv}cdotd{bfr}=2pi,r,v_theta(r)={mitGamma},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1705.png"
 ALT="$\displaystyle {\mit\Gamma}_r = \oint {\bf v}\cdot d{\bf r}= 2\pi r v_\theta(r) = {\mit\Gamma},$">|; 

$key = q/displaystylefrac{partialxi}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6010.png"
 ALT="$\displaystyle \frac{\partial\xi}{\partial x}$">|; 

$key = q/displaystyle=p_0-rho',g,zeta+p_1(x,0_+,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4103.png"
 ALT="$\displaystyle = p_0-\rho' g \zeta + p_1(x,0_+,t)$">|; 

$key = q/displaystyler,frac{d}{dr}!left(r,frac{dR}{dr}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1732.png"
 ALT="$\displaystyle r \frac{d}{dr}\!\left(r \frac{dR}{dr}\right)$">|; 

$key = q/{bfa},cos({bfk}cdot{bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6757.png"
 ALT="$ {\bf a} \cos({\bf k}\cdot {\bf r})$">|; 

$key = q/displaystylep=p_0+frac{1}{2},rholeft(V_0^{,2}-v^{,2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1778.png"
 ALT="$\displaystyle p = p_0 + \frac{1}{2} \rho\left(V_0^{ 2}-v^{ 2}\right),$">|; 

$key = q/muslash(rho,g,a)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4644.png"
 ALT="$ \mu/(\rho g a)=0$">|; 

$key = q/displaystylerightarrowfrac{1}{2},V,r^{,2},sin^2theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3764.png"
 ALT="$\displaystyle \rightarrow \frac{1}{2} V r^{ 2} \sin^2\theta.$">|; 

$key = q/displaystyleP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4837.png"
 ALT="$\displaystyle P$">|; 

$key = q/{bfv}=-V,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3754.png"
 ALT="$ {\bf v}=-V {\bf e}_z$">|; 

$key = q/displaystylephi(x,z,t)=(V-c),zeta_0,frac{cosh[k,(z+d)]}{sinh(k,d)},cos(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4093.png"
 ALT="$\displaystyle \phi(x,z,t) = (V-c) \zeta_0 \frac{\cosh[k (z+d)]}{\sinh(k d)} \cos(\omega t-k x).$">|; 

$key = q/displaystyle=Vleft(-cosalpha,{bfe}_xi+sinalpha,{bfe}_etaright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3298.png"
 ALT="$\displaystyle = V\left(-\cos\alpha {\bf e}_\xi + \sin\alpha {\bf e}_\eta\right),$">|; 

$key = q/T_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6996.png"
 ALT="$ T_{ij}$">|; 

$key = q/displaystyleV'_c=frac{(1+s)}{s^{1slash2}},c_0=6.6,{rmmslashs}=12.8,{rmkts};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4197.png"
 ALT="$\displaystyle V'_c = \frac{(1+s)}{s^{1/2}} c_0=6.6 {\rm m/s} = 12.8 {\rm kts}$">|; 

$key = q/displaystyle({bfa}times{bfb})^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6372.png"
 ALT="$\displaystyle ({\bf a}\times{\bf b})^2$">|; 

$key = q/displaystyle{bfe}_icdot{bfe}_j=delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7037.png"
 ALT="$\displaystyle {\bf e}_i\cdot {\bf e}_j = \delta_{ij}.$">|; 

$key = q/yslashd=3slash4+1slash(2pi)=0.909;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2523.png"
 ALT="$ y/d=3/4+1/(2\pi)= 0.909$">|; 

$key = q/displaystylefrac{d}{dx}left(rho,u^{,2}+pright)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5454.png"
 ALT="$\displaystyle \frac{d}{dx}\left(\rho u^{ 2}+p\right) = 0,$">|; 

$key = q/displaystyleepsilon_{ijk},{calR}_{li},{calR}_{mj},{calR}_{nk}=epsilon_{lmn}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6833.png"
 ALT="$\displaystyle \epsilon_{ijk} {\cal R}_{li} {\cal R}_{mj} {\cal R}_{nk} = \epsilon_{lmn}.$">|; 

$key = q/displaystyle=b_i,epsilon_{ijk},frac{partiala_k}{partialx_j}-a_i,epsilon_{ijk},fralx_j}={bfb}cdot(nablatimes{bfa})-{bfa}cdot(nablatimes{bfb}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="441" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6891.png"
 ALT="$\displaystyle =b_i  \epsilon_{ijk} \frac{\partial a_k}{\partial x_j}-a_i \ep...
...x_j} ={\bf b}\cdot(\nabla \times {\bf a}) - {\bf a}\cdot(\nabla\times {\bf b}).$">|; 

$key = q/widehat{E}=Eslash(G,M^{,2}slasha_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img894.png"
 ALT="$ \widehat{E}= E/(G M^{ 2}/a_0)$">|; 

$key = q/displaystyle{mitGamma}_{n,,r}^{,m}=lambda_rint_{mitOmega}skew{5}hat{P}_n^{,m}(coheta),{rme}^{,{rmi},m,phi},{mitPhi}_r(theta,phi),d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4919.png"
 ALT="$\displaystyle {\mit\Gamma}_{n, r}^{ m} =\lambda_r\int_{\mit\Omega} \skew{5}\h...
...theta) {\rm e}^{ {\rm i} m \phi} {\mit\Phi}_r(\theta,\phi) d{\mit\Omega}.$">|; 

$key = q/displaystyle=c,sinhxi_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2756.png"
 ALT="$\displaystyle =c \sinh\xi_0.$">|; 

$key = q/displaystyle=frac{4,alpha_0^{,2}}{sqrt{{rmMa}_1^{,2}-1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5806.png"
 ALT="$\displaystyle = \frac{4 \alpha_0^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}.$">|; 

$key = q/{calS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img523.png"
 ALT="$ {\cal S}$">|; 

$key = q/P_1(costheta)=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2643.png"
 ALT="$ P_1(\cos\theta)=\cos\theta$">|; 

$key = q/displaystyleu(x,y)simequ_0(y)+alpha(y)sqrt{x_0-x},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3116.png"
 ALT="$\displaystyle u(x,y)\simeq u_0(y) +\alpha(y)\sqrt{x_0-x},$">|; 

$key = q/r<a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3841.png"
 ALT="$ r&lt;a$">|; 

$key = q/overline{mu};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3842.png"
 ALT="$ \overline{\mu}$">|; 

$key = q/displaystyle=costheta,frac{partial{mitPhi}_r}{partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4836.png"
 ALT="$\displaystyle = \cos\theta \frac{\partial{\mit\Phi}_r}{\partial\theta},$">|; 

$key = q/displaystyletimes(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img809.png"
 ALT="$\displaystyle \times ($">|; 

$key = q/m(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5172.png"
 ALT="$ m(r)$">|; 

$key = q/displaystyle|u_2|=c_1left(frac{p_2}{p_1}-1right)left[frac{2slashgamma_1}{2,gamma_1+(gamma_1+1),(p_2slashp_1-1)}right]^{,1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5640.png"
 ALT="$\displaystyle \vert u_2\vert= c_1\left(\frac{p_2}{p_1}-1\right)\left[\frac{2/\gamma_1}{2 \gamma_1+ (\gamma_1+1) (p_2/p_1-1)}\right]^{ 1/2},
$">|; 

$key = q/displaystyleR=frac{a}{costheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1139.png"
 ALT="$\displaystyle R = \frac{a}{\cos\theta}.$">|; 

$key = q/displaystyle=a_i,b_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6800.png"
 ALT="$\displaystyle = a_i b_i,$">|; 

$key = q/displaystyle=alpha^{,2},cos^2phi+beta^{,2},sin^2phi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1298.png"
 ALT="$\displaystyle = \alpha^{ 2} \cos^2\phi + \beta^{ 2} \sin^2\phi,$">|; 

$key = q/displaystyle=widehat{U},frac{dwidehat{U}}{partialX}+frac{partial^{,2}V_y}{partialY^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2900.png"
 ALT="$\displaystyle = \widehat{U} \frac{d\widehat{U}}{\partial X} + \frac{\partial^{ 2} V_y}{\partial Y^{ 2}},$">|; 

$key = q/overline{A}=overline{B}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3847.png"
 ALT="$ \overline{A}=\overline{B}=0$">|; 

$key = q/18.3,{rmkm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4651.png"
 ALT="$ 18.3 {\rm km}$">|; 

$key = q/displaystyle=sum_{n=1,infty}left[h_2,skew{5}bar{zeta}_2,delta_{n,2}+h_n',alpha_n,zeta_nright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4504.png"
 ALT="$\displaystyle = \sum_{n=1,\infty}\left[h_2 \skew{5}\bar{\zeta}_2 \delta_{n 2} + h_n' \alpha_n \zeta_n\right],$">|; 

$key = q/displaystyle{calA}_e={calA},sqrt{{rmMa}_infty^{,2}-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6084.png"
 ALT="$\displaystyle {\cal A}_e = {\cal A} \sqrt{{\rm Ma}_\infty^{ 2}-1}$">|; 

$key = q/displaystyle{mitPsi}_r'(theta,phi_pm);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4859.png"
 ALT="$\displaystyle {\mit\Psi}_r'(\theta,\phi_\pm)$">|; 

$key = q/x=a,lnrslashpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2093.png"
 ALT="$ x=a \ln r/\pi$">|; 

$key = q/k=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6811.png"
 ALT="$ k=l$">|; 

$key = q/C_{L0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6076.png"
 ALT="$ C_{L0}$">|; 

$key = q/displaystylev_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5089.png"
 ALT="$\displaystyle v_0$">|; 

$key = q/{mitDelta}pslashp_1rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5519.png"
 ALT="$ {\mit\Delta}p/p_1\rightarrow\infty$">|; 

$key = q/theta<theta_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5711.png"
 ALT="$ \theta&lt;\theta_{\rm max}$">|; 

$key = q/beta=mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5708.png"
 ALT="$ \beta=\mu$">|; 

$key = q/displaystyle=-pi,rholeft.int_{eta=0}^{eta=pi}phi,dpsiright|_{xi=xi_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2810.png"
 ALT="$\displaystyle = -\pi \rho\left.\int_{\eta=0}^{\eta=\pi} \phi d\psi\right\vert _{\xi=\xi_0}$">|; 

$key = q/[phi];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2510.png"
 ALT="$ [\phi]$">|; 

$key = q/duslashdt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img760.png"
 ALT="$ du/dt=0$">|; 

$key = q/displaystyle=lim_{epsilonrightarrow0}left(frac{1}{sinphi},lnleft[frac{sin(phi-epsilonslash2)}{sin(phi+epsilonslash2)}right]right)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3499.png"
 ALT="$\displaystyle =\lim_{\epsilon\rightarrow 0}\left(\frac{1}{\sin\phi} \ln\left[\frac{\sin(\phi-\epsilon/2)}{\sin(\phi+\epsilon/2)}\right]\right) = 0,$">|; 

$key = q/-(zeta_1+zeta_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4526.png"
 ALT="$ -(\zeta_1 + \zeta_2)$">|; 

$key = q/+1.72times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4628.png"
 ALT="$ +1.72\times 10^{-2}$">|; 

$key = q/Ox'y'z';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6274.png"
 ALT="$ Ox'y'z'$">|; 

$key = q/displaystyle=frac{1}{sintheta}left[frac{partial}{partialtheta},(sintheta,eta)-frac{partialxi}{partialphi}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4762.png"
 ALT="$\displaystyle = \frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta} (\sin\theta \eta) -\frac{\partial\xi}{\partial \phi}\right],$">|; 

$key = q/displaystyle{mitDelta}p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5748.png"
 ALT="$\displaystyle {\mit\Delta} p$">|; 

$key = q/displaystyle=-frac{1}{rho},frac{partialp}{partialx}+nuleft(frac{partial^{,2}v_x}{partialx^{,2}}+frac{partial^{,2}v_x}{partialy^{,2}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2862.png"
 ALT="$\displaystyle =-\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu\left(\frac{...
... 2} v_x}{\partial x^{ 2}}+\frac{\partial^{ 2} v_x}{\partial y^{ 2}}\right),$">|; 

$key = q/displaystylenu,frac{partial^{,3}psi}{partialy^{,3}}-frac{partialpsi}{partialx},faly},frac{partial^{,2}psi}{partialx,partialy}=U,frac{dU}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2907.png"
 ALT="$\displaystyle \nu \frac{\partial^{ 3}\psi}{\partial y^{ 3}}-\frac{\partial\p...
...rtial y} \frac{\partial^{ 2}\psi}{\partial x  \partial y}= U \frac{dU}{dx},$">|; 

$key = q/displaystyle=Xleft(1-frac{1}{2},sin^2betaright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4038.png"
 ALT="$\displaystyle = X\left(1-\frac{1}{2} \sin^2\beta\right),$">|; 

$key = q/displaystyle{bfa},%,{bfb}equiva_x,b_z+a_y,b_x+a_z,b_y={rmscalar~number},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6292.png"
 ALT="$\displaystyle {\bf a} \% {\bf b} \equiv a_x  b_z + a_y  b_x + a_z  b_y = {\rm scalar&nbsp;number},$">|; 

$key = q/displaystylewidehat{omega}^{,2}=2left[frac{F(gamma,alpha)-E(gamma,alpha)}{tanbetmma,cos^2alpha}-frac{cos^2beta}{tan^2gamma,cos^2alpha}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="502" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img942.png"
 ALT="$\displaystyle \widehat{\omega}^{ 2} =2\left[\frac{F(\gamma,\alpha)-E(\gamma,\a...
...n^3\gamma \cos^2\alpha}-\frac{\cos^2\beta}{\tan^2\gamma \cos^2\alpha}\right],$">|; 

$key = q/overline{p}=(p-p_0)slash(rho_0,V_0^{,2}+rho_0,g,L+rho_0,nu,V_0slashL);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="$ \overline{p} = (p-p_0)/(\rho_0 V_0^{ 2} +\rho_0 g L+ \rho_0 \nu V_0/L)$">|; 

$key = q/displaystylefrac{dwidehat{mitGamma}}{dhat{t}}=-widehat{mitGamma}+sinalpha,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3337.png"
 ALT="$\displaystyle \frac{d\widehat{\mit\Gamma}}{d\hat{t}}=-\widehat{\mit\Gamma}+\sin\alpha,$">|; 

$key = q/displaystyle=tan^{-1}left(sqrt{frac{chi_s}{1-chi_s}},right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6102.png"
 ALT="$\displaystyle = \tan^{-1}\left(\sqrt{\frac{\chi_s}{1-\chi_s}} \right),$">|; 

$key = q/displaystylel=int_A^B(dx^{,2}+dy^{,2})^{1slash2}=int_a^b[1+y'^{,2}(x)]^{1slash2},dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7298.png"
 ALT="$\displaystyle l = \int_A^B (dx^{ 2} + dy^{ 2})^{1/2} = \int_a^b [1 + y'^{ 2}(x)]^{1/2} dx,$">|; 

$key = q/{bfv}=U_0,{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2977.png"
 ALT="$ {\bf v}= U_0 {\bf e}_x$">|; 

$key = q/displaystyle=f'(z_0),dz',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2071.png"
 ALT="$\displaystyle = f'(z_0) dz',$">|; 

$key = q/displaystyleepsilon_{mis},a_{sj}+epsilon_{mjs},a_{is}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6901.png"
 ALT="$\displaystyle \epsilon_{mis} a_{sj} + \epsilon_{mjs} a_{is}=0.$">|; 

$key = q/alpha_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1753.png"
 ALT="$ \alpha_1$">|; 

$key = q/({bfa}cdot{bfb})cdot{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6309.png"
 ALT="$ ({\bf a}\cdot{\bf b}) \cdot{\bf c}$">|; 

$key = q/displaystylegamma=frac{{calC}_p}{{calC}_V};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5327.png"
 ALT="$\displaystyle \gamma = \frac{{\cal C}_p}{{\cal C}_V}$">|; 

$key = q/displaystyle=lambda,delta_{kl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6950.png"
 ALT="$\displaystyle =\lambda \delta_{kl},$">|; 

$key = q/y_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7366.png"
 ALT="$ y_k$">|; 

$key = q/dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6454.png"
 ALT="$ dy$">|; 

$key = q/-widehat{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img886.png"
 ALT="$ -\widehat{E}$">|; 

$key = q/displaystyle=left(frac{2}{gamma+1}right)^{1slash(gamma-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5400.png"
 ALT="$\displaystyle = \left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)}.$">|; 

$key = q/gamma=7.28times10^{-2},{rmN,m^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1083.png"
 ALT="$ \gamma=7.28\times 10^{-2} {\rm N m^{-1}}$">|; 

$key = q/rho=Mslash[(4slash3),pi,a_0^{,3}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1019.png"
 ALT="$ \rho=M/[(4/3) \pi a_0^{ 3}]$">|; 

$key = q/p_2,S_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1394.png"
 ALT="$ p_2 S_2$">|; 

$key = q/P,dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2379.png"
 ALT="$ P dy$">|; 

$key = q/displaystyleleft[pright]_{z=0_-}^{z=0_+}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4106.png"
 ALT="$\displaystyle \left[p\right]_{z=0_-}^{z=0_+}=0.$">|; 

$key = q/displaystyle=-left[h_2+frac{h_2',(1+k_2-h_2),alpha_2}{1-(1+k_2'-h_2'),alpha_2}right](zeta_{mitOmega}+zeta_M+zeta_E),P_2(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="476" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4639.png"
 ALT="$\displaystyle = - \left[h_2+ \frac{h_2' (1+k_2-h_2) \alpha_2}{1-(1+k_2'-h_2') \alpha_2}\right](\zeta_{\mit\Omega}+\zeta_M+\zeta_E) P_2(\cos\theta),$">|; 

$key = q/displaystyle=a_{sskl},delta_{ij}+a_{sjsl},delta_{ik}+a_{sjks},delta_{il},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6939.png"
 ALT="$\displaystyle =a_{sskl} \delta_{ij} + a_{sjsl} \delta_{ik}+ a_{sjks} \delta_{il},$">|; 

$key = q/displaystyleQ=h_1,v_1=h_2,v_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1467.png"
 ALT="$\displaystyle Q = h_1 v_1 = h_2 v_2,$">|; 

$key = q/deltaz={rmi},deltay;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1964.png"
 ALT="$ \delta z
= {\rm i} \delta y$">|; 

$key = q/displaystylelambda_1,lambda_2=frac{1}{V_1,V_2}left[A,kappa^{,2}-left(frac{V_1,V_2}{V_1+V_2}right)H_1H_2right]^{,2}geq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img698.png"
 ALT="$\displaystyle \lambda_1 \lambda_2 = \frac{1}{V_1 V_2}\left[A \kappa^{ 2}- \left(\frac{V_1 V_2}{V_1+V_2}\right) H_1H_2\right]^{ 2}\geq 0,$">|; 

$key = q/displaystyleA_r=int_{mitOmega}costhetaleft(-eta,frac{partial{mitPhi}_r}{partialtntheta},frac{partial{mitPhi}_r}{partialphi}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4901.png"
 ALT="$\displaystyle A_r= \int_{\mit\Omega}\cos\theta\left(-\eta \frac{\partial{\mit\...
...xi}{\sin\theta} \frac{\partial{\mit\Phi}_r}{\partial\phi}\right)d{\mit\Omega}.$">|; 

$key = q/int_0^piy'^{,2},dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7372.png"
 ALT="$ \int_0^\pi y'^{ 2} dx$">|; 

$key = q/F(gamma,alpha)rightarrowgamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img953.png"
 ALT="$ F(\gamma,\alpha)\rightarrow \gamma$">|; 

$key = q/lambda(theta_pm)=-0.1567;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3363.png"
 ALT="$ \lambda(\theta_\pm)=-0.1567$">|; 

$key = q/displaystyle=-frac{3,mu,V,a,sintheta}{2,r^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3789.png"
 ALT="$\displaystyle = -\frac{3 \mu V a \sin\theta}{2 r^{ 2}},$">|; 

$key = q/displaystyle{rmRe}=frac{U_0,d}{nu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3008.png"
 ALT="$\displaystyle {\rm Re} = \frac{U_0 d}{\nu}.$">|; 

$key = q/displaystylerho(z)=left{{array}{ccc}0&mbox{hspace{0.5cm}}&z>0[0.5ex]rho_0&&zleq0{array}right.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img561.png"
 ALT="$\displaystyle \rho(z) = \left\{ \begin{array}{ccc} 0&amp;\mbox{\hspace{0.5cm}}&amp;z&gt;0\ [0.5ex] \rho_0&amp;&amp;z\leq 0 \end{array}\right.,$">|; 

$key = q/-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$ -$">|; 

$key = q/displaystyle[({bfA}cdotnabla),{bfA}]_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7187.png"
 ALT="$\displaystyle [({\bf A}\cdot\nabla) {\bf A}]_\phi$">|; 

$key = q/displaystylelambda=-left(frac{k,V}{bar{F}}right)^{1slash2}={rmi},sinh{mitOmega}=}left(V,frac{dz}{dbar{F}}-frac{1}{V},frac{dbar{F}}{dz}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2260.png"
 ALT="$\displaystyle \lambda =- \left(\frac{k V}{\bar{F}}\right)^{1/2}= {\rm i} \sin...
...\rm i}}{2}\left(V \frac{dz}{d\bar{F}}-\frac{1}{V} \frac{d\bar{F}}{dz}\right).$">|; 

$key = q/displaystylesigma_{rr}(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3839.png"
 ALT="$\displaystyle \sigma_{rr}(r,\theta)$">|; 

$key = q/displaystylev_x(y)=frac{G}{2,mu},y,(d-y).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3582.png"
 ALT="$\displaystyle v_x(y) = \frac{G}{2 \mu} y (d-y).$">|; 

$key = q/displaystyle=p_0-rho',g,zeta_0,sin(omega,t-k,x)-rho',k,(V'-c),A,sin(omega,t-k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4104.png"
 ALT="$\displaystyle =p_0 -\rho' g \zeta_0 \sin(\omega t-k x)-\rho' k (V'-c) A \sin(\omega t-k x)$">|; 

$key = q/displaystyle=frac{partial}{partialx_i}!left(epsilon_{ijk},a_j,b_kright)=epsilon_artialx_i},b_k+epsilon_{ijk},a_j,frac{partialb_k}{partialx_i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6890.png"
 ALT="$\displaystyle = \frac{\partial}{\partial x_i}\!\left(\epsilon_{ijk} a_j b_k\r...
..._j}{\partial x_i} b_k + \epsilon_{ijk} a_j \frac{\partial b_k}{\partial x_i}$">|; 

$key = q/A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$ A$">|; 

$key = q/partial^{,2}uslashpartialy^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3122.png"
 ALT="$ \partial^{ 2} u/\partial y^{ 2}$">|; 

$key = q/k=alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5973.png"
 ALT="$ k=\alpha$">|; 

$key = q/{bfr}cdot{bfn}=c,|{bfr}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6723.png"
 ALT="$ {\bf r}\cdot{\bf n} = c \vert{\bf r}\vert$">|; 

$key = q/displaystyleomega_z=frac{partial^{,2}psi}{partialx^{,2}}+frac{partial^{,2}psi}{partialy^{,2}}=nabla^{,2}psi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1641.png"
 ALT="$\displaystyle \omega_z=\frac{\partial^{ 2}\psi}{\partial x^{ 2}}+\frac{\partial^{ 2}\psi}{\partial y^{ 2}}=\nabla^{ 2}\psi.$">|; 

$key = q/displaystylefrac{beta_{n,,n'}^{,m,,m'}}{c_n^{,m},c_{n'}^{,m'}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4958.png"
 ALT="$\displaystyle \frac{\beta_{n, n'}^{ m, m'}}{c_n^{ m} c_{n'}^{ m'}}$">|; 

$key = q/partialv_1slashpartialx_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="$ \partial v_1/\partial x_2$">|; 

$key = q/epsilon_{n,,n'}^{,m,,m'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4987.png"
 ALT="$ \epsilon_{n, n'}^{ m, m'}=0$">|; 

$key = q/S_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1356.png"
 ALT="$ S_2$">|; 

$key = q/displaystyle=-left(frac{{rmMa}_1^{,2}+2}{{rmMa}_1^{,2}}+gamma,sin^2thetaright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6089.png"
 ALT="$\displaystyle =-\left(\frac{{\rm Ma}_1^{ 2}+2}{{\rm Ma}_1^{ 2}}+\gamma \sin^2\theta\right),$">|; 

$key = q/displaystylepsi(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1973.png"
 ALT="$\displaystyle \psi(r,\theta)$">|; 

$key = q/displaystylesigma_{yy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="$\displaystyle \sigma_{yy}$">|; 

$key = q/displaystyle=sigma_{rtheta}(a,theta)=muleft(frac{1}{r},frac{partialv_r}{partialt+frac{partialv_theta}{partialr}-frac{v_theta}{r}right)_{r=a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3799.png"
 ALT="$\displaystyle =\sigma_{r\theta}(a,\theta) = \mu\left(\frac{1}{r} \frac{\partia...
... \theta}+\frac{\partial v_\theta}{\partial r} -\frac{v_\theta}{r}\right)_{r=a}.$">|; 

$key = q/displaystylelambda={rmi},sinh{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2198.png"
 ALT="$\displaystyle \lambda = {\rm i} \sinh{\mit\Omega}$">|; 

$key = q/14.3,{rmkm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4653.png"
 ALT="$ 14.3 {\rm km}$">|; 

$key = q/displaystyleh_c=left(frac{Q^{,2}}{g,W^{,2}}right)^{1slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1605.png"
 ALT="$\displaystyle h_c=\left(\frac{Q^{ 2}}{g W^{ 2}}\right)^{1/3},
$">|; 

$key = q/_y)_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$ _y)_x$">|; 

$key = q/|zeta|slash|{muskipz@5mudividemuskipz@tw@mkernmuskipz@mathaccentV{tilde}07E{mkern-muskipz@{zeta}mkernmuskipz@}mkern-muskipz@}{}_6|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$ \vert\zeta \vert/\vert{\muskip \z@ 5mu\divide \muskip \z@ \tw@ \mkern \muskip ...
...{\mkern -\muskip \z@ {\zeta }\mkern \muskip \z@ }\mkern -\muskip \z@ }{}_6\vert$">|; 

$key = q/displaystyleleft.frac{partial}{partialx}right|_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2923.png"
 ALT="$\displaystyle \left.\frac{\partial}{\partial x}\right\vert _y$">|; 

$key = q/epsilon_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5017.png"
 ALT="$ \epsilon_{n, n'}^{ m, m'}$">|; 

$key = q/langlecdotsrangle=int_{z_1}^{z_2}(cdots),dzslashl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3685.png"
 ALT="$ \langle \cdots\rangle = \int_{z_1}^{z_2} (\cdots) dz/l$">|; 

$key = q/{rmRe}=(aslash4times10^{-5})^{,3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3823.png"
 ALT="$ {\rm Re} = (a/4\times 10^{-5})^{ 3}$">|; 

$key = q/displaystylebeta_{-n,,n'}^{,-m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4950.png"
 ALT="$\displaystyle \beta_{-n, n'}^{ -m, m'}$">|; 

$key = q/h_2'=-5slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4646.png"
 ALT="$ h_2'=-5/3$">|; 

$key = q/displaystyleu,frac{partialv}{partialx}+v,frac{partialv}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5877.png"
 ALT="$\displaystyle u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}$">|; 

$key = q/a_1>a_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img872.png"
 ALT="$ a_1&gt;a_3$">|; 

$key = q/t=k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6914.png"
 ALT="$ t=k$">|; 

$key = q/intx_i^{,2},dV=(1slash5),a_i^{,2},V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7275.png"
 ALT="$ \int x_i^{ 2} dV = (1/5) a_i^{ 2} V$">|; 

$key = q/displaystyleoint{bfA}cdotd{bfr}=oint{bfA}cdotd{bfr}_1+oint{bfA}cdotd{bfr}_2+oint{bfA}cdotd{bfr}_3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6643.png"
 ALT="$\displaystyle \oint {\bf A} \cdot d{\bf r} = \oint {\bf A}\cdot d{\bf r}_1 + \oint {\bf A}\cdot d{\bf r}_2+\oint {\bf A}\cdot d{\bf r}_3,$">|; 

$key = q/psi=psi_0,{rme}^{,{rmi},(sigma,t-{bfk}cdot{bfr})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5110.png"
 ALT="$ \psi=\psi_0 {\rm e}^{ {\rm i} (\sigma t-{\bf k}\cdot{\bf r})}$">|; 

$key = q/z+dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3465.png"
 ALT="$ z+dz$">|; 

$key = q/displaystylexi_z(x,z,t,)=(V-c)^{-1}left[A,sinh(k,z)+B,cosh(k,z)right]sin(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="529" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4081.png"
 ALT="$\displaystyle \xi_z(x,z,t,) = (V-c)^{-1}\left[A \sinh(k z)+ B \cosh(k z)\right]\sin(\omega t-k x).$">|; 

$key = q/phi=f(|{bfr}|);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6759.png"
 ALT="$ \phi=f(\vert{\bf r} \vert)$">|; 

$key = q/_x)_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$ _x)_y$">|; 

$key = q/R';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2536.png"
 ALT="$ R'$">|; 

$key = q/displaystylec^{,2}<frac{1}{2},a^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1049.png"
 ALT="$\displaystyle c^{ 2}&lt; \frac{1}{2} a^{ 2}.
$">|; 

$key = q/displaystyleP_i=int_Vrho,v_i,dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="$\displaystyle P_i = \int_V \rho v_i dV.$">|; 

$key = q/A={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4789.png"
 ALT="$ A={\rm constant}$">|; 

$key = q/displaystylefrac{p_1}{rho_1}+{calT}_1=frac{p_2}{rho_2}+{calT}_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1372.png"
 ALT="$\displaystyle \frac{p_1}{\rho_1}+ {\cal T}_1= \frac{p_2}{\rho_2}+{\cal T}_2,$">|; 

$key = q/displaystyleK=frac{1}{4},rho,g,a^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3977.png"
 ALT="$\displaystyle K = \frac{1}{4} \rho g a^{ 2}.$">|; 

$key = q/displaystylezeta_j(mu,phi,t)=sum_{n=1,infty}z_n,P_n^{,m_j}(mu),{rme}^{,{rmi},(m_j,phi+sigma_j,t)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4682.png"
 ALT="$\displaystyle \zeta_j(\mu,\phi,t)=\sum_{n=1,\infty} z_n P_n^{ m_j}(\mu) {\rm e}^{ {\rm i} (m_j \phi+\sigma_j t)}.$">|; 

$key = q/(h_1^{,2}slashh),nablau_1=nablau_2timesnablau_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7058.png"
 ALT="$ (h_1^{ 2}/h) \nabla u_1= \nabla u_2\times \nabla u_3$">|; 

$key = q/lambda=2pislashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3925.png"
 ALT="$ \lambda=2\pi/k$">|; 

$key = q/phi_n({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4414.png"
 ALT="$ \phi_n({\bf r})$">|; 

$key = q/k,d'rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4187.png"
 ALT="$ k d'\rightarrow \infty$">|; 

$key = q/displaystyle=-left.left(frac{2,n+1}{3}right)rightslashleft[1+frac{(2,n+1),(2,n^{,2}+4,n+3)}{(6+n^{,2})},frac{mu}{rho,g,a}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="464" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4436.png"
 ALT="$\displaystyle = -\left.\left(\frac{2 n+1}{3}\right)\right/\left[1+\frac{(2 n+1) (2 n^{ 2}+4 n+3)}{(6+n^{ 2})} \frac{\mu}{\rho g a}\right].$">|; 

$key = q/9.8,{rmK,km^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5149.png"
 ALT="$ 9.8 {\rm K km^{-1}}$">|; 

$key = q/h_eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7226.png"
 ALT="$ h_\eta$">|; 

$key = q/displaystyle{bfx}''={bfx}+r'',{bfn}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7239.png"
 ALT="$\displaystyle {\bf x}'' = {\bf x} + r'' {\bf n}.$">|; 

$key = q/m>m';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1864.png"
 ALT="$ m&gt;m'$">|; 

$key = q/eta={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7228.png"
 ALT="$ \eta={\rm constant}$">|; 

$key = q/{calS}_n(theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4312.png"
 ALT="$ {\cal S}_n(\theta,\varphi)$">|; 

$key = q/widehat{omega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img884.png"
 ALT="$ \widehat{\omega}$">|; 

$key = q/Q-V,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3604.png"
 ALT="$ Q-V h$">|; 

$key = q/X,cos^3theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4219.png"
 ALT="$ X \cos^3\theta$">|; 

$key = q/displaystyle=-frac{5}{2},(zeta_{mitOmega}+zeta_M+zeta_E),P_2(costheta)=-18.3,P_2(costheta),{rmkm},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4649.png"
 ALT="$\displaystyle = - \frac{5}{2} (\zeta_{\mit\Omega}+\zeta_M+\zeta_E) P_2(\cos\theta)= -18.3 P_2(\cos\theta) {\rm km},$">|; 

$key = q/displaystyleb;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2129.png"
 ALT="$\displaystyle b$">|; 

$key = q/displaystylefrac{partialomega}{partialt}+[psi,,omega]=nu,nabla^{,2}omega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img538.png"
 ALT="$\displaystyle \frac{\partial\omega}{\partial t}+[\psi, \omega] = \nu \nabla^{ 2}\omega,
$">|; 

$key = q/G(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2087.png"
 ALT="$ G(z)$">|; 

$key = q/displaystyle(A_{ij}-lambda,delta_{ij}),x_j=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7004.png"
 ALT="$\displaystyle (A_{ij}-\lambda \delta_{ij}) x_j = 0.
$">|; 

$key = q/displaystylefrac{dP_i}{dt}+{mitPhi}_i=f_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="$\displaystyle \frac{dP_i}{dt} + {\mit\Phi}_i = f_i,$">|; 

$key = q/stackrel{displaystylerightarrow}{QR};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6227.png"
 ALT="$ \stackrel{\displaystyle \rightarrow}{QR}$">|; 

$key = q/s>0.2113;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img767.png"
 ALT="$ s&gt;0.2113$">|; 

$key = q/displaystylefrac{1}{2}left(1pmsqrt{1-2,frac{a^{,2}}{h^{,2}}},right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1043.png"
 ALT="$\displaystyle \frac{1}{2}\left(1\pm \sqrt{1-2 \frac{a^{ 2}}{h^{ 2}}} \right).
$">|; 

$key = q/displaystylesimn,({mitDelta}theta)^3simtheta,({mitDelta}theta)^2simfrac{theta^{,3}}{n^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5756.png"
 ALT="$\displaystyle \sim n ({\mit\Delta}\theta)^3\sim \theta ({\mit\Delta}\theta)^2\sim\frac{\theta^{ 3}}{n^{ 2}}.$">|; 

$key = q/P'BQ';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img773.png"
 ALT="$ P'BQ'$">|; 

$key = q/displaystyle=int_{rmfluid}frac{1}{2},rho,v^{,2},dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2677.png"
 ALT="$\displaystyle = \int_{\rm fluid}\frac{1}{2} \rho v^{ 2} dV$">|; 

$key = q/sin(theta_2-theta_1)=2,sin[(theta_2-theta_1)slash2],cos[(theta_2-theta_1)slash2];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="392" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2623.png"
 ALT="$ \sin(\theta_2-\theta_1)= 2 \sin[(\theta_2-\theta_1)/2] \cos[(\theta_2-\theta_1)/2]$">|; 

$key = q/b>a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2038.png"
 ALT="$ b&gt;a$">|; 

$key = q/(partialhslashpartialx)_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6527.png"
 ALT="$ (\partial h/\partial x)_y$">|; 

$key = q/displaystylenabla^{,2}{mitPhi}_0=left{{array}{lll}4pi,G,rho+4pi,G,skew{3}bar{rholta(r-a)&mbox{hspace{0.5cm}}&rleqa[0.5ex]0&&r>a{array}right.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4355.png"
 ALT="$\displaystyle \nabla^{ 2}{\mit\Phi}_0 =\left\{ \begin{array}{lll} 4\pi G \rh...
...,\delta(r-a)&amp;\mbox{\hspace{0.5cm}}&amp;r\leq a\ [0.5ex] 0&amp;&amp;r&gt;a \end{array}\right.,$">|; 

$key = q/displaystylefrac{A}{A_1}=frac{rho_1}{rho},frac{u_1}{u}=frac{1}{{rmMa}},frac{c_1}left(frac{T_1}{T}right)^{1slash2}left(frac{rho_1}{rho}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5414.png"
 ALT="$\displaystyle \frac{A}{A_1} =\frac{\rho_1}{\rho} \frac{u_1}{u} = \frac{1}{{\rm...
...{1}{{\rm Ma}} \left(\frac{T_1}{T}\right)^{1/2}\left(\frac{\rho_1}{\rho}\right),$">|; 

$key = q/displaystyletimes{bfr}_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6660.png"
 ALT="$\displaystyle \times {\bf r}_0.$">|; 

$key = q/displaystylenablaphicdotnablapsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1969.png"
 ALT="$\displaystyle \nabla\phi\cdot\nabla\psi$">|; 

$key = q/rho(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5440.png"
 ALT="$ \rho(x)$">|; 

$key = q/+4.07times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4634.png"
 ALT="$ +4.07\times 10^{-2}$">|; 

$key = q/gamma_{13};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1124.png"
 ALT="$ \gamma_{13}$">|; 

$key = q/z_H=-h=-s,cslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img670.png"
 ALT="$ z_H = -h=-s c/2$">|; 

$key = q/displaystyleV,frac{dz}{dbar{F}}=left[-{rmi},lambda+(1-lambda^{,2})^{1slash2}right]^{,2,alphaslashpi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2243.png"
 ALT="$\displaystyle V \frac{dz}{d\bar{F}} = \left[-{\rm i} \lambda + (1-\lambda^{ 2})^{1/2}\right]^{ 2 \alpha/\pi}.$">|; 

$key = q/1slash(gamma-1)simeq2.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5155.png"
 ALT="$ 1/(\gamma-1)\simeq 2.5$">|; 

$key = q/displaystyleleft(frac{h_1}{h_2}right)^2+frac{h_1}{h_2}-2,{rmFr}_2^{,2}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1488.png"
 ALT="$\displaystyle \left(\frac{h_1}{h_2}\right)^2+\frac{h_1}{h_2}-2 {\rm Fr}_2^{ 2} = 0,$">|; 

$key = q/displaystyle(m,{calT}_2+T)-(m,{calT}_1+T)=m,({calT}_2-{calT}_1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1370.png"
 ALT="$\displaystyle (m {\cal T}_2+T)-(m {\cal T}_1+T)=m ({\cal T}_2-{\cal T}_1).$">|; 

$key = q/displaystylesigma_{thetavarphi}=sigma_{varphitheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img500.png"
 ALT="$\displaystyle \sigma_{\theta \varphi} = \sigma_{\varphi\theta}$">|; 

$key = q/displaystyle=frac{2,m,m'}{m'^{,2}-m^{,2}},I_{n,,n'}^{,m,,m'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4979.png"
 ALT="$\displaystyle =\frac{2 m m'}{m'^{ 2}-m^{ 2}} I_{n, n'}^{ m, m'}.$">|; 

$key = q/displaystylephi(x,z,t)=-(V'-c),zeta_0,frac{cosh[k,(z-d')]}{sinh(k,d')},cos(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4097.png"
 ALT="$\displaystyle \phi(x,z,t) = -(V'-c) \zeta_0 \frac{\cosh[k (z-d')]}{\sinh(k d')} \cos(\omega t-k x).$">|; 

$key = q/displaystyleint_S(p_1-p_2),{bfn},dS=-gammaoint_C{bft}times{bfn},dr.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1091.png"
 ALT="$\displaystyle \int_S (p_1-p_2) {\bf n} dS =- \gamma \oint_C {\bf t}\times {\bf n} dr.$">|; 

$key = q/alpha=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2151.png"
 ALT="$ \alpha=0$">|; 

$key = q/displaystyle=-left[frac{nu,U_0,x^{,m-1}}{2,(m+1)}right]^{1slash2}[(m+1),f+(m-1),eta,f'],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2927.png"
 ALT="$\displaystyle = -\left[\frac{\nu U_0 x^{ m-1}}{2 (m+1)}\right]^{1/2}[(m+1) f+(m-1) \eta f'],$">|; 

$key = q/Ox;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6253.png"
 ALT="$ Ox$">|; 

$key = q/displaystylez={rmi},b,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2267.png"
 ALT="$\displaystyle z={\rm i} b,$">|; 

$key = q/overline{D};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3855.png"
 ALT="$ \overline{D}$">|; 

$key = q/theta=180^circ-30^circ=150^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1207.png"
 ALT="$ \theta=180^\circ-30^\circ=150^\circ$">|; 

$key = q/0.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$ 0.8$">|; 

$key = q/(a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$ (a$">|; 

$key = q/{bfcurl},{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6628.png"
 ALT="$ {\bf curl} {\bf A}$">|; 

$key = q/dslashHll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1462.png"
 ALT="$ d/H\ll 1$">|; 

$key = q/dS_3=h_1,h_2,du_1,du_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7053.png"
 ALT="$ dS_3 = h_1 h_2 du_1 du_2$">|; 

$key = q/lambda_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4824.png"
 ALT="$ \lambda_2$">|; 

$key = q/displaystyle=frac{u_1}{u_2}=frac{(gamma+1),{rmMa}_1^{,2}}{2+(gamma-1),{rmMa}_1^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5486.png"
 ALT="$\displaystyle =\frac{u_1}{u_2}=\frac{(\gamma+1) {\rm Ma}_1^{ 2}}{2+(\gamma-1) {\rm Ma}_1^{ 2}},$">|; 

$key = q/theta_0>theta_->-pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3366.png"
 ALT="$ \theta_0&gt;\theta_- &gt;-\pi$">|; 

$key = q/{bfv}(r,theta,varphi)rightarrowV,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2636.png"
 ALT="$ {\bf v}(r,\theta,\varphi)\rightarrow V {\bf e}_z$">|; 

$key = q/deltap=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img713.png"
 ALT="$ \delta p =0$">|; 

$key = q/beta_{-n,,-n'}^{,-m,,-m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5016.png"
 ALT="$ \beta_{-n, -n'}^{ -m, -m'}$">|; 

$key = q/displaystyleX_i(a,theta,phi)=gleft[rho,zeta_b+skew{3}bar{rho},(d+zeta)right]left(frac{x_i}{r}right)_{r=a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4405.png"
 ALT="$\displaystyle X_i(a,\theta,\phi) = g\left[\rho \zeta_b + \skew{3}\bar{\rho} (d+\zeta)\right]\left(\frac{x_i}{r}\right)_{r=a}.$">|; 

$key = q/v^{,2}={rmMa}^{,2},c^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5382.png"
 ALT="$ v^{ 2}= {\rm Ma}^{ 2} c^{ 2}$">|; 

$key = q/displaystylegamma_{n,,n'}^{,m,,m'}=(1+k_n'-h_n'),alpha_n,(a_n^{,m})^{-2},delta_{m,,m'}-{rmi},epsilon_{n,,n'}^{,m,,m'}+theta_{n,,n'}^{,m,,m'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="472" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4986.png"
 ALT="$\displaystyle \gamma_{n, n'}^{ m, m'} = (1+k_n'-h_n') \alpha_n (a_n^{ m})...
...{ m, m'}- {\rm i} \epsilon_{n, n'}^{ m, m'} + \theta_{n, n'}^{ m, m'}.$">|; 

$key = q/displaystyle=gleft(frac{sigma,k_x-{rmi},f,k_y}{sigma^{,2}-f^{,2}}right)zeta_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5088.png"
 ALT="$\displaystyle = g\left(\frac{\sigma k_x-{\rm i} f k_y}{\sigma^{ 2}-f^{ 2}}\right)\zeta_0,$">|; 

$key = q/displaystylefrac{d^{,2}theta}{dt^{,2}}=-omega^{,2},theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img685.png"
 ALT="$\displaystyle \frac{d^{ 2}\theta}{dt^{ 2}} = -\omega^{ 2} \theta,$">|; 

$key = q/rho,V,{mitGamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3288.png"
 ALT="$ \rho V {\mit\Gamma}$">|; 

$key = q/j=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$ j=2$">|; 

$key = q/displaystyleu(x,y)=U(x)left{{array}{lcl}sin(alpha,y)&mbox{hspace{1cm}}&0leqyleqpislash(2,alpha)[0.5ex]1&&y>pislash(2,alpha){array}right.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3262.png"
 ALT="$\displaystyle u(x,y) = U(x)\left\{\begin{array}{lcl}
\sin(\alpha y)&amp;\mbox{\hsp...
...0\leq y\leq \pi/(2 \alpha)\ [0.5ex]
1&amp;&amp;y&gt;\pi/(2 \alpha)
\end{array}\right.,
$">|; 

$key = q/a_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6766.png"
 ALT="$ a_{ij}$">|; 

$key = q/1.0times10^{-6},{rmm^2slashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="$ 1.0\times 10^{-6} {\rm m^2/s}$">|; 

$key = q/displaystylev_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img534.png"
 ALT="$\displaystyle v_x$">|; 

$key = q/+x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5367.png"
 ALT="$ +x$">|; 

$key = q/displaystyleF(z)=-V_0,z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1970.png"
 ALT="$\displaystyle F(z)=-V_0 z,$">|; 

$key = q/W_0=W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img597.png"
 ALT="$ W_0=W$">|; 

$key = q/displaystyle={bfr}times{bfF}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6386.png"
 ALT="$\displaystyle = {\bf r}\times{\bf F}.$">|; 

$key = q/k_2'=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4648.png"
 ALT="$ k_2'=-1$">|; 

$key = q/displaystyle=nabla^{,2}(psi,nablavarphi)+nabla{mitOmega}_varphitimesnablavarphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2562.png"
 ALT="$\displaystyle = \nabla^{ 2}(\psi \nabla\varphi)+\nabla{\mit\Omega}_\varphi\times \nabla\varphi,$">|; 

$key = q/u=u_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1277.png"
 ALT="$ u=u_c$">|; 

$key = q/displaystyle=+k^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5965.png"
 ALT="$\displaystyle = +k^{ 2}.$">|; 

$key = q/displaystyle({bfa}times{bfb})times{bfc}equiv({bfa}cdot{bfc}),{bfb}-({bfb}cdot{bfc}),{bfa}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6422.png"
 ALT="$\displaystyle ({\bf a}\times{\bf b}) \times {\bf c}\equiv ({\bf a}\cdot {\bf c})  {\bf b} - ({\bf b}\cdot {\bf c}) {\bf a}.$">|; 

$key = q/displaystylenablau_icdotnablau_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7059.png"
 ALT="$\displaystyle \nabla u_i\cdot \nabla u_j$">|; 

$key = q/displaystyleP_n^{,m}(x)=(-1)^m,(1-x^{,2})^{mslash2},frac{d^{,m}[P_n(x)]}{dx^{,m}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4263.png"
 ALT="$\displaystyle P_n^{ m}(x)=(-1)^m (1-x^{ 2})^{m/2} \frac{d^{ m}[P_n(x)]}{dx^{ m}},$">|; 

$key = q/displaystyle=frac{4,alpha_0}{sqrt{{rmMa}_1^{,2}-1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5805.png"
 ALT="$\displaystyle = \frac{4 \alpha_0}{\sqrt{{\rm Ma}_1^{ 2}-1}},$">|; 

$key = q/displaystyle=frac{2}{sqrt{{rmMa}_infty^{,2}-1}}left(-frac{dy_L}{dx}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6059.png"
 ALT="$\displaystyle = \frac{2}{\sqrt{{\rm Ma}_\infty^{ 2}-1}}\left(-\frac{dy_L}{dx}\right).$">|; 

$key = q/(partialv_thetaslashpartialr)_{r=a}=omega_phi(a,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3802.png"
 ALT="$ (\partial v_\theta/\partial r)_{r=a}=\omega_\phi(a,\theta)$">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1161.png"
 ALT="$ f$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{AppendixAslashfigA.12.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="164" BORDER="0"
 SRC="|."$dir".q|img6414.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{AppendixA/figA.12.eps}}
\end{figure}">|; 

$key = q/T({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6472.png"
 ALT="$ T({\bf r})$">|; 

$key = q/e=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2785.png"
 ALT="$ e=1$">|; 

$key = q/displaystyle=frac{u_1}{u_2}=frac{(gamma+1),{rmMa}_1^{,2},sin^2beta}{2+(gamma-1),{rmMa}_1^{,2},sin^2beta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5688.png"
 ALT="$\displaystyle =\frac{u_1}{u_2}=\frac{(\gamma+1) {\rm Ma}_1^{ 2} \sin^2\beta}{2+(\gamma-1) {\rm Ma}_1^{ 2} \sin^2\beta},$">|; 

$key = q/displaystylefrac{v_g}{v_p}simeqfrac{1}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4159.png"
 ALT="$\displaystyle \frac{v_g}{v_p} \simeq \frac{1}{2}.$">|; 

$key = q/displaystylew(phi)=frac{{mitGamma}_0}{2pi,b}int_0^pifrac{cosphi',dphi'}{cosphi'-(1+frac{cosphi}{pi}int_0^pifrac{dphi'}{cosphi'-cosphi}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="559" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3493.png"
 ALT="$\displaystyle w(\phi) = \frac{{\mit\Gamma}_0}{2\pi b}\int_0^\pi\frac{\cos\phi'...
...left( 1+\frac{\cos\phi}{\pi}\int_0^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}\right).$">|; 

$key = q/(partialv_rslashpartialr)_{r=a}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3801.png"
 ALT="$ (\partial v_r/\partial r)_{r=a}=0$">|; 

$key = q/n_{rmmax}=20;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5034.png"
 ALT="$ n_{\rm max}=20$">|; 

$key = q/mu_rrightarrowmu_n^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4941.png"
 ALT="$ \mu_r\rightarrow\mu_n^{ m}$">|; 

$key = q/displaystyleI,frac{d^{,2}theta}{dt^{,2}}=-W,lambda,sinthetasimeq-W,lambda,theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img683.png"
 ALT="$\displaystyle I \frac{d^{ 2}\theta}{dt^{ 2}} = - W \lambda \sin\theta\simeq - W \lambda \theta,$">|; 

$key = q/5slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5238.png"
 ALT="$ 5/3$">|; 

$key = q/r=sqrt{x^{,2}+y^{,2}+z^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7166.png"
 ALT="$ r=\sqrt{x^{ 2}+y^{ 2}+z^{ 2}}$">|; 

$key = q/displaystyle-!!left.gammaleft(frac{overline{p}_0+overline{p}}{overline{rho}}righ2}!left(frac{overline{p}_0+overline{p}}{overline{rho}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="527" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="$\displaystyle -\!\! \left.\gamma\left(\frac{\overline{p}_0+\overline{p}}{\overl...
...bla^{ 2}\!\left( \frac{\overline{p}_0 + \overline{p}}{\overline{\rho}}\right),$">|; 

$key = q/displaystylep(x)-p_0=frac{6,mu}{alpha}left[Uleft(frac{1}{d}-frac{1}{d_1}right)-Qleft(frac{1}{d^{,2}}-frac{1}{d_1^{,2}}right)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3701.png"
 ALT="$\displaystyle p(x)-p_0 = \frac{6 \mu}{\alpha}\left[U\left(\frac{1}{d}-\frac{1}{d_1}\right)-Q\left(\frac{1}{d^{ 2}}-\frac{1}{d_1^{ 2}}\right)\right],$">|; 

$key = q/displaystylerho,frac{partial{bfv}}{partialt}+rho,({bfv}cdotnabla),{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3895.png"
 ALT="$\displaystyle \rho \frac{\partial {\bf v}}{\partial t}+ \rho ({\bf v}\cdot\nabla) {\bf v}$">|; 

$key = q/displaystyle={bfV}+{bfv}_1({bfr},t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4067.png"
 ALT="$\displaystyle = {\bf V} + {\bf v}_1({\bf r},t),$">|; 

$key = q/r_H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img824.png"
 ALT="$ r_H$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter09slashwing.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="500" HEIGHT="314" BORDER="0"
 SRC="|."$dir".q|img3402.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter09/wing.eps}}
\end{figure}">|; 

$key = q/displaystyleS=m,{calC}_Vln(p,v^{,gamma})=m,{calC}_Vlnleft(frac{p}{rho^{,gamma}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5326.png"
 ALT="$\displaystyle S = m {\cal C}_V\ln(p v^{ \gamma})= m {\cal C}_V\ln\left(\frac{p}{\rho^{ \gamma}}\right),$">|; 

$key = q/sigma_{11};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$ \sigma_{11}$">|; 

$key = q/displaystyleP_1(mu);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2586.png"
 ALT="$\displaystyle P_1(\mu)$">|; 

$key = q/displaystyleh=left(frac{3,nu,Q}{g,sinalpha}right)^{1slash3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3600.png"
 ALT="$\displaystyle h =\left(\frac{3 \nu Q}{g \sin\alpha}\right)^{1/3}.$">|; 

$key = q/displaystyle=frac{1}{4},a_i^{,2},A,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img841.png"
 ALT="$\displaystyle = \frac{1}{4} a_i^{ 2} A,$">|; 

$key = q/displaystylefrac{partial{bfv}}{partialt}=-nablaleft(frac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi}right)+{bfv}times;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1564.png"
 ALT="$\displaystyle \frac{\partial {\bf v}}{\partial t} = -\nabla\left(\frac{p}{\rho}+\frac{1}{2} v^{ 2}+{\mit\Psi}\right)+ {\bf v}\times$">|; 

$key = q/z=b,sin(k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4119.png"
 ALT="$ z=b \sin(k x)$">|; 

$key = q/displaystylef'''+f,f''=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2985.png"
 ALT="$\displaystyle f''' + f f'' = 0$">|; 

$key = q/m_s=-0.0125;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3231.png"
 ALT="$ m_s=-0.0125$">|; 

$key = q/displaystyledu=c,frac{drho}{rho}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5368.png"
 ALT="$\displaystyle du = c \frac{d\rho}{\rho}.$">|; 

$key = q/{bfc}={bfp}+nu,{bfq};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6732.png"
 ALT="$ {\bf c} = {\bf p} + \nu {\bf q}$">|; 

$key = q/displaystyle2,a_{iklj}+(a_{ikjl}+a_{ilkj}+a_{ijlk});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6966.png"
 ALT="$\displaystyle 2 a_{iklj}+(a_{ikjl}+a_{ilkj}+a_{ijlk})$">|; 

$key = q/{bfa}times({bfb}times{bfc})equiv(a_y,b_x,c_y,,-a_x,b_x,c_y,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6426.png"
 ALT="$ {\bf a}\times({\bf b}\times{\bf c}) \equiv (a_y  b_x  c_y,  -a_x  b_x  c_y,  0)$">|; 

$key = q/f_x=1slashtantheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1197.png"
 ALT="$ f_x=1/\tan\theta$">|; 

$key = q/d'rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4116.png"
 ALT="$ d'\rightarrow \infty$">|; 

$key = q/{rmMa}_infty>5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5923.png"
 ALT="$ {\rm Ma}_\infty&gt;5$">|; 

$key = q/displaystylezeta_0+dzeta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2060.png"
 ALT="$\displaystyle \zeta_0+d\zeta'$">|; 

$key = q/{bfU};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3410.png"
 ALT="$ {\bf U}$">|; 

$key = q/left.v_zright|_{z=d'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4094.png"
 ALT="$ \left.v_z\right\vert _{z=d'}=0$">|; 

$key = q/displaystylesimeqgamma,{rmMa}_p,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5541.png"
 ALT="$\displaystyle \simeq \gamma {\rm Ma}_p,$">|; 

$key = q/rho,{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="$ \rho {\mit\Psi}$">|; 

$key = q/displaystylepsi(x,y)=-6,nu,x^{,1slash3},F(yslashx^{,2slash3}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3253.png"
 ALT="$\displaystyle \psi(x,y)=-6 \nu x^{ 1/3} F(y/x^{ 2/3}).
$">|; 

$key = q/dl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3392.png"
 ALT="$ dl$">|; 

$key = q/displaystyleH_L=frac{(h_2slashh_1-1)^3}{4,h_2slashh_1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1507.png"
 ALT="$\displaystyle H_L = \frac{(h_2/h_1-1)^3}{4 h_2/h_1}.$">|; 

$key = q/y=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6744.png"
 ALT="$ y=a$">|; 

$key = q/e=(1-b^{,2}slasha^{,2})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2768.png"
 ALT="$ e=(1-b^{ 2}/a^{ 2})^{1/2}$">|; 

$key = q/displaystyle={bfB}cdotnablatimes{bfA}-{bfA}cdotnablatimes{bfB},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6691.png"
 ALT="$\displaystyle = {\bf B}\cdot \nabla\times{\bf A} - {\bf A}\cdot \nabla\times{\bf B},$">|; 

$key = q/displaystyleU_1=frac{delta}{L},U_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2888.png"
 ALT="$\displaystyle U _1= \frac{\delta}{L} U_0.$">|; 

$key = q/{bfW}=-W,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3554.png"
 ALT="$ {\bf W} = -W {\bf e}_y$">|; 

$key = q/overline{rho}slashrho=780;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3821.png"
 ALT="$ \overline{\rho}/\rho=780$">|; 

$key = q/displaystyleh_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4433.png"
 ALT="$\displaystyle h_n$">|; 

$key = q/{bfa}ast{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6351.png"
 ALT="$ {\bf a} \ast {\bf b}$">|; 

$key = q/b_{jk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6857.png"
 ALT="$ b_{jk}$">|; 

$key = q/a_0=(a_1,a_2,a_3)^{1slash3}=(3,Vslash4pi)^{1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img891.png"
 ALT="$ a_0=(a_1 a_2 a_3)^{1/3}= (3 V/4\pi)^{1/3}$">|; 

$key = q/displaystyley(xi)=-xi^{,2},frac{dtheta}{dxi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5219.png"
 ALT="$\displaystyle y(\xi) = -\xi^{ 2} \frac{d\theta}{d\xi}.$">|; 

$key = q/|u_2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5636.png"
 ALT="$ \vert u_2\vert$">|; 

$key = q/displaystylefrac{c(x)-c_2}{c_1-c_2}=frac{x-x_2}{x_1-x_2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5570.png"
 ALT="$\displaystyle \frac{c(x)-c_2}{c_1-c_2} = \frac{x-x_2}{x_1-x_2}$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.15.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="200" BORDER="0"
 SRC="|."$dir".q|img6461.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.15.eps}}
\end{figure}">|; 

$key = q/epsilonllmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5732.png"
 ALT="$ \epsilon\ll \mu$">|; 

$key = q/displaystylefrac{partialphi}{partialeta}=-frac{1}{c,coshxi,sineta},frac{partialpsi}{partialxi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2776.png"
 ALT="$\displaystyle \frac{\partial\phi}{\partial\eta} = -\frac{1}{c \cosh\xi \sin\eta} \frac{\partial\psi}{\partial\xi},$">|; 

$key = q/displaystyle{mitPhi}_{rmmoon}({bfr})=-frac{G,m'}{|{bfr}-{bfr}'|}=-frac{G,m'}{r'}sum_{n=0,infty}left(frac{r}{r'}right)^nP_n(cosgamma),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4245.png"
 ALT="$\displaystyle {\mit\Phi}_{\rm moon}({\bf r}) = - \frac{G m'}{\vert{\bf r}-{\bf...
... -\frac{G m'}{r'}\sum_{n=0,\infty}\left(\frac{r}{r'}\right)^n P_n(\cos\gamma),$">|; 

$key = q/displaystyle={calH}_2+frac{1}{2},u_2^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5467.png"
 ALT="$\displaystyle = {\cal H}_2+\frac{1}{2} u_2^{ 2},$">|; 

$key = q/displaystyleJ=ointfrac{d{mitOmega}}{sum_{i=1,3}n_i^{,2}slasha_i^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7253.png"
 ALT="$\displaystyle J = \oint \frac{d{\mit\Omega}}{\sum_{i=1,3}n_i^{ 2}/a_i^{ 2}}.$">|; 

$key = q/displaystyle=-frac{1}{h_xi},frac{partialphi}{partialxi}=-frac{1}{varpi,h_eta},frac{partialpsi}{partialeta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2694.png"
 ALT="$\displaystyle =-\frac{1}{h_\xi} \frac{\partial\phi}{\partial\xi} =-\frac{1}{\varpi h_\eta} \frac{\partial\psi}{\partial \eta},$">|; 

$key = q/displaystyleV>(g,d)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4019.png"
 ALT="$\displaystyle V &gt;(g d)^{1/2}.$">|; 

$key = q/H=7.99,{rmkm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5130.png"
 ALT="$ H= 7.99 {\rm km}$">|; 

$key = q/displaystyle=int_0^gamma(1-sin^2alpha,sin^2theta)^{1slash2},dtheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img944.png"
 ALT="$\displaystyle = \int_0^\gamma (1-\sin^2\alpha  \sin^2\theta)^{1/2} d\theta,$">|; 

$key = q/displaystylefrac{partialrho}{partialt}+v_j,frac{partialrho}{partialx_j}+rho,frac{partialv_j}{partialx_j}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="$\displaystyle \frac{\partial\rho}{\partial t} + v_j \frac{\partial\rho}{\partial x_j} + \rho \frac{\partial v_j}{\partial x_j}=0.$">|; 

$key = q/displaystyleF(phi,k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1182.png"
 ALT="$\displaystyle F(\phi,k)$">|; 

$key = q/displaystylefrac{z-{rmi},d}{b}=frac{2}{pi}int_{sin^{-1}(lambda)}^{pislash2}expleft(-{rmi},frac{2,alpha}{pi},betaright),frac{dbeta}{tanbeta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2248.png"
 ALT="$\displaystyle \frac{z-{\rm i} d}{b} = \frac{2}{\pi}\int_{\sin^{-1}(\lambda)}^{...
...p\left(-{\rm i} \frac{2 \alpha}{\pi} \beta\right) \frac{d\beta}{\tan\beta}.$">|; 

$key = q/displaystylev_z(r)=frac{G}{4,mu},(a^{,2}-r^{,2}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3615.png"
 ALT="$\displaystyle v_z(r) = \frac{G}{4 \mu} (a^{ 2}-r^{ 2}).$">|; 

$key = q/displaystyleleft(frac{partial}{partialt}+{bfV}cdotnablaright){bfv}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4071.png"
 ALT="$\displaystyle \left(\frac{\partial}{\partial t} + {\bf V}\cdot\nabla\right){\bf v}_1$">|; 

$key = q/displaystyleint_{-h}^hu(y),dy=2,int_{-l}^lv(x),dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3047.png"
 ALT="$\displaystyle \int_{-h}^h u(y) dy = 2 \int_{-l}^l v(x) dx.$">|; 

$key = q/I_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7285.png"
 ALT="$ I_1$">|; 

$key = q/displaystyleF(z)={rmi},frac{{mitGamma}}{2pi},ln(z-{rmi},a)-{rmi},frac{{mitGamma}rac{{mitGamma}}{2pi},lnleft(frac{z-{rmi},a}{z+{rmi},a}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="507" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2033.png"
 ALT="$\displaystyle F(z) = {\rm i}  \frac{{\mit\Gamma}}{2\pi} \ln(z-{\rm i} a) - {...
... \frac{{\mit\Gamma}}{2\pi} \ln\left(\frac{z-{\rm i} a}{z+{\rm i} a}\right).$">|; 

$key = q/displaystyle=-oint_CP,dy,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2389.png"
 ALT="$\displaystyle =-\oint_C P dy,$">|; 

$key = q/displaystylefrac{1}{2},{rmi},rhooint_Cleft(frac{dF}{dz}right)^{,2}dz=X-{rmi},Y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3284.png"
 ALT="$\displaystyle \frac{1}{2} {\rm i} \rho\oint_C \left(\frac{dF}{dz}\right)^{ 2}dz = X-{\rm i} Y,$">|; 

$key = q/(p_0slashp_{rmatm})_{rmcrit}=1.89;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5429.png"
 ALT="$ (p_0/p_{\rm atm})_{\rm crit} = 1.89$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter05slashjet1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="537" HEIGHT="401" BORDER="0"
 SRC="|."$dir".q|img1894.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter05/jet1.eps}}
\end{figure}">|; 

$key = q/{rmRe}^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3101.png"
 ALT="$ {\rm Re}^{1/2}$">|; 

$key = q/displaystyleD,{mitPsi}_r';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4855.png"
 ALT="$\displaystyle D {\mit\Psi}_r'$">|; 

$key = q/w_1=(u_1^{,2}+v^{,2})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5680.png"
 ALT="$ w_1=(u_1^{ 2}+v^{ 2})^{1/2}$">|; 

$key = q/displaystylezeta(x,t)=zeta_0,sin(omega,t-k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4088.png"
 ALT="$\displaystyle \zeta(x,t)= \zeta_0 \sin(\omega t-k x)$">|; 

$key = q/displaystyle=2pi,gamma,rho,V_0^{,2},a=rho,V_0,(-{mitGamma}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1785.png"
 ALT="$\displaystyle = 2\pi \gamma \rho V_0^{ 2} a= \rho V_0 (-{\mit\Gamma}).$">|; 

$key = q/int_0^1left(y'^{,2}+x^2right),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7370.png"
 ALT="$ \int_0^1\left(y'^{ 2}+x^2\right) dx$">|; 

$key = q/displaystyledelta_2(x)=int_0^inftyfrac{u}{U}left(1-frac{u}{U}right)dy.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3086.png"
 ALT="$\displaystyle \delta_2(x) = \int_0^\infty \frac{u}{U}\left(1-\frac{u}{U}\right)dy.$">|; 

$key = q/p({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4375.png"
 ALT="$ p({\bf r})$">|; 

$key = q/omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img530.png"
 ALT="$ \omega$">|; 

$key = q/displaystyleM={rmRe}left[-frac{1}{2},rhooint_Cleft(frac{dF}{dz}right)^{,2}z,dzright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2412.png"
 ALT="$\displaystyle M = {\rm Re}\left[-\frac{1}{2} \rho \oint_C \left(\frac{dF}{dz}\right)^{ 2} z dz\right],$">|; 

$key = q/displaystyle=rho,V,{mitGamma},{bfe}_perp,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3319.png"
 ALT="$\displaystyle = \rho V {\mit\Gamma} {\bf e}_\perp,$">|; 

$key = q/displaystyler;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1318.png"
 ALT="$\displaystyle r$">|; 

$key = q/displaystyleskew{5}bar{zeta}_2(theta,phi,t)=skew{5}tilde{zeta}_j,P_2^{,m_j}(costheta),{rme}^{,{rmi},(m_j,phi+sigma_j,t)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4923.png"
 ALT="$\displaystyle \skew{5}\bar{\zeta}_2(\theta,\phi,t) = \skew{5}\tilde{\zeta}_j P_2^{ m_j}(\cos\theta) {\rm e}^{ {\rm i} (m_j \phi+\sigma_j t)}.$">|; 

$key = q/displaystyley=frac{k}{sin^4alpha}left[2,cosomega+cosalpha,sin^2omega+sinalpha,sinomega,cosomega+sinalphaleft(frac{pi}{2}-omegaright)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="579" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2313.png"
 ALT="$\displaystyle y = \frac{k}{\sin^4\alpha}\left[2 \cos\omega+\cos\alpha \sin^2\...
...pha \sin\omega \cos\omega+\sin\alpha\left(\frac{\pi}{2}-\omega\right)\right],$">|; 

$key = q/displaystylea_{ijk}=mu,epsilon_{ijk}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6934.png"
 ALT="$\displaystyle a_{ijk} = \mu \epsilon_{ijk}.$">|; 

$key = q/displaystylephantom{=}-frac{f_j,(n+2)^2,n,(n-m_j+1),(n+m_j+1)}{(n+1),(2,n+1),(2,n+3),[m_j-(n+1),(n+2),f_j]},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="456" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4705.png"
 ALT="$\displaystyle \phantom{=}-\frac{f_j (n+2)^2 n (n-m_j+1) (n+m_j+1)}{(n+1) (2 n+1) (2 n+3) [m_j-(n+1) (n+2) f_j]},$">|; 

$key = q/displaystyleskew{5}ddot{mitPsi}-2,{mitOmega},skew{5}dot{mitPsi}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4793.png"
 ALT="$\displaystyle \skew{5}\ddot{\mit\Psi} - 2 {\mit\Omega} \skew{5}\dot{\mit\Psi}'$">|; 

$key = q/displaystylefrac{G''}{G};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5964.png"
 ALT="$\displaystyle \frac{G''}{G}$">|; 

$key = q/displaystyle=frac{h}{2},(1+sinalpha),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1891.png"
 ALT="$\displaystyle = \frac{h}{2} (1+\sin\alpha),$">|; 

$key = q/displaystylebar{F}=skew{3}bar{phi}=V,(k+s),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2270.png"
 ALT="$\displaystyle \bar{F}=\skew{3}\bar{\phi}=V (k+s),$">|; 

$key = q/|{bfA}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6597.png"
 ALT="$ \vert{\bf A}\vert$">|; 

$key = q/{bfF}={bfg}times{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1095.png"
 ALT="$ {\bf F} = {\bf g}\times {\bf b}$">|; 

$key = q/h_2-h_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1479.png"
 ALT="$ h_2-h_1$">|; 

$key = q/gamma=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img888.png"
 ALT="$ \gamma=\pi/2$">|; 

$key = q/displaystyle(widetilde{nabla{bfA}})_{rz}=(widetilde{nabla{bfA}})_{zr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7155.png"
 ALT="$\displaystyle (\widetilde{\nabla {\bf A}})_{rz}=(\widetilde{\nabla {\bf A}})_{zr}$">|; 

$key = q/displaystyleomega_z=left{{array}{lll}{mitGamma}slash(pi,a^{,2})&mbox{hspace{1cm}}&rleqa[0.5ex]0&&r>a{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1710.png"
 ALT="$\displaystyle \omega_z = \left\{\begin{array}{lll} {\mit\Gamma}/(\pi a^{ 2})&amp;\mbox{\hspace{1cm}}&amp;r\leq a\ [0.5ex] 0&amp;&amp;r&gt;a \end{array}\right..$">|; 

$key = q/{bfg}=gamma,{bfn};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1102.png"
 ALT="$ {\bf g} = \gamma {\bf n}$">|; 

$key = q/d{bfr}equiv(dx,,dy);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6520.png"
 ALT="$ d{\bf r}\equiv(dx, dy)$">|; 

$key = q/displaystylefrac{partialphi}{partialr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1755.png"
 ALT="$\displaystyle \frac{\partial\phi}{\partial r}$">|; 

$key = q/nabla^{,2}phi=nabla^{,2}psi=nablaphicdotnablapsi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2083.png"
 ALT="$ \nabla^{ 2}\phi=\nabla^{ 2}\psi=\nabla\phi\cdot\nabla\psi=0$">|; 

$key = q/skew{5}bar{zeta}_2(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4408.png"
 ALT="$ \skew{5}\bar{\zeta}_2(\theta,\phi)$">|; 

$key = q/displaystyle{mitPsi}=-frac{1}{2},G,rhoointleft[frac{2left(sum_{i=1,3}x_i,n_islasa_i^{,2}}{sum_{i=1,3}n_i^{,2}slasha_i^{,2}}right]d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="98" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7249.png"
 ALT="$\displaystyle {\mit\Psi} = -\frac{1}{2} G \rho\oint\left[ \frac{2\left(\sum_{...
...,3}x_i^{ 2}/a_i^{ 2}}{\sum_{i=1,3}n_i^{ 2}/a_i^{ 2}} \right] d{\mit\Omega}.$">|; 

$key = q/displaystyleb(n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4702.png"
 ALT="$\displaystyle b(n)$">|; 

$key = q/l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1056.png"
 ALT="$ l$">|; 

$key = q/{mitPi}=nablacdot{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5074.png"
 ALT="$ {\mit\Pi} = \nabla\cdot{\bf v}$">|; 

$key = q/displaystylefrac{d}{dx}!left(frac{partialF}{partialy'}right)-frac{partialF}{part}{partialy'}right)-frac{partial[lambda,G]}{partialy}right]=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7337.png"
 ALT="$\displaystyle \frac{d}{dx}\!\left(\frac{\partial F}{\partial y'}\right)-\frac{\...
...da G]}{\partial y'}\right)-\frac{\partial [\lambda G]}{\partial y}\right]= 0.$">|; 

$key = q/|{bfr}|^{,n},{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6755.png"
 ALT="$ \vert{\bf r}\vert^{ n} {\bf r}$">|; 

$key = q/displaystylev_xi={bfV}cdot{bfe}_xi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2707.png"
 ALT="$\displaystyle v_\xi = {\bf V}\cdot{\bf e}_\xi,$">|; 

$key = q/displaystylenuleft.frac{partialu}{partialy}right|_{y=0}=U^{,2},frac{ddelta_2}{dx}+U,frac{dU}{dx},(delta_1+2,delta_2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3087.png"
 ALT="$\displaystyle \nu\left.\frac{\partial u}{\partial y}\right\vert _{y=0} = U^{ 2} \frac{d\delta_2}{dx} + U \frac{dU}{dx} (\delta_1+2 \delta_2).$">|; 

$key = q/displaystyle=frac{2,m}{(m'^{,2}-m^{,2}),(2,n'+1)}left[(n'-1),(n'+1),(n'+m'),I_{n,,n'-1}^{,m,,m'}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="477" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4977.png"
 ALT="$\displaystyle =\frac{2 m}{(m'^{ 2}-m^{ 2}) (2 n'+1)} \left[(n'-1) (n'+1) (n'+m') I_{n, n'-1}^{ m, m'} \right.$">|; 

$key = q/{bfb}times{bfc}equiv(0,,0,,b_x,c_y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6425.png"
 ALT="$ {\bf b}\times{\bf c} \equiv (0, 0, b_x  c_y)$">|; 

$key = q/eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2057.png"
 ALT="$ \eta)$">|; 

$key = q/displaystyledT={bfgrad},Tcdotd{bfr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6544.png"
 ALT="$\displaystyle dT = {\bf grad} T \cdot d{\bf r}.$">|; 

$key = q/displaystyleb=V^{,-1},B.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4124.png"
 ALT="$\displaystyle b= V^{ -1} B.$">|; 

$key = q/psi_1=psi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1626.png"
 ALT="$ \psi_1=\psi_2$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partialP(theta,phi_pm,t)}{partialphi}=-Y(theta,phi_pm,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4736.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial P(\theta,\phi_\pm,t)}{\partial\phi}=-Y(\theta,\phi_\pm,t),$">|; 

$key = q/displaystylez=x+{rmi},y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1933.png"
 ALT="$\displaystyle z = x + {\rm i} y,$">|; 

$key = q/alpha,pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1867.png"
 ALT="$ \alpha \pi$">|; 

$key = q/nabla^{,2}phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6758.png"
 ALT="$ \nabla^{ 2}\phi$">|; 

$key = q/lambda<lambda_s=-0.1567;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3228.png"
 ALT="$ \lambda&lt;\lambda_s=-0.1567$">|; 

$key = q/displaystyleU=langlefrac{1}{2},rho,g,(zeta^{,2}-d^{,2})rangle+frac{1}{2},rho,g,d^{,2}=frac{1}{2},rho,g,langlezeta^{,2}rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3979.png"
 ALT="$\displaystyle U = \langle \frac{1}{2} \rho g (\zeta^{ 2}-d^{ 2})\rangle + \frac{1}{2} \rho g d^{ 2} = \frac{1}{2} \rho g \langle\zeta^{ 2}\rangle,$">|; 

$key = q/nablacdot{bfA}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6589.png"
 ALT="$ \nabla\cdot{\bf A} = 0$">|; 

$key = q/displaystyle=nu,frac{partial^{,2}u}{partialy^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3075.png"
 ALT="$\displaystyle = \nu \frac{\partial^{ 2} u}{\partial y^{ 2}},$">|; 

$key = q/phi=tan^{-1}(yslashx);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7167.png"
 ALT="$ \phi=\tan^{-1}(y/x)$">|; 

$key = q/Y=-costheta,xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4766.png"
 ALT="$ Y=-\cos\theta \xi$">|; 

$key = q/phi({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1585.png"
 ALT="$ \phi({\bf r})$">|; 

$key = q/z={rmi},d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2214.png"
 ALT="$ z={\rm i} d$">|; 

$key = q/displaystyle=gleft(frac{sigma,k_y+{rmi},f,k_x}{sigma^{,2}-f^{,2}}right)zeta_0,,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5090.png"
 ALT="$\displaystyle = g\left(\frac{\sigma k_y+{\rm i} f k_x}{\sigma^{ 2}-f^{ 2}}\right)\zeta_0,,$">|; 

$key = q/displaystyle=-frac{1}{rho},frac{partialp}{partialr}-frac{partial{mitPsi}}{partialr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="$\displaystyle = - \frac{1}{\rho} \frac{\partial p}{\partial r} - \frac{\partial{\mit\Psi}}{\partial r}$">|; 

$key = q/displaystyleF(z)=3,tanh^2left(alpha+frac{z}{sqrt{2}}right)-2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3240.png"
 ALT="$\displaystyle F(z) = 3 \tanh^2\left(\alpha+\frac{z}{\sqrt{2}}\right)-2
$">|; 

$key = q/displaystyleX(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1853.png"
 ALT="$\displaystyle X(t)$">|; 

$key = q/skew{5}tilde{zeta}_j(rmm);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4608.png"
 ALT="$ \skew{5}\tilde{\zeta}_j (\rm m)$">|; 

$key = q/displaystyle=-frac{nablap}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="$\displaystyle = -\frac{\nabla p}{\rho},$">|; 

$key = q/displaystyleC-frac{f^{,2}}{2,l^{,2}}=-cos(2,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1170.png"
 ALT="$\displaystyle C - \frac{f^{ 2}}{2 l^{ 2}} =-\cos(2 \phi),$">|; 

$key = q/pislashK'=w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2160.png"
 ALT="$ \pi/K'=w$">|; 

$key = q/F(lambda)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3199.png"
 ALT="$ F(\lambda)=0$">|; 

$key = q/displaystylefrac{b^{,2}}{6,c^{,2}}-s,(1-s)>0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img676.png"
 ALT="$\displaystyle \frac{b^{ 2}}{6  c^{ 2}} - s (1-s)&gt; 0.$">|; 

$key = q/{calF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4796.png"
 ALT="$ {\cal F}$">|; 

$key = q/displaystyle{calC}_P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5386.png"
 ALT="$\displaystyle {\cal C}_P$">|; 

$key = q/displaystylev_r=frac{Q}{4pi,r^{,2}}=-frac{1}{r^{,2},sintheta},frac{partialpsi}{partialtheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2610.png"
 ALT="$\displaystyle v_r = \frac{Q}{4\pi r^{ 2}} = -\frac{1}{r^{ 2} \sin\theta} \frac{\partial\psi}{\partial\theta}.$">|; 

$key = q/gamma>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1082.png"
 ALT="$ \gamma&gt;0$">|; 

$key = q/displaystylefrac{partial}{partialx},(rho,u)+frac{partial}{partialy},(rho,v);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5874.png"
 ALT="$\displaystyle \frac{\partial}{\partial x} (\rho u) + \frac{\partial}{\partial y} (\rho v)$">|; 

$key = q/{mitOmega}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3330.png"
 ALT="$ {\mit\Omega}_z$">|; 

$key = q/displaystyled{bfx}=frac{du_1}{|nablau_1|},{bfe}_1+frac{du_2}{|nablau_2|},{bfe}_2+frac{du_3}{|nablau_3|},{bfe}_3=sum_{i=1,3}h_i,du_i,{bfe}_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="434" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7045.png"
 ALT="$\displaystyle d {\bf x} = \frac{du_1}{\vert\nabla u_1\vert} {\bf e}_1 + \frac{...
...\frac{du_3}{\vert\nabla u_3\vert} {\bf e}_3=\sum_{i=1,3} h_i du_i {\bf e}_i.$">|; 

$key = q/theta=1-xi^{,2}slash6+{calO}(xi^{,4});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5220.png"
 ALT="$ \theta= 1-\xi^{ 2}/6 + {\cal O}(\xi^{ 4})$">|; 

$key = q/1slash294;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img917.png"
 ALT="$ 1/294$">|; 

$key = q/displaystylev_x,frac{partialv_x}{partialx}+v_y,frac{partialv_x}{partialy}=-frac{x}{partialx^{,2}}+frac{partial^{,2}v_x}{partialy^{,2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3021.png"
 ALT="$\displaystyle v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\...
...2} v_x}{\partial x^{ 2}} + \frac{\partial^{ 2} v_x}{\partial y^{ 2}}\right).$">|; 

$key = q/displaystyle=frac{partialskew{3}bar{phi}}{partialeta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6019.png"
 ALT="$\displaystyle =\frac{\partial\skew{3}\bar{\phi}}{\partial \eta},$">|; 

$key = q/c_p=c_V+R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="$ c_p=c_V+R$">|; 

$key = q/displaystylechi=frac{partialv_i}{partialx_j},d_{ij}=2,muleft(e_{ij},e_{ij}-frac{ac{partialv_i}{partialx_i},frac{partialv_j}{partialx_j}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="631" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="$\displaystyle \chi = \frac{\partial v_i}{\partial x_j} d_{ij} = 2 \mu\left(e_...
...3} \frac{\partial v_i}{\partial x_i} \frac{\partial v_j}{\partial x_j}\right)$">|; 

$key = q/k,drightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3994.png"
 ALT="$ k d\rightarrow 0$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashsource1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img2120.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/source1.eps}}
\end{figure}">|; 

$key = q/theta_0=-2,alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3355.png"
 ALT="$ \theta_0=-2 \alpha$">|; 

$key = q/displaystyleleft.wright|_{z=0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4470.png"
 ALT="$\displaystyle \left.w\right\vert _{z=0}$">|; 

$key = q/gamma_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1123.png"
 ALT="$ \gamma_{12}$">|; 

$key = q/displaystyle=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="$\displaystyle = 0,$">|; 

$key = q/m=-0.05;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$ m=-0.05$">|; 

$key = q/v_varphi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3750.png"
 ALT="$ v_\varphi=0$">|; 

$key = q/sqrt{g,h};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1442.png"
 ALT="$ \sqrt{g h}$">|; 

$key = q/displaystyle=(a_y,c_y,b_x,,-a_x,b_x,c_y,,0)={rmLHS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6431.png"
 ALT="$\displaystyle = (a_y  c_y  b_x,  -a_x  b_x  c_y,  0 ) = {\rm LHS},$">|; 

$key = q/displaystyle=-left(frac{m-m'}{m+m'}right),frac{g}{omega^{,2}}left[1-cos(omega,t)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1856.png"
 ALT="$\displaystyle = -\left(\frac{m-m'}{m+m'}\right) \frac{g}{\omega^{ 2}}\left[1-\cos(\omega t)\right].$">|; 

$key = q/displaystylesimeq-frac{3}{4},V,sintheta,frac{a}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3890.png"
 ALT="$\displaystyle \simeq -\frac{3}{4} V \sin\theta \frac{a}{r},$">|; 

$key = q/displaystyleC_D=frac{1}{l}int_0^lC_p,frac{dy}{dx},dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5998.png"
 ALT="$\displaystyle C_D=\frac{1}{l}\int_0^lC_p \frac{dy}{dx} dx,$">|; 

$key = q/|{bfa}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6224.png"
 ALT="$ \vert{\bf a}\vert$">|; 

$key = q/displaystyle2,a_{ijkl}+a_{jikl}+a_{kjil}+a_{ljki};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6938.png"
 ALT="$\displaystyle 2 a_{ijkl} + a_{jikl} + a_{kjil}+a_{ljki}$">|; 

$key = q/psi=c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1627.png"
 ALT="$ \psi=c$">|; 

$key = q/displaystylefrac{c^{,2}}{c_0^{,2}}=frac{1}{2}left(frac{1}{k,l}+k,lright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4178.png"
 ALT="$\displaystyle \frac{c^{ 2}}{c_0^{ 2}} = \frac{1}{2}\left(\frac{1}{k l} + k l\right).$">|; 

$key = q/_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$ _x$">|; 

$key = q/M_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img588.png"
 ALT="$ M_0$">|; 

$key = q/displaystyle{bfc}={bfa}-{bfb}={bfa}+(-{bfb}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6240.png"
 ALT="$\displaystyle {\bf c} = {\bf a} - {\bf b} = {\bf a} +(-{\bf b}).$">|; 

$key = q/displaystyled{mitPsi}=-int_0^{r'}frac{G,rho}{r},dV-int_{r''}^0frac{G,rho}{(-r)},dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7232.png"
 ALT="$\displaystyle d{\mit\Psi} = - \int_0^{r'}\frac{G \rho}{r} dV - \int_{r''}^0 \frac{G \rho}{(-r)} dV,$">|; 

$key = q/v_perp,deltas;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1631.png"
 ALT="$ v_\perp \delta s$">|; 

$key = q/displaystyle{mitPhi}(theta,phi,t)=sum_{r=1,infty}p_r(t),{mitPhi}_r(theta,phi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4890.png"
 ALT="$\displaystyle {\mit\Phi}(\theta,\phi,t) = \sum_{r=1,\infty} p_r(t) {\mit\Phi}_r(\theta,\phi).$">|; 

$key = q/displaystyle{bfv}({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4066.png"
 ALT="$\displaystyle {\bf v}({\bf r},t)$">|; 

$key = q/lambda=exp[-(pislash2),(sslashb)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2249.png"
 ALT="$ \lambda = \exp[-(\pi/2) (s/b)]$">|; 

$key = q/displaystylefrac{d}{dt}left({bfa}cdot{bfb}right)=frac{d{bfa}}{dt}cdot{bfb}+{bfa}cdotfrac{d{bfb}}{dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6439.png"
 ALT="$\displaystyle \frac{d}{dt}\left({\bf a}\cdot{\bf b}\right) = \frac{d{\bf a}}{dt}\cdot {\bf b} +{\bf a}\cdot\frac{d{\bf b}}{dt},$">|; 

$key = q/displaystyle=rho,{mitGamma},V_x-m',frac{dV_y}{dt}+m',g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1841.png"
 ALT="$\displaystyle = \rho {\mit\Gamma} V_x - m' \frac{dV_y}{dt} + m' g.$">|; 

$key = q/1.2leq{rmMa}_inftyleq5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5921.png"
 ALT="$ 1.2\leq {\rm Ma}_\infty \leq 5$">|; 

$key = q/displaystyle{rmFr}_1^{,2}=frac{1}{2},frac{h_2}{h_1}left(frac{h_2}{h_1}+1right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1505.png"
 ALT="$\displaystyle {\rm Fr}_1^{ 2} = \frac{1}{2} \frac{h_2}{h_1}\left(\frac{h_2}{h_1}+1\right).$">|; 

$key = q/r^{-(l+1)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2593.png"
 ALT="$ r^{-(l+1)}$">|; 

$key = q/displaystylefrac{partialpsi}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1658.png"
 ALT="$\displaystyle \frac{\partial\psi}{\partial x}$">|; 

$key = q/z_0={rmi},skew{3}bar{psi}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2205.png"
 ALT="$ z_0={\rm i} \skew{3}\bar{\psi}_1$">|; 

$key = q/-0.05;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$ -0.05$">|; 

$key = q/deltaslashc=1.0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$ \delta /c = 1.0$">|; 

$key = q/displaystyle=frac{pi}{8},rho,V^{,2},(c^{,2}-delta^{,2}),sin(2,alpha),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3320.png"
 ALT="$\displaystyle =\frac{\pi}{8} \rho V^{ 2} (c^{ 2}-\delta^{ 2}) \sin(2 \alpha),$">|; 

$key = q/displaystylea(n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4700.png"
 ALT="$\displaystyle a(n)$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partial{mitPhi}'(theta,phi_pm,t)}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4773.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial{\mit\Phi}'(\theta,\phi_\pm,t)}{\partial \phi}$">|; 

$key = q/displaystyle{rmMa}_1^{,2}=1+left(frac{gamma+1}{2,gamma}right)frac{{mitDelta}p}{p_1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5521.png"
 ALT="$\displaystyle {\rm Ma}_1^{ 2} = 1+\left(\frac{\gamma+1}{2 \gamma}\right)\frac{{\mit\Delta} p}{p_1}.$">|; 

$key = q/displaystylefrac{partialphi(x,infty)}{partialx}=frac{partialphi(x,infty)}{partialy}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5956.png"
 ALT="$\displaystyle \frac{\partial \phi(x,\infty)}{\partial x} = \frac{\partial \phi(x,\infty)}{\partial y} =0.$">|; 

$key = q/displaystyle{bfa}cdot{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6302.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b}$">|; 

$key = q/c^{,2}>(1slash2),a^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1050.png"
 ALT="$ c^{ 2}&gt;(1/2) a^{ 2}$">|; 

$key = q/n=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4699.png"
 ALT="$ n=2$">|; 

$key = q/displaystylefrac{beta_{-n,,n'}^{,-m,,m'}}{c_n^{,m},c_{n'}^{,m'}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4960.png"
 ALT="$\displaystyle \frac{\beta_{-n, n'}^{ -m, m'}}{c_n^{ m} c_{n'}^{ m'}}$">|; 

$key = q/displaystylefrac{dp}{dr}+rho,frac{d{mitPsi}}{dr}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5177.png"
 ALT="$\displaystyle \frac{dp}{dr} + \rho \frac{d{\mit\Psi}}{dr} = 0,$">|; 

$key = q/{bfa}ast{bfb}=(1slash2,,-1slash2,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6357.png"
 ALT="$ {\bf a}\ast {\bf b} = (1/2,  -1/2, 0)$">|; 

$key = q/displaystyle{rmRHS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6429.png"
 ALT="$\displaystyle {\rm RHS}$">|; 

$key = q/S_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1355.png"
 ALT="$ S_1$">|; 

$key = q/displaystylephantom{=}left.+frac{1}{2}left(frac{1}{r,sintheta},frac{partialv_therphi}{partialtheta}-frac{cottheta,v_varphi}{r}right)^2right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img516.png"
 ALT="$\displaystyle \phantom{=}\left.+\frac{1}{2}\left(\frac{1}{r \sin\theta} \frac...
...al v_\varphi}{\partial \theta}-\frac{\cot\theta v_\varphi}{r}\right)^2\right].$">|; 

$key = q/displaystylefrac{{mitDelta}{rmMa}}{{rmMa}_1}simeq-left[frac{1+[(gamma-1)slash2],{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right]theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6117.png"
 ALT="$\displaystyle \frac{{\mit\Delta}{\rm Ma}}{{\rm Ma}_1}\simeq -\left[\frac{1+[(\gamma-1)/2] {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right]\theta.
$">|; 

$key = q/displaystylec^{,2}=g,lleft(frac{s-1}{s+1}right)left(k,l-frac{1}{k,l}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4182.png"
 ALT="$\displaystyle c^{ 2} = g l\left(\frac{s-1}{s+1}\right)\left(k l-\frac{1}{k l}\right).$">|; 

$key = q/{bfe}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7112.png"
 ALT="$ {\bf e}_r$">|; 

$key = q/displaystylefrac{partialphi}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1656.png"
 ALT="$\displaystyle \frac{\partial\phi}{\partial x}$">|; 

$key = q/lambda_2gtreqqless0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img700.png"
 ALT="$ \lambda_2\gtreqqless 0$">|; 

$key = q/displaystylefrac{y}{d};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2232.png"
 ALT="$\displaystyle \frac{y}{d}$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}=frac{p_0}{rho}+frac{1}{2},rho,V^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2652.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} = \frac{p_0}{\rho}+\frac{1}{2} \rho V^{ 2},$">|; 

$key = q/dUslashdx=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3200.png"
 ALT="$ dU/dx=0$">|; 

$key = q/displaystyle=sum_{r=1,infty}left(p_{-r},frac{partial{mitPsi}_r}{partialtheta}-p_,frac{1}{sintheta},frac{partial{mitPhi}_r}{partialphi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4893.png"
 ALT="$\displaystyle = \sum_{r=1,\infty}\left(p_{-r} \frac{\partial{\mit\Psi}_r}{\par...
... p_{r} \frac{1}{\sin\theta} \frac{\partial{\mit\Phi}_r}{\partial\phi}\right),$">|; 

$key = q/-(1-ln2)slash(2pi)=-0.153;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2522.png"
 ALT="$ -(1-\ln 2)/(2\pi)=-0.153$">|; 

$key = q/I=I_{rmmax}costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6627.png"
 ALT="$ I= I_{\rm max}\cos\theta$">|; 

$key = q/displaystylefrac{1}{4},M,g-frac{3}{16},M,omega^{,2},a,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1067.png"
 ALT="$\displaystyle \frac{1}{4} M g - \frac{3}{16} M \omega^{ 2} a,
$">|; 

$key = q/sinthetasimeqtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6395.png"
 ALT="$ \sin\theta \simeq \theta$">|; 

$key = q/displaystylesigma_{thetatheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="$\displaystyle \sigma_{\theta\theta}$">|; 

$key = q/displaystyleI_n(phi)=int_0^pifrac{cos(n,phi'),dphi'}{cosphi'-cosphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3538.png"
 ALT="$\displaystyle I_n(\phi) = \int_0^\pi \frac{\cos(n \phi') d\phi'}{\cos\phi'-\cos\phi},
$">|; 

$key = q/displaystyleD=left(frac{2pi}{pi+4}right)rho,V^{,2},b.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2284.png"
 ALT="$\displaystyle D= \left(\frac{2\pi}{\pi+4}\right)\rho V^{ 2} b.$">|; 

$key = q/[(K+4muslash3)slashrho]^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7024.png"
 ALT="$ [(K+4\mu/3)/\rho]^{1/2}$">|; 

$key = q/displaystyle=muleft(frac{1}{r},frac{partialv_z}{partialtheta}+frac{partialv_theta}{partialz}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="$\displaystyle = \mu\left(\frac{1}{r} \frac{\partial v_z}{\partial\theta}+\frac{\partial v_\theta}{\partial z}\right),$">|; 

$key = q/p_0';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img856.png"
 ALT="$ p_0'$">|; 

$key = q/p_e>p_{rmatm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5434.png"
 ALT="$ p_e&gt;p_{\rm atm}$">|; 

$key = q/displaystyle{bfgrad},(phi,psi)=phi,,{bfgrad},psi+psi,,{bfgrad},phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6564.png"
 ALT="$\displaystyle {\bf grad} (\phi  \psi) = \phi   {\bf grad} \psi +\psi   {\bf grad} \phi$">|; 

$key = q/displaystylex_i,T_{ij},x_j=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6995.png"
 ALT="$\displaystyle x_i T_{ij} x_j = 1,
$">|; 

$key = q/displaystyle=left[frac{1}{r,sintheta},frac{partial}{partialtheta},(sintheta,A_ph{1}{r,sintheta},frac{partialA_theta}{partialphi}right]{bfe}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7180.png"
 ALT="$\displaystyle = \left[\frac{1}{r \sin\theta} \frac{\partial}{\partial \theta}...
...frac{1}{r \sin\theta} \frac{\partial A_\theta}{\partial \phi}\right]{\bf e}_r$">|; 

$key = q/a_{ijkl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6946.png"
 ALT="$ a_{ijkl}$">|; 

$key = q/displaystylefrac{H_L}{h_1}=-left(frac{h_2}{h_1}-1right)+frac{1}{2},{rmFr}_1^{,2}left(1-frac{h_1^{,2}}{h_2^{,2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1504.png"
 ALT="$\displaystyle \frac{H_L}{h_1}=-\left(\frac{h_2}{h_1}-1\right) +\frac{1}{2} {\rm Fr}_1^{ 2}\left(1-\frac{h_1^{ 2}}{h_2^{ 2}}\right).$">|; 

$key = q/displaystyle{mitPsi}(x_1,x_2,x_3)=-frac{3}{4},G,Mleft(alpha_0-sum_{i=1,3}alpha_i,x_i^{,2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7293.png"
 ALT="$\displaystyle {\mit\Psi}(x_1,x_2,x_3) = - \frac{3}{4} G M\left(\alpha_0-\sum_{i=1,3}\alpha_i x_i^{ 2}\right),$">|; 

$key = q/127^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1137.png"
 ALT="$ 127^\circ$">|; 

$key = q/|z|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2132.png"
 ALT="$ \vert z\vert$">|; 

$key = q/displaystyle=a,omega,frac{sinh[k,(z+d)]}{sinh(k,d)},cos(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3971.png"
 ALT="$\displaystyle =a \omega \frac{\sinh[k (z+d)]}{\sinh(k d)} \cos(\omega t-k x).$">|; 

$key = q/h<2,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1598.png"
 ALT="$ h &lt; 2 a$">|; 

$key = q/|lambda|geq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2296.png"
 ALT="$ \vert\lambda\vert\geq 1$">|; 

$key = q/displaystylerho,g,z=-gamma,nablacdot{bfn},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1118.png"
 ALT="$\displaystyle \rho g z = -\gamma \nabla\cdot{\bf n},$">|; 

$key = q/displaystylenabla^{,2}psi=frac{1}{r},frac{partial}{partialr}left(r,frac{partialppsi}{partialtheta^{,2}}+frac{partial^{,2}psi}{partialz^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7133.png"
 ALT="$\displaystyle \nabla^{ 2} \psi = \frac{1}{r} \frac{\partial}{\partial r}\left...
...,2} \psi}{\partial\theta^{ 2}} + \frac{\partial^{ 2} \psi}{\partial z^{ 2}}.$">|; 

$key = q/displaystylefrac{h_2}{h_1}=frac{1}{2}left(-1+sqrt{1+8,{rmFr}_1^{,2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1483.png"
 ALT="$\displaystyle \frac{h_2}{h_1}=\frac{1}{2}\left(-1+\sqrt{1+8 {\rm Fr}_1^{ 2}}\right).$">|; 

$key = q/d{bfS}={bfn},dS;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="$ d{\bf S} = {\bf n} dS$">|; 

$key = q/{bfB}=W_0,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img595.png"
 ALT="$ {\bf B}=W_0 {\bf e}_z$">|; 

$key = q/displaystylefrac{d{rmMa}}{{rmMa}}=left[frac{1+(1slash2),(gamma-1),{rmMa}^{,2}}{1-{rmMa}^{,2}}right]frac{dA}{A}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5626.png"
 ALT="$\displaystyle \frac{d{\rm Ma}}{{\rm Ma}} = \left[\frac{1+(1/2) (\gamma-1) {\rm Ma}^{ 2}}{1-{\rm Ma}^{ 2}}\right]\frac{dA}{A}.
$">|; 

$key = q/bslashc=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$ b/c = 4$">|; 

$key = q/displaystyleH=frac{1}{2},v^{,2}+frac{p}{rho}+{mitPsi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img555.png"
 ALT="$\displaystyle H = \frac{1}{2} v^{ 2} + \frac{p}{\rho} + {\mit\Psi}.
$">|; 

$key = q/{bfb}times{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6409.png"
 ALT="$ {\bf b}\times {\bf c}$">|; 

$key = q/displaystylef(eta)=F(eta)+{mitLambda},G(eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3165.png"
 ALT="$\displaystyle f(\eta) = F(\eta)+{\mit\Lambda} G(\eta)$">|; 

$key = q/displaystylefrac{partialv}{partialt}+2,{mitOmega},costheta,u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4492.png"
 ALT="$\displaystyle \frac{\partial v}{\partial t} + 2 {\mit\Omega} \cos\theta u$">|; 

$key = q/hgga;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1145.png"
 ALT="$ h\gg a$">|; 

$key = q/displaystyleleft({array}{ccc}frac{1}{3}sigma_{ii},&0,&0[0.5ex]0,&frac{1}{3}sigma_{ii},&0[0.5ex]0,&0,&frac{1}{3}sigma_{ii}{array}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="98" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="$\displaystyle \left( \begin{array}{ccc} \frac{1}{3}\sigma_{ii},&amp; 0, &amp; 0\ [0.5e...
...{3}\sigma_{ii}, &amp; 0 \ [0.5ex] 0,&amp;0,&amp;\frac{1}{3}\sigma_{ii} \end{array}\right),$">|; 

$key = q/displaystyle=frac{dU}{dx}int_0^infty(U-u),dy+frac{d}{dx}int_0^inftyu,(U-u),dy.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3084.png"
 ALT="$\displaystyle = \frac{dU}{dx}\int_0^\infty (U-u) dy + \frac{d}{dx}\int_0^\infty u (U-u) dy.$">|; 

$key = q/k,{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3967.png"
 ALT="$ k {\bf e}_x$">|; 

$key = q/{bfa}=(1slash!sqrt{2},,1slash!sqrt{2},,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6355.png"
 ALT="$ {\bf a} = (1/\!\sqrt{2},  1/\!\sqrt{2},  0)$">|; 

$key = q/|theta|(z_2)=|theta_2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6141.png"
 ALT="$ \vert\theta\vert(z_2)=\vert\theta_2\vert$">|; 

$key = q/displaystyle2,a_{ijkl}+a_{kjil}+a_{ikjl}+a_{ijlk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6942.png"
 ALT="$\displaystyle 2 a_{ijkl} +a_{kjil} +a_{ikjl}+a_{ijlk}$">|; 

$key = q/zeta_bslasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4487.png"
 ALT="$ \zeta_b/a$">|; 

$key = q/varpi=coshxi,sineta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2793.png"
 ALT="$ \varpi=\cosh\xi \sin\eta$">|; 

$key = q/displaystylef''(0)=-{mitLambda}(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3157.png"
 ALT="$\displaystyle f''(0) = -{\mit\Lambda}(x),$">|; 

$key = q/displaystylecsimeq(g,d)^{1slash2}left[frac{tanh(k,d)}{k,d}right]^{1slash2}left{1-frac{1}{2},s,[1+tanh(k,d)]right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4117.png"
 ALT="$\displaystyle c\simeq (g d)^{1/2}\left[\frac{\tanh(k d)}{k d}\right]^{1/2}\left\{1-\frac{1}{2} s [1+\tanh(k d)]\right\}.$">|; 

$key = q/displaystyleG(x)=left(frac{1-gamma+2,gamma,x}{gamma+1}right)left[frac{2+(gamma-1),x}{(gamma+1),x}right]^{,gamma}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5495.png"
 ALT="$\displaystyle G(x) = \left(\frac{1-\gamma+2 \gamma x}{\gamma+1}\right)\left[\frac{2+(\gamma-1) x}{(\gamma+1) x}\right]^{ \gamma}.$">|; 

$key = q/H_1G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img693.png"
 ALT="$ H_1G$">|; 

$key = q/c,|{rmcosec},xi_0|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2466.png"
 ALT="$ c \vert{\rm cosec} \xi_0\vert$">|; 

$key = q/displaystylefrac{partialrho}{partialt}+nablacdot(rho,{bfv})=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="$\displaystyle \frac{\partial\rho}{\partial t} + \nabla\cdot(\rho {\bf v}) = 0$">|; 

$key = q/1ll{rmRe}<1.2times10^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3001.png"
 ALT="$ 1\ll {\rm Re}&lt;1.2\times 10^5$">|; 

$key = q/displaystyle=epsilon_{lmn}-deltatheta_aleft(epsilon_{imn},epsilon_{ial}+epsilon_{inl},epsilon_{iam}+epsilon_{ilm},epsilon_{ian}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6837.png"
 ALT="$\displaystyle = \epsilon_{lmn} -\delta\theta_a\left(\epsilon_{imn} \epsilon_{ial}+\epsilon_{inl} \epsilon_{iam} +\epsilon_{ilm} \epsilon_{ian}\right)$">|; 

$key = q/2,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2487.png"
 ALT="$ 2 c$">|; 

$key = q/delta>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4286.png"
 ALT="$ \delta&gt;0$">|; 

$key = q/displaystyle=-sum_{n=1,infty}^{n,{rmodd}}sum_{m=0,n}zeta_n^{,m},{mitPhi}_n^m(theta,phi),sin(n,pislash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5026.png"
 ALT="$\displaystyle = -\sum_{n=1,\infty}^{n {\rm odd}}\sum_{m=0,n}\zeta_n^{ m} {\mit\Phi}_n^m(\theta,\phi) \sin(n \pi/2),$">|; 

$key = q/displaystyleoverline{y}equivfrac{1}{c}int_{x_L}^{x_R}y(x),dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5850.png"
 ALT="$\displaystyle \overline{y} \equiv \frac{1}{c}\int_{x_L}^{x_R} y(x) dx,$">|; 

$key = q/B_2=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5970.png"
 ALT="$ B_2=0$">|; 

$key = q/v=1slashrho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5289.png"
 ALT="$ v=1/\rho$">|; 

$key = q/displaystylec_0=sqrt{frac{gamma,p_0}{rho_0}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5357.png"
 ALT="$\displaystyle c_0 =\sqrt{\frac{\gamma p_0}{\rho_0}}.$">|; 

$key = q/6.5times10^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3619.png"
 ALT="$ 6.5\times 10^3$">|; 

$key = q/displaystylegamma=-frac{{mitGamma}}{2pi,a,V_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1769.png"
 ALT="$\displaystyle \gamma = -\frac{{\mit\Gamma}}{2\pi a V_0},$">|; 

$key = q/displaystyleA,r^{,2}+B,r+C=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7241.png"
 ALT="$\displaystyle A r^{ 2}+ B r+C = 0,$">|; 

$key = q/displaystyle=frac{1}{2},(3,x^{,2}-1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4266.png"
 ALT="$\displaystyle = \frac{1}{2} (3 x^{ 2}-1),$">|; 

$key = q/displaystyle=rho_0left(1-frac{gamma-1}{gamma},frac{z}{H}right)^{1slash(gamma-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5153.png"
 ALT="$\displaystyle =\rho_0\left(1-\frac{\gamma-1}{\gamma} \frac{z}{H}\right)^{1/(\gamma-1)}.$">|; 

$key = q/psi(xi,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2085.png"
 ALT="$ \psi(\xi,\eta)$">|; 

$key = q/y=-h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3037.png"
 ALT="$ y=-h$">|; 

$key = q/v_z(y=0_-);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3379.png"
 ALT="$ v_z(y=0_-)$">|; 

$key = q/displaystylev_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img536.png"
 ALT="$\displaystyle v_y$">|; 

$key = q/P=(varpi,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2836.png"
 ALT="$ P = (\varpi, z)$">|; 

$key = q/displaystylenablacdot(nablaT);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6612.png"
 ALT="$\displaystyle \nabla\cdot(\nabla T)$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{AppendixDslashfig5.06.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="493" HEIGHT="250" BORDER="0"
 SRC="|."$dir".q|img7237.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{AppendixD/fig5.06.eps}}
\end{figure}">|; 

$key = q/displaystyley';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2328.png"
 ALT="$\displaystyle y'$">|; 

$key = q/f'(eta)>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2965.png"
 ALT="$ f'(\eta)&gt;0$">|; 

$key = q/displaystylefrac{h}{a}=2,costheta,frac{l^{,2}}{a^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1143.png"
 ALT="$\displaystyle \frac{h}{a} = 2 \cos\theta \frac{l^{ 2}}{a^{ 2}}.$">|; 

$key = q/displaystyletau_2=-2pi,r_2^{,2},sigma_{rtheta}(r_2)=4pi,muleft(frac{{mitOmega}_1-{mitOmega}_2}{r_1^{,-2}-r_2^{,-2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3643.png"
 ALT="$\displaystyle \tau_2 = -2\pi r_2^{ 2} \sigma_{r\theta}(r_2) = 4\pi \mu\left(\frac{{\mit\Omega}_1-{\mit\Omega}_2}{r_1^{ -2}-r_2^{ -2}}\right).$">|; 

$key = q/nabla^{,2}p_n=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4394.png"
 ALT="$ \nabla^{ 2} p_n=0$">|; 

$key = q/gamma=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$ \gamma =0$">|; 

$key = q/displaystyle=omega_1,t+2,e,sin[(omega_1-omega_2),t]+{calO}(e^{,2})+{calO}(sin^2delta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4302.png"
 ALT="$\displaystyle = \omega_1 t + 2 e \sin[(\omega_1-\omega_2) t] +{\cal O}(e^{ 2})+{\cal O}(\sin^2\delta).$">|; 

$key = q/C(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1592.png"
 ALT="$ C(t)$">|; 

$key = q/displaystyleK=-pi,rholeft.int_{eta=0}^{eta=pi}phi,dpsiright|_{xi=xi_0}=frac{2}{3{1slash2},sin^{-1}e}{sin^{-1}e-e,(1-e^{,2})^{1slash2}}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="544" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2780.png"
 ALT="$\displaystyle K = -\pi \rho\left.\int_{\eta=0}^{\eta=\pi} \phi d\psi\right\ve...
...rac{e-(1-e^{ 2})^{1/2} \sin^{-1} e}{\sin^{-1} e-e (1-e^{ 2})^{1/2}}\right].$">|; 

$key = q/rho,S_2,v_2,{bfv}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1390.png"
 ALT="$ \rho S_2 v_2 {\bf v}_2$">|; 

$key = q/a_{ip},b_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6777.png"
 ALT="$ a_{ip} b_p$">|; 

$key = q/displaystyle=frac{3}{2},X_1,sectheta,tantheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4222.png"
 ALT="$\displaystyle = \frac{3}{2} X_1 \sec\theta \tan\theta.$">|; 

$key = q/nabla^{,2}{bfA}=(nabla^{,2}A_x,,nabla^{,2}A_y,,nabla^{,2}A_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6672.png"
 ALT="$ \nabla^{ 2}{\bf A} = (\nabla^{ 2} A_x,  \nabla^{ 2} A_y, \nabla^{ 2} A_z)$">|; 

$key = q/-w(z),{bfe}_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3454.png"
 ALT="$ -w(z) {\bf e}_\perp$">|; 

$key = q/displaystyle=g(x+lambda,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6050.png"
 ALT="$\displaystyle =g(x+\lambda y)$">|; 

$key = q/displaystylefrac{x^{,2}}{a^{,2}}+frac{y^{,2}}{b^{,2}}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3878.png"
 ALT="$\displaystyle \frac{x^{ 2}}{a^{ 2}}+ \frac{y^{ 2}}{b^{ 2}} =1.
$">|; 

$key = q/displaystyle=-frac{G,m',a^{,2}}{r'^{,3}}left[P_2^{,0}(costheta'),{calR}_2^{,(0)}(r,theta)right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4280.png"
 ALT="$\displaystyle = - \frac{G m' a^{ 2}}{r'^{ 3}}\left[P_2^{ 0}(\cos\theta') {\cal R}_2^{ (0)}(r,\theta)\right.$">|; 

$key = q/j=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$ j=4$">|; 

$key = q/displaystyledelta_0=left[frac{int_{A_0}delta^{,2}(x,y),dx,dy}{A_0}right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img634.png"
 ALT="$\displaystyle \delta_0 = \left[\frac{\int_{A_0}\delta^{ 2}(x,y) dx dy}{A_0}\right]^{1/2}.$">|; 

$key = q/displaystyleJ=8int_0^{pislash2}sintheta,dthetaint_0^{pislash2}left(frac{sin^2thein^2phi}{a_2^{,2}}+frac{cos^2theta}{a_3^{,2}}right)^{-1}dphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="570" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7258.png"
 ALT="$\displaystyle J= 8\int_0^{\pi/2}\sin\theta d\theta\int_0^{\pi/2}\left(\frac{\s...
...theta \sin^2\phi}{a_2^{ 2}}+ \frac{\cos^2\theta}{a_3^{ 2}}\right)^{-1}d\phi.$">|; 

$key = q/(x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1980.png"
 ALT="$ (x_0$">|; 

$key = q/e=0.81267;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1075.png"
 ALT="$ e=0.81267$">|; 

$key = q/f(eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2984.png"
 ALT="$ f(\eta)$">|; 

$key = q/CEDF;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2845.png"
 ALT="$ CEDF$">|; 

$key = q/displaystyle=-g,frac{partial(zeta-skew{5}bar{zeta}')}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5064.png"
 ALT="$\displaystyle = -g \frac{\partial(\zeta-\skew{5}\bar{\zeta}')}{\partial x},$">|; 

$key = q/rho_2>rho(z_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5164.png"
 ALT="$ \rho_2&gt;\rho(z_2)$">|; 

$key = q/{bfk}=k,{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3915.png"
 ALT="$ {\bf k}=k {\bf e}_x$">|; 

$key = q/CC';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img972.png"
 ALT="$ CC'$">|; 

$key = q/displaystylev_perp=frac{dpsi}{ds}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1635.png"
 ALT="$\displaystyle v_\perp = \frac{d\psi}{ds}.$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi}=C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1382.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} + {\mit\Psi} = C$">|; 

$key = q/varphi=tan^{-1}(yslashx);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2526.png"
 ALT="$ \varphi= \tan^{-1}(y/x)$">|; 

$key = q/{bfa}ast{bfb}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6354.png"
 ALT="$ {\bf a} \ast {\bf b}= {\bf0}$">|; 

$key = q/MslashM'=0.01230;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1027.png"
 ALT="$ M/M'=0.01230$">|; 

$key = q/displaystyle=-frac{1}{a},frac{partial}{partialtheta}left(frac{p}{skew{3}bar{rho}}+{mitPhi}+chiright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4462.png"
 ALT="$\displaystyle =- \frac{1}{a} \frac{\partial}{\partial\theta}\left(\frac{p}{\skew{3}\bar{\rho}} + {\mit\Phi} + \chi\right),$">|; 

$key = q/displaystyle={bfA}cdotnablaA_phi+frac{A_r,A_phi+cottheta,A_theta,A_phi}{r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7188.png"
 ALT="$\displaystyle = {\bf A}\cdot\nabla A_\phi+ \frac{A_r A_\phi+\cot\theta A_\theta A_\phi}{r}.$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{Chapter04slashweir.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="532" HEIGHT="165" BORDER="0"
 SRC="|."$dir".q|img1519.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{Chapter04/weir.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{dlnp}{dlnrho}<gamma,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5169.png"
 ALT="$\displaystyle \frac{d\ln p}{d\ln\rho} &lt; \gamma,$">|; 

$key = q/sigma_{11}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$ \sigma_{11}'$">|; 

$key = q/{mitGamma}_C=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1795.png"
 ALT="$ {\mit\Gamma}_C=0$">|; 

$key = q/DslashDt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$ D/Dt$">|; 

$key = q/theta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2828.png"
 ALT="$ \theta'$">|; 

$key = q/displaystylefrac{2,A,gamma,costheta}{d}+L,gamma,sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1341.png"
 ALT="$\displaystyle \frac{2 A \gamma \cos\theta}{d} + L \gamma \sin\theta,
$">|; 

$key = q/displaystyle=frac{v_x}{U_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2880.png"
 ALT="$\displaystyle = \frac{v_x}{U_0},$">|; 

$key = q/beta_w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6100.png"
 ALT="$ \beta_w$">|; 

$key = q/displaystyle=-frac{partialphi}{partialy}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1653.png"
 ALT="$\displaystyle =-\frac{\partial\phi}{\partial y}.$">|; 

$key = q/displaystyle{bfg}_{,rmtide}({bfr})={bfg}_{,rmmoon}({bfr})-{bfg}_{,rmcentripetal}=-nabla{mitPhi}_{rmtide},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4247.png"
 ALT="$\displaystyle {\bf g}_{ \rm tide}({\bf r}) = {\bf g}_{ \rm moon}({\bf r})-{\bf g}_{ \rm centripetal} = - \nabla{\mit\Phi}_{\rm tide},$">|; 

$key = q/';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5961.png"
 ALT="$ '$">|; 

$key = q/displaystyle=int_lambda^inftyfrac{du}{mitDelta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7294.png"
 ALT="$\displaystyle =\int_\lambda^\infty \frac{du}{\mit\Delta},$">|; 

$key = q/K_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$ K_1$">|; 

$key = q/m'slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2682.png"
 ALT="$ m'/2$">|; 

$key = q/{mitDelta}pslashp_1ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5508.png"
 ALT="$ {\mit\Delta}p/p_1\ll 1$">|; 

$key = q/C_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3719.png"
 ALT="$ C_f$">|; 

$key = q/{calC}>1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1422.png"
 ALT="$ {\cal C} &gt; 1/2$">|; 

$key = q/displaystyle=frac{partialA_z}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7152.png"
 ALT="$\displaystyle = \frac{\partial A_z}{\partial z},$">|; 

$key = q/displaystylesigma_{ij}=-p,delta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img575.png"
 ALT="$\displaystyle \sigma_{ij} = - p \delta_{ij}$">|; 

$key = q/displaystyle=-sum_{n=1,infty}frac{z_n}{n,(n+1)},P_n^{,m_j}(mu),{rme}^{,{rmi},(m_j,phi+sigma_j,t)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4676.png"
 ALT="$\displaystyle =- \sum_{n=1,\infty}\frac{z_n}{n (n+1)} P_n^{ m_j}(\mu) {\rm e}^{ {\rm i} (m_j \phi+\sigma_j t)},$">|; 

$key = q/displaystylefrac{Dv_i}{Dt}=frac{F_i}{rho}+frac{1}{rho}frac{partialsigma_{ij}}{partialx_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="$\displaystyle \frac{D v_i}{Dt} = \frac{F_i}{\rho} + \frac{1}{\rho}\frac{\partial\sigma_{ij}}{\partial x_j}.$">|; 

$key = q/displaystyle=r,frac{partial{calR}_n}{partialr}=n,{calR}_n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4331.png"
 ALT="$\displaystyle = r \frac{\partial {\cal R}_n}{\partial r} = n {\cal R}_n,$">|; 

$key = q/displaystylefrac{partial}{partialz}left(frac{p}{skew{3}bar{rho}}+{mitPhi}+chiright)simeq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4483.png"
 ALT="$\displaystyle \frac{\partial}{\partial z}\left(\frac{p}{\skew{3}\bar{\rho}}+{\mit\Phi}+\chi\right)\simeq 0.$">|; 

$key = q/phi_P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1577.png"
 ALT="$ \phi_P$">|; 

$key = q/Q(theta,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4734.png"
 ALT="$ Q(\theta,\phi,t)$">|; 

$key = q/theta=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5209.png"
 ALT="$ \theta=1$">|; 

$key = q/lambda=lambda_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4166.png"
 ALT="$ \lambda=\lambda_c$">|; 

$key = q/u_2(x_1,x_2,x_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7027.png"
 ALT="$ u_2(x_1, x_2, x_3)$">|; 

$key = q/T({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="$ T({\bf r},t)$">|; 

$key = q/displaystyleU(x)=U_0,x^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2939.png"
 ALT="$\displaystyle U(x) = U_0 x^{ m}$">|; 

$key = q/P'Q';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img733.png"
 ALT="$ P'Q'$">|; 

$key = q/displaystyleU=-frac{3}{10},G,M^{,2},alpha_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img869.png"
 ALT="$\displaystyle U =-\frac{3}{10} G M^{ 2} \alpha_0.$">|; 

$key = q/displaystylepsi(xi,eta)=-frac{1}{2},V,a^{,2},e^{,2}left[frac{sinhxi-cosh^2xi,tanlashsinhxi)}{e,(1-e^{,2})^{1slash2}-sin^{-1}e}right]sin^2eta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="512" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2775.png"
 ALT="$\displaystyle \psi(\xi,\eta) =-\frac{1}{2} V a^{ 2} e^{ 2}\left[\frac{\sin...
...\xi \tan^{-1}(1/\sinh\xi)} {e (1-e^{ 2})^{1/2}-\sin^{-1}e}\right]\sin^2\eta.$">|; 

$key = q/displaystylec_f=frac{left.sigma_{xy}right|_{y=0}}{(1slash2),rho,U^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2947.png"
 ALT="$\displaystyle c_f = \frac{\left.\sigma_{xy}\right\vert _{y=0}}{(1/2) \rho U^{ 2}}.$">|; 

$key = q/displaystylefrac{T_3}{T_4}=left(frac{p_2slashp_1}{p_4slashp_1}right)^{,(gamma_4-1)slashgamma_4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5651.png"
 ALT="$\displaystyle \frac{T_3}{T_4} = \left(\frac{p_2/p_1}{p_4/p_1}\right)^{ (\gamma_4-1)/\gamma_4}.
$">|; 

$key = q/displaystyle=-frac{partialpsi}{partialr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1758.png"
 ALT="$\displaystyle =-\frac{\partial\psi}{\partial r}.$">|; 

$key = q/psi=psi(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2609.png"
 ALT="$ \psi=\psi(\theta)$">|; 

$key = q/displaystylecoshxi,frac{dF}{dxi}-2,sinhxi,F=B,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2762.png"
 ALT="$\displaystyle \cosh\xi \frac{dF}{d\xi}-2 \sinh\xi F = B,$">|; 

$key = q/displaystyle=a_{ssij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6956.png"
 ALT="$\displaystyle =a_{ssij},$">|; 

$key = q/{bfe}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7049.png"
 ALT="$ {\bf e}_2$">|; 

$key = q/displaystyleF(z)=-frac{Q}{2pi},lnz-V_0,z,{rme}^{-{rmi},theta_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2425.png"
 ALT="$\displaystyle F(z)=-\frac{Q}{2\pi} \ln z - V_0 z {\rm e}^{-{\rm i} \theta_0}.$">|; 

$key = q/displaystylefrac{D}{L}=frac{2}{A},alphaleft(1+sum_{nu=3,5,7,cdots}frac{nu,c_nu^{,2}}{c_1^{,2}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3551.png"
 ALT="$\displaystyle \frac{D}{L} = \frac{2}{A} \alpha\left(1+\sum_{\nu=3,5,7,\cdots}\frac{\nu c_\nu^{ 2}}{c_1^{ 2}}\right),
$">|; 

$key = q/P(theta,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4733.png"
 ALT="$ P(\theta,\phi,t)$">|; 

$key = q/displaystyledelta_{ik},delta_{kj};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6795.png"
 ALT="$\displaystyle \delta_{ik} \delta_{kj}$">|; 

$key = q/displaystylefrac{partial}{partialt}!left[rholeft({calE}+frac{1}{2},v_i,v_iright)j}!left(v_i,sigma_{ij}+kappa,frac{partialT}{partialx_j}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="636" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="$\displaystyle \frac{\partial}{\partial t}\!\left[\rho\left({\cal E}+ \frac{1}{2...
...l x_j}\!\left(v_i \sigma_{ij} + \kappa \frac{\partial T}{\partial x_j}\right)$">|; 

$key = q/nablacdot(rho,{bfv})={bfv}cdotnablarho+rho,nablacdot{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="$ \nabla \cdot(\rho {\bf v}) = {\bf v}\cdot\nabla \rho + \rho \nabla \cdot{\bf v}$">|; 

$key = q/P'ABQ';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img737.png"
 ALT="$ P'ABQ'$">|; 

$key = q/lambdagglambda_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4156.png"
 ALT="$ \lambda \gg \lambda_c$">|; 

$key = q/displaystylepsi(r,theta)=-frac{1}{2},V,r^{,2},sin^2theta+frac{Q}{4pi},costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2815.png"
 ALT="$\displaystyle \psi(r,\theta) = -\frac{1}{2} V r^{ 2} \sin^2\theta + \frac{Q}{4\pi} \cos\theta,$">|; 

$key = q/displaystyle=alphaleft[E(phi,k)-(1-k^{,2})^{1slash2},F(phi,k)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1328.png"
 ALT="$\displaystyle = \alpha\left[E(\phi,k) -(1-k^{ 2})^{1/2} F(\phi,k)\right].$">|; 

$key = q/C_fsim{calO}(d_0slashl)ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3727.png"
 ALT="$ C_f\sim {\cal O}(d_0/l)\ll 1$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashimage2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="337" BORDER="0"
 SRC="|."$dir".q|img2031.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/image2.eps}}
\end{figure}">|; 

$key = q/displaystyle=-left(1+frac{1}{{rmFr}^{,2}}+frac{1}{{rmRe}}right)frac{overline{nab}{3},overline{nabla}(overline{nabla}cdotoverline{bfv})right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="515" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="$\displaystyle = - \left(1+\frac{1}{{\rm Fr}^{ 2}}+\frac{1}{{\rm Re}}\right)\fr...
...-\frac{1}{3} \overline{\nabla}(\overline{\nabla}\cdot\overline{\bf v})\right],$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashcylinder2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1787.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/cylinder2.eps}}
\end{figure}">|; 

$key = q/V_0ll10^{-1},{rmm,s^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="$ V_0\ll 10^{-1} {\rm m s^{-1}}$">|; 

$key = q/displaystyleK=frac{1}{4},(rho,g+gamma,k^{,2}),a^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4212.png"
 ALT="$\displaystyle K = \frac{1}{4} (\rho g+\gamma k^{ 2}) a^{ 2},
$">|; 

$key = q/displaystyle(n+m),{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6248.png"
 ALT="$\displaystyle (n+m) {\bf a}$">|; 

$key = q/displaystyleI=frac{W}{g,V}int_{-aslash2}^{aslash2}int_{-bslash2}^{bslash2}int_{-y,dz=frac{W}{12,g}left(b^{,2}+4,[(1-s)^3+s^{,3}],c^{,2}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="630" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img684.png"
 ALT="$\displaystyle I = \frac{W}{g V}\int_{-a/2}^{a/2}\int_{-b/2}^{b/2}\int_{-s c}^...
...,dx dy dz= \frac{W}{12 g}\left(b^{ 2}+ 4 [(1-s)^3+s^{ 3}] c^{ 2}\right)$">|; 

$key = q/I_{n,,n'}^{,m,,m'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4967.png"
 ALT="$ I_{n, n'}^{ m, m'}=0$">|; 

$key = q/displaystyleh,sinalpha=h_1-h_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1889.png"
 ALT="$\displaystyle h \sin\alpha = h_1-h_2.$">|; 

$key = q/displaystylevsimfrac{delta}{x},u.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3019.png"
 ALT="$\displaystyle v\sim \frac{\delta}{x} u.$">|; 

$key = q/C'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img969.png"
 ALT="$ C''$">|; 

$key = q/dV=4pi,r^{,2},dr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5270.png"
 ALT="$ dV = 4\pi r^{ 2} dr$">|; 

$key = q/displaystyle=int_Vleft[v_i,F_i+frac{partial}{partialx_j}!left(v_i,sigma_{ij}+kappa,frac{partialT}{partialx_j}right)right]dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="$\displaystyle = \int_V\left[v_i F_i + \frac{\partial}{\partial x_j}\!\left(v_i \sigma_{ij} + \kappa \frac{\partial T}{\partial x_j}\right)\right]dV.$">|; 

$key = q/displaystylemu_rint_{mitOmega}{mitPsi}',{mitPsi}_r,d{mitOmega}=-sum_{s=1,infty}left(beta_{-r,,s},p_s+beta_{-r,,-s},p_{-s}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4909.png"
 ALT="$\displaystyle \mu_r\int_{\mit\Omega}{\mit\Psi}' {\mit\Psi}_r d{\mit\Omega} = -\sum_{s=1,\infty} \left(\beta_{-r, s} p_s + \beta_{-r, -s} p_{-s}\right).$">|; 

$key = q/displaystyled_{12}=d_{21}=mu,frac{partialv_1}{partialx_2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="$\displaystyle d_{12} = d_{21} = \mu \frac{\partial v_1}{\partial x_2}.$">|; 

$key = q/displaystylenuleft.frac{partialu}{partialy}right|_{y=0}=U^{,2},frac{ddelta_2}{dx}+U,frac{dU}{dx},(delta_1+2,delta_2)+U,v_s.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3261.png"
 ALT="$\displaystyle \nu\left.\frac{\partial u}{\partial y}\right\vert _{y=0} = U^{ 2} \frac{d\delta_2}{dx} + U \frac{dU}{dx} (\delta_1+2 \delta_2) + U v_s.
$">|; 

$key = q/displaystyle=frac{G(z)}{4,mu}left[a^{,2}(z)-r^{,2}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3678.png"
 ALT="$\displaystyle = \frac{G(z)}{4 \mu}\left[a^{ 2}(z)-r^{ 2}\right],$">|; 

$key = q/w=v_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4458.png"
 ALT="$ w=v_r$">|; 

$key = q/displaystylefrac{p}{p_0}=exp!left(-frac{z}{H}-frac{1}{2},frac{beta,z^{,2}}{H}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5262.png"
 ALT="$\displaystyle \frac{p}{p_0} = \exp\!\left(-\frac{z}{H}-\frac{1}{2} \frac{\beta z^{ 2}}{H}\right).
$">|; 

$key = q/displaystyleI_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7281.png"
 ALT="$\displaystyle I_2$">|; 

$key = q/displaystyle-f_j^{,2},p_{-r}+{rmi}sum_{s=1,infty}(f_j,beta_{-r,,s},p_s+f_j,beta_{-r,,-s},p_{-s})=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4928.png"
 ALT="$\displaystyle -f_j^{ 2} p_{-r} + {\rm i}\sum_{s=1,\infty}(f_j \beta_{-r, s} p_s + f_j \beta_{-r, -s} p_{-s}) =0,$">|; 

$key = q/{mitOmega}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3623.png"
 ALT="$ {\mit\Omega}_2$">|; 

$key = q/t=tanphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7259.png"
 ALT="$ t=\tan\phi$">|; 

$key = q/{bfT};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3518.png"
 ALT="$ {\bf T}$">|; 

$key = q/displaystyle=omega_1,t+2,e,sin[(omega_1-omega_2),t]+{calO}(e^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4294.png"
 ALT="$\displaystyle = \omega_1 t + 2 e \sin[(\omega_1-\omega_2) t] + {\cal O}(e^{ 2}),$">|; 

$key = q/displaystyle[({bfA}cdotnabla),{bfA}]_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7139.png"
 ALT="$\displaystyle [({\bf A}\cdot\nabla) {\bf A}]_z$">|; 

$key = q/displaystyleU=frac{1}{2},int{mitPsi},rho,dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7273.png"
 ALT="$\displaystyle U = \frac{1}{2} \int{\mit\Psi} \rho dV,$">|; 

$key = q/u(x_0,y)=u_0(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3111.png"
 ALT="$ u(x_0,y)=u_0(y)$">|; 

$key = q/delta_{jk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6921.png"
 ALT="$ \delta_{jk}$">|; 

$key = q/displaystylezeta(theta,phi)=zeta_{t}(theta,phi)-zeta_{b}(theta,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4357.png"
 ALT="$\displaystyle \zeta(\theta,\phi)= \zeta_{t}(\theta,\phi)-\zeta_{b}(\theta,\phi),$">|; 

$key = q/displaystyle=frac{partialA_r}{partialr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7148.png"
 ALT="$\displaystyle = \frac{\partial A_r}{\partial r},$">|; 

$key = q/(X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4218.png"
 ALT="$ (X$">|; 

$key = q/I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7308.png"
 ALT="$ I$">|; 

$key = q/displaystylev_p=v_g=(g,d)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3954.png"
 ALT="$\displaystyle v_p = v_g = (g d)^{1/2},$">|; 

$key = q/displaystyle={calR}_{ik},{calR}_{jl},b_{kl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6851.png"
 ALT="$\displaystyle ={\cal R}_{ik} {\cal R}_{jl} b_{kl},$">|; 

$key = q/{rmRe}gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="$ {\rm Re}\gg 1$">|; 

$key = q/displaystyle({bfa}times{bfb})_i=epsilon_{ijk},a_j,b_k.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6786.png"
 ALT="$\displaystyle ({\bf a}\times {\bf b})_i = \epsilon_{ijk} a_j b_k.$">|; 

$key = q/displaystylenabla^{,2}psi=frac{1}{r^{,2}},frac{partial}{partialr}left(r^{,2},fra1}{r^{,2},sin^2theta},frac{partial^{,2}psi}{partialphi^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="551" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7183.png"
 ALT="$\displaystyle \nabla^{ 2} \psi= \frac{1}{r^{ 2}} \frac{\partial}{\partial r}...
...rac{1}{r^{ 2} \sin^2\theta} \frac{\partial^{ 2} \psi}{\partial \phi^{ 2}}.$">|; 

$key = q/displaystylecostheta,frac{partial{mitPhi}_r}{partialtheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4848.png"
 ALT="$\displaystyle \cos\theta \frac{\partial{\mit\Phi}_r}{\partial\theta}$">|; 

$key = q/{rmMa}_1>[4slash(3-gamma)]^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6173.png"
 ALT="$ {\rm Ma}_1&gt; [4/(3-\gamma)]^{1/2}$">|; 

$key = q/displaystylesimfrac{delta{mitPhi}}{a,{mitOmega}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4498.png"
 ALT="$\displaystyle \sim \frac{\delta{\mit\Phi}}{a {\mit\Omega}},$">|; 

$key = q/displaystylefrac{T}{T_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5401.png"
 ALT="$\displaystyle \frac{T}{T_1}$">|; 

$key = q/epsilon_{mit};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6909.png"
 ALT="$ \epsilon_{mit}$">|; 

$key = q/displaystylefrac{d}{dxi}left(coshxi,frac{dF}{dxi}-2,sinhxi,Fright)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2761.png"
 ALT="$\displaystyle \frac{d}{d\xi}\left(\cosh\xi \frac{dF}{d\xi}-2 \sinh\xi F\right)=0.$">|; 

$key = q/S,p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1413.png"
 ALT="$ S p$">|; 

$key = q/p(l)=p_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3703.png"
 ALT="$ p(l)=p_0$">|; 

$key = q/displaystyle=mu,sinthetaleft(frac{6,A}{r^{,4}}+6,D,rright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3838.png"
 ALT="$\displaystyle = \mu \sin\theta\left(\frac{6 A}{r^{ 4}} + 6 D r\right),$">|; 

$key = q/displaystylealpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5596.png"
 ALT="$\displaystyle \alpha$">|; 

$key = q/displaystylepsi(r,theta)=-frac{Q}{2pi},tan^{-1}left[frac{(r-b,cslashr),sintheta}{(r+b,cslashr),costheta-(b+c)}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2044.png"
 ALT="$\displaystyle \psi(r,\theta) = -\frac{Q}{2\pi} \tan^{-1}\left[\frac{(r-b c/r) \sin\theta}{(r+b c/r) \cos\theta-(b+c)}\right].$">|; 

$key = q/displaystyleomega=left(frac{2,gamma}{rho,a,c^{,2}}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1350.png"
 ALT="$\displaystyle \omega = \left(\frac{2 \gamma}{\rho a c^{ 2}}\right)^{1/2},
$">|; 

$key = q/displaystyle=-frac{G,m'}{r'},frac{r}{r'},P_1(cosgamma),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4235.png"
 ALT="$\displaystyle = - \frac{G m'}{r'} \frac{r}{r'} P_1(\cos\gamma),$">|; 

$key = q/{multline}sum_{n'=1,infty}sum_{m'=0,n'}left(left[left(beta^{,-1},[lambda_n^{,m}-_{2,,n}^{,m_j,,m}+skew{3}hat{kappa}_{2,,n}^{,m_j,,m}right),{multline};LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="612" HEIGHT="165" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4994.png"
 ALT="\begin{multline}
\sum_{n'=1,\infty}\sum_{m'=0,n'}\left(\left[\left(\beta^{ -1}...
...n}^{ m_j, m}+ \skew{3}\hat{\kappa}_{2, n}^{ m_j, m}\right),
\end{multline}">|; 

$key = q/displaystylezeta_s(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5025.png"
 ALT="$\displaystyle \zeta_s(\theta,\phi)$">|; 

$key = q/displaystyleskew{3}hat{epsilon}_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5006.png"
 ALT="$\displaystyle \skew{3}\hat{\epsilon}_{n, n'}^{ m, m'}$">|; 

$key = q/theta_1ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6169.png"
 ALT="$ \theta_1\ll 1$">|; 

$key = q/partialslashpartialz=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1792.png"
 ALT="$ \partial/\partial z = 0$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{AppendixAslashfigA.11.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="342" BORDER="0"
 SRC="|."$dir".q|img6391.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{AppendixA/figA.11.eps}}
\end{figure}">|; 

$key = q/displaystyle{bfr}=alpha,{bfa}+beta,{bfb}+gamma,{bfc}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6415.png"
 ALT="$\displaystyle {\bf r} = \alpha  {\bf a} + \beta {\bf b} + \gamma  {\bf c}.$">|; 

$key = q/displaystyle=frac{1}{r^{,2}},frac{d}{dr}left(r^{,3},frac{d{mitOmega}}{dr}right),{bfe}_theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3634.png"
 ALT="$\displaystyle =\frac{1}{r^{ 2}} \frac{d}{dr}\left(r^{ 3} \frac{d{\mit\Omega}}{dr}\right) {\bf e}_\theta.$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter14slashpiston1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="341" BORDER="0"
 SRC="|."$dir".q|img5550.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter14/piston1.eps}}
\end{figure}">|; 

$key = q/y=a,(thetaslashpi-1slash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2094.png"
 ALT="$ y=a (\theta/\pi-1/2)$">|; 

$key = q/|zeta|slashskew{5}tilde{zeta}_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5043.png"
 ALT="$ \vert\zeta\vert/\skew{5}\tilde{\zeta}_3$">|; 

$key = q/displaystylefrac{p}{p_infty}=left[1+frac{1}{2},(gamma-1),{rmMa}_infty^{,2}left(1-frac{q^{,2}}{U^{,2}}right)right]^{,gammaslash(gamma-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="382" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5929.png"
 ALT="$\displaystyle \frac{p}{p_\infty} = \left[1+\frac{1}{2} (\gamma-1) {\rm Ma}_\infty^{ 2}\left(1-\frac{q^{ 2}}{U^{ 2}}\right)\right]^{ \gamma/(\gamma-1)}.$">|; 

$key = q/M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="$ M$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi}=C_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1379.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} + {\mit\Psi} = C_2,$">|; 

$key = q/2,omega_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4553.png"
 ALT="$ 2 \omega_M$">|; 

$key = q/{bfe}_2=h_2,nablau_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7034.png"
 ALT="$ {\bf e}_2=h_2 \nabla u_2$">|; 

$key = q/{bfb}cdot(nabla{bfg})cdot{bfn}equivb_i,(partialg_jslashpartialx_i),n_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1100.png"
 ALT="$ {\bf b} \cdot(\nabla {\bf g})\cdot{\bf n} \equiv b_i (\partial g_j/\partial x_i) n_j$">|; 

$key = q/pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$ \pi$">|; 

$key = q/displaystylep_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5188.png"
 ALT="$\displaystyle p_g$">|; 

$key = q/displaystyleint_P^Q{bfA}cdotd{bfr}=V(Q),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6553.png"
 ALT="$\displaystyle \int_P^Q {\bf A}\cdot d{\bf r} = V(Q),$">|; 

$key = q/rhoslashrho_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$ \rho /\rho _1$">|; 

$key = q/r_H>r_G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img830.png"
 ALT="$ r_H&gt;r_G$">|; 

$key = q/omega_1-omega_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4528.png"
 ALT="$ \omega_1-\omega_2$">|; 

$key = q/displaystylefrac{f,f_z}{a}=frac{d}{dz}!left[frac{f}{(1+f_z^{,2})^{1slash2}}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1236.png"
 ALT="$\displaystyle \frac{f f_z}{a} = \frac{d}{dz}\!\left[\frac{f}{(1+f_z^{ 2})^{1/2}}\right],$">|; 

$key = q/displaystyleint_{mitOmega}{mitPhi}_s,{mitPhi}_r,d{mitOmega}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4814.png"
 ALT="$\displaystyle \int_{\mit\Omega} {\mit\Phi}_s {\mit\Phi}_r d{\mit\Omega} = 0$">|; 

$key = q/j=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6808.png"
 ALT="$ j=l$">|; 

$key = q/displaystyleF_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2656.png"
 ALT="$\displaystyle F_z$">|; 

$key = q/displaystyle{bfv}({bfr})=frac{1}{4pi}intfrac{mbox{boldmathOmega}({bfr}')times({bfr}-{bfr}')}{|{bfr}-{bfr}'|^{,3}},dS',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3393.png"
 ALT="$\displaystyle {\bf v}({\bf r}) = \frac{1}{4\pi}\int\frac{\mbox{\boldmath$\Omega$}({\bf r}')\times ({\bf r}-{\bf r}')}{\vert{\bf r}-{\bf r}'\vert^{ 3}} dS',$">|; 

$key = q/nablatimes{bfA}cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6663.png"
 ALT="$ \nabla\times{\bf A}
\cdot d{\bf S}$">|; 

$key = q/displaystyleoint_C{bfB}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2358.png"
 ALT="$\displaystyle \oint_C {\bf B}\cdot d{\bf r}$">|; 

$key = q/phi=phi_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4723.png"
 ALT="$ \phi=\phi_-$">|; 

$key = q/n=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3775.png"
 ALT="$ n=-1$">|; 

$key = q/displaystyle=frac{partialxi}{partialy},frac{partial}{partialxi}+frac{partialeta}tialy},frac{partial}{partialeta}=m,frac{partial}{partialeta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6016.png"
 ALT="$\displaystyle = \frac{\partial\xi}{\partial y} \frac{\partial}{\partial\xi} +\...
...}{\partial y} \frac{\partial}{\partial\eta} =m \frac{\partial}{\partial\eta}.$">|; 

$key = q/displaystyleleft.wright|_{z=d};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4472.png"
 ALT="$\displaystyle \left. w\right\vert _{z=d}$">|; 

$key = q/displaystylep(z)=left{{array}{llc}p_0&mbox{hspace{1cm}}&z>0[0.5ex]p_0-rho_0,g,z&&zleq0{array}right.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img566.png"
 ALT="$\displaystyle p(z)=\left\{\begin{array}{llc} p_0&amp;\mbox{\hspace{1cm}}&amp;z&gt;0\ [0.5ex] p_0-\rho_0 g z&amp;&amp;z\leq 0 \end{array}\right.,$">|; 

$key = q/{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="$ {\bf v}$">|; 

$key = q/r=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1713.png"
 ALT="$ r=0$">|; 

$key = q/u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1511.png"
 ALT="$ u_1$">|; 

$key = q/displaystyleK=frac{1}{2},rhointv^{,2},dV=frac{1}{2},rhoint(nablaphi)^2,dV=frac{1}{2},rhointnablacdot(phi,nablaphi),dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="489" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2713.png"
 ALT="$\displaystyle K = \frac{1}{2} \rho\int v^{ 2} dV = \frac{1}{2} \rho\int(\nabla\phi)^2 dV = \frac{1}{2} \rho\int\nabla\cdot(\phi \nabla\phi) dV,$">|; 

$key = q/phi'=phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3495.png"
 ALT="$ \phi'=\phi$">|; 

$key = q/H_L>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1508.png"
 ALT="$ H_L&gt;0$">|; 

$key = q/a_{ij}({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6873.png"
 ALT="$ a_{ij}({\bf x})$">|; 

$key = q/displaystyleF(Q)=F(P)+int_P^Qleft(f,dx+g,dyright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6464.png"
 ALT="$\displaystyle F(Q) = F(P) + \int_P^Q \left(f dx + g dy\right)$">|; 

$key = q/displaystyleleft.frac{partial}{partialy}right|_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2925.png"
 ALT="$\displaystyle \left.\frac{\partial}{\partial y}\right\vert _x$">|; 

$key = q/displaystylenabla^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="$\displaystyle \nabla^{ 2}$">|; 

$key = q/displaystyle{rmMa}=frac{v}{c},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5380.png"
 ALT="$\displaystyle {\rm Ma} = \frac{v}{c},$">|; 

$key = q/displaystyleU=-rho,g,inty,ds=-rho,g,int_{-aslash2}^{aslash2}y,(1+y'^{,2})^{1slash2},dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7342.png"
 ALT="$\displaystyle U = - \rho g \int y ds = - \rho g \int_{-a/2}^{a/2} y (1+y'^{ 2})^{1/2} dx,$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter04slashorifice.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="281" BORDER="0"
 SRC="|."$dir".q|img1405.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter04/orifice.eps}}
\end{figure}">|; 

$key = q/{rmMa}sim{calO}(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="$ {\rm Ma}\sim {\cal O}(1)$">|; 

$key = q/displaystylefrac{2pi}{S,(2,g)^{1slash2}}left(frac{2}{3},a,h^{,3slash2}-frac{1}{5},h^{,5slash2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1599.png"
 ALT="$\displaystyle \frac{2\pi}{S (2 g)^{1/2}}\left(\frac{2}{3} a h^{ 3/2} - \frac{1}{5} h^{ 5/2}\right).
$">|; 

$key = q/overline{chi}=(LslashV_0)^2,chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="$ \overline{\chi} = (L/V_0)^2 \chi$">|; 

$key = q/{rmRe}ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="$ {\rm Re}\ll 1$">|; 

$key = q/displaystyley',frac{partialF}{partialy'}-F={rmconst}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7325.png"
 ALT="$\displaystyle y' \frac{\partial F}{\partial y'} - F = {\rm const}.$">|; 

$key = q/alpha(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5832.png"
 ALT="$ \alpha(x)$">|; 

$key = q/mu_1=sin^{-1}(1slash{rmMa}_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5762.png"
 ALT="$ \mu_1=\sin^{-1}(1/{\rm Ma}_1)$">|; 

$key = q/{bfv}=Vtmspace+thinmuskip{.1667em}{rme}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$ {\bf v}= V\tmspace +\thinmuskip {.1667em}{\rm e}_z$">|; 

$key = q/displaystyleF(zeta)=Vleft(zeta,{rme}^{,{rmi},alpha}+frac{a^{,2}}{zeta},{rme}^{-{ight)+{rmi},frac{{mitGamma}}{2pi},lnleft(frac{zeta}{a}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3293.png"
 ALT="$\displaystyle F(\zeta) = V\left(\zeta {\rm e}^{ {\rm i} \alpha} + \frac{a^{\...
...}\right) + {\rm i} \frac{{\mit\Gamma}}{2\pi} \ln\left(\frac{\zeta}{a}\right),$">|; 

$key = q/displaystyleh_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7116.png"
 ALT="$\displaystyle h_r$">|; 

$key = q/displaystyleskew{3}bar{phi}=V,s,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2220.png"
 ALT="$\displaystyle \skew{3}\bar{\phi}=V s,$">|; 

$key = q/{calC};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1407.png"
 ALT="$ {\cal C}$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{AppendixCslashfigC.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="309" BORDER="0"
 SRC="|."$dir".q|img7168.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixC/figC.02.eps}}
\end{figure}">|; 

$key = q/displaystylem=frac{alpha'}{1+alpha'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1881.png"
 ALT="$\displaystyle m=\frac{\alpha'}{1+\alpha'},$">|; 

$key = q/displaystylerightarrowfrac{gamma+1}{gamma-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5517.png"
 ALT="$\displaystyle \rightarrow \frac{\gamma+1}{\gamma-1},$">|; 

$key = q/oint{bfA}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6623.png"
 ALT="$ \oint {\bf A}\cdot d{\bf r}$">|; 

$key = q/{bfB}=(psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2354.png"
 ALT="$ {\bf B} = (\psi$">|; 

$key = q/r_1leqrleqr_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3621.png"
 ALT="$ r_1\leq r\leq r_2$">|; 

$key = q/displaystyle=V_x(t),frac{a^{,2}}{r^{,2}},sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1807.png"
 ALT="$\displaystyle =V_x(t) \frac{a^{ 2}}{r^{ 2}} \sin\theta.$">|; 

$key = q/{bfv}=V,{rme}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2650.png"
 ALT="$ {\bf v}= V {\rm e}_z$">|; 

$key = q/displaystyle=c,x+d,y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1900.png"
 ALT="$\displaystyle =c x+d y$">|; 

$key = q/sintheta_2-sintheta_1=2,cos[(theta_2+theta_1)slash2],sin[(theta_2-theta_1)slash2];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2625.png"
 ALT="$ \sin\theta_2-\sin\theta_1 = 2 \cos[(\theta_2+\theta_1)/2] \sin[(\theta_2-\theta_1)/2]$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{AppendixCslashfigC.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="306" BORDER="0"
 SRC="|."$dir".q|img7108.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixC/figC.01.eps}}
\end{figure}">|; 

$key = q/displaystyle=-p+2,mu,frac{partialv_r}{partialr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="$\displaystyle =-p + 2 \mu \frac{\partial v_r}{\partial r},$">|; 

$key = q/displaystylenablacdot,(nablatimes{bfA})equiv0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6667.png"
 ALT="$\displaystyle \nabla\cdot (\nabla\times{\bf A}) \equiv 0.$">|; 

$key = q/displaystyleQ,costheta-Q',costheta'={rmconstant},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2827.png"
 ALT="$\displaystyle Q \cos\theta - Q' \cos\theta' = {\rm constant},
$">|; 

$key = q/{rmMa}_infty^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5919.png"
 ALT="$ {\rm Ma}_\infty^{ 2}$">|; 

$key = q/z=cosh(pi,zetaslasha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2491.png"
 ALT="$ z=\cosh(\pi \zeta/a)$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partial{mitPhi}_r(theta,phi_pm)}{partialphi}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4806.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial {\mit\Phi}_r(\theta,\phi_\pm)}{\partial \phi} =0.$">|; 

$key = q/k=0.9406;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1206.png"
 ALT="$ k=0.9406$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter15slashoblique.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="334" BORDER="0"
 SRC="|."$dir".q|img5682.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter15/oblique.eps}}
\end{figure}">|; 

$key = q/|theta|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6126.png"
 ALT="$ \vert\theta\vert$">|; 

$key = q/|lambda|rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2288.png"
 ALT="$ \vert\lambda\vert\rightarrow\infty$">|; 

$key = q/displaystylec'=u+c.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5365.png"
 ALT="$\displaystyle c'=u+c.$">|; 

$key = q/{bfv}({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="$ {\bf v}({\bf r},t)$">|; 

$key = q/displaystylefrac{pi}{2},frac{V}{skew{3}bar{psi}_1},(z-z_1)={rmi},(lambda-1)-(1-lambda^{,2})^{1slash2}+tanh^{-1}[(1-lambda^{,2})^{1slash2}],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2211.png"
 ALT="$\displaystyle \frac{\pi}{2} \frac{V}{\skew{3}\bar{\psi}_1} (z-z_1)= {\rm i} (\lambda-1)-(1-\lambda^{ 2})^{1/2}+\tanh^{-1}[(1-\lambda^{ 2})^{1/2}],$">|; 

$key = q/displaystylembox{boldmath{{Omega{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3382.png"
 ALT="$\displaystyle \mbox{\boldmath$\Omega$}$">|; 

$key = q/displaystyleskew{5}hat{zeta}_j(vartheta)=sum_{n=2,infty}z_n,P_n^{,m_j}(sinvartheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4711.png"
 ALT="$\displaystyle \skew{5}\hat{\zeta}_j(\vartheta)=\sum_{n=2,\infty} z_n P_n^{ m_j}(\sin\vartheta).$">|; 

$key = q/displaystyle{bfq}=U,{bfe}_x-nablaphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5942.png"
 ALT="$\displaystyle {\bf q} = U {\bf e}_x -\nabla\phi,$">|; 

$key = q/displaystylea_{sjsl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6951.png"
 ALT="$\displaystyle a_{sjsl}$">|; 

$key = q/displaystyleintx_1,x_2,dA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img842.png"
 ALT="$\displaystyle \int x_1 x_2 dA$">|; 

$key = q/displaystylefrac{drho}{rho}+frac{dA}{A}+frac{du}{u}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5624.png"
 ALT="$\displaystyle \frac{d\rho}{\rho} + \frac{dA}{A}+\frac{du}{u} =0.
$">|; 

$key = q/displaystyle={bfA}cdotnablaA_theta+frac{A_r,A_theta}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7138.png"
 ALT="$\displaystyle = {\bf A}\cdot\nabla A_\theta + \frac{A_r A_\theta}{r},$">|; 

$key = q/displaystylefrac{partial^{,2}{mitPi}}{partialt^{,2}}+f^{,2},{mitPi}-g,d,nabla^{,abla^{,2}left(frac{partialskew{5}bar{zeta}'}{partialt}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5076.png"
 ALT="$\displaystyle \frac{\partial^{ 2}{\mit\Pi}}{\partial t^{ 2}} + f^{ 2} {\mit...
...i}=g \nabla^{ 2}\left(\frac{\partial\skew{5}\bar{\zeta}'}{\partial t}\right),$">|; 

$key = q/k=gslashV^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4000.png"
 ALT="$ k=g/V^{ 2}$">|; 

$key = q/displaystylev^{,2}=V^{,2}-2,g,h.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1456.png"
 ALT="$\displaystyle v^{ 2}= V^{ 2}-2 g h.$">|; 

$key = q/zeta=zeta_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1278.png"
 ALT="$ \zeta=\zeta_c$">|; 

$key = q/displaystylepsi({bfr})=psi_Q({bfr}-{bfd}slash2)-psi_Q({bfr}+{bfd}slash2)simeq-{bfd}cdotnablapsi_Q({bfr}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1688.png"
 ALT="$\displaystyle \psi({\bf r}) = \psi_Q({\bf r}-{\bf d}/2)-\psi_Q({\bf r}+{\bf d}/2)\simeq - {\bf d}\cdot\nabla\psi_Q({\bf r}),$">|; 

$key = q/(a_1-a_3)slasha_1simeqe_{13}^{,2}slash2=2.81times10^{-5};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1031.png"
 ALT="$ (a_1-a_3)/a_1 \simeq e_{13}^{ 2}/2 = 2.81\times 10^{-5}$">|; 

$key = q/displaystyletanbetasimeqtanmu=frac{1}{sqrt{{rmMa}_1^{,2}-1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5725.png"
 ALT="$\displaystyle \tan\beta\simeq \tan\mu =\frac{1}{\sqrt{{\rm Ma}_1^{ 2}-1}}.$">|; 

$key = q/displaystylef_i=sigma_{ij},dS_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$\displaystyle f_i = \sigma_{ij} dS_j,$">|; 

$key = q/displaystyle=(1-beta),p,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5189.png"
 ALT="$\displaystyle =(1-\beta) p,$">|; 

$key = q/displaystylenabla^{,2}{bfv}=-frac{G}{mu},{bfe}_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3611.png"
 ALT="$\displaystyle \nabla^{ 2}{\bf v} = -\frac{G}{\mu} {\bf e}_z.$">|; 

$key = q/{mitPsi}=g,z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img560.png"
 ALT="$ {\mit\Psi}=g z$">|; 

$key = q/nablatimes{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6647.png"
 ALT="$ \nabla\times {\bf A}$">|; 

$key = q/displaystyleC_{p0}=-frac{2,alpha_0}{sqrt{{rmMa}_infty^{,2}-1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6062.png"
 ALT="$\displaystyle C_{p0}=-\frac{2 \alpha_0}{\sqrt{{\rm Ma}_\infty^{ 2}-1}}.$">|; 

$key = q/displaystyleintsqrt{{rmMa}^{,2}-1},frac{dw}{w}=mu-frac{1}{b},tan^{-1}left(frac{1}{b},tanmuright)+{rmconst.};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6209.png"
 ALT="$\displaystyle \int \sqrt{{\rm Ma}^{ 2}-1} \frac{dw}{w} = \mu -\frac{1}{b} \tan^{-1}\left(\frac{1}{b} \tan\mu\right)+{\rm const.}
$">|; 

$key = q/{calE}=Eslashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5299.png"
 ALT="$ {\cal E} = E/m$">|; 

$key = q/sin^2theta=[a_1^{,2}slash(a_1^{,2}+u)],sin^2gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img935.png"
 ALT="$ \sin^2\theta = [a_1^{ 2}/(a_1^{ 2}+u)] \sin^2\gamma$">|; 

$key = q/displaystylet^{,2}=12,s,(1-s)-2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img766.png"
 ALT="$\displaystyle t^{ 2}= 12 s (1-s)-2.$">|; 

$key = q/chi=pslashrho+(1slash2),v^{,2}+{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1930.png"
 ALT="$ \chi = p/\rho+(1/2) v^{ 2}+{\mit\Psi}$">|; 

$key = q/displaystylealpha_0=frac{2,(cosbeta,cosgamma)^{1slash3}}{a_0,singamma},F(gamma,alpha).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img947.png"
 ALT="$\displaystyle \alpha_0 = \frac{2 (\cos\beta \cos\gamma)^{1/3}}{a_0 \sin\gamma} F(\gamma,\alpha).$">|; 

$key = q/displaystyle=r,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7118.png"
 ALT="$\displaystyle = r,$">|; 

$key = q/displaystylefrac{partialv_x}{partialx}+frac{partialv_y}{partialy}+frac{partialv_z}{partialz}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6596.png"
 ALT="$\displaystyle \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z} {\partial z}=0.$">|; 

$key = q/displaystyleC_D=frac{16}{3sqrt{{rmMa}_1^{,2}-1}}left(frac{t}{c}right)^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6213.png"
 ALT="$\displaystyle C_D = \frac{16}{3\sqrt{{\rm Ma}_1^{ 2}-1}}\left(\frac{t}{c}\right)^2,
$">|; 

$key = q/displaystyleint_{A_0}x,dx,dy=int_{A_0}y,dx,dy=int_{A_0}x,y,dx,dy=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img627.png"
 ALT="$\displaystyle \int_{A_0}x dx dy = \int_{A_0}y dx dy=\int_{A_0}x y dx dy=0,$">|; 

$key = q/displaystyle=costheta,xi(theta,phi_pm,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4774.png"
 ALT="$\displaystyle =\cos\theta \xi(\theta,\phi_\pm,t),$">|; 

$key = q/T_c=1.58times10^7,{rmK};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5247.png"
 ALT="$ T_c = 1.58\times 10^7 {\rm K}$">|; 

$key = q/displaystylesimeqfrac{gamma-1}{gamma},frac{{mitDelta}p}{p_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6162.png"
 ALT="$\displaystyle \simeq \frac{\gamma-1}{\gamma} \frac{{\mit\Delta} p}{p_1},$">|; 

$key = q/x+{rmi},y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2511.png"
 ALT="$ x+{\rm i} y$">|; 

$key = q/displaystylefrac{d}{dr}left(r^{,2},frac{dR}{dr}right)-l,(l+1),R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2575.png"
 ALT="$\displaystyle \frac{d}{dr}\left(r^{ 2} \frac{dR}{dr}\right) -l (l+1) R$">|; 

$key = q/f(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1247.png"
 ALT="$ f(z)$">|; 

$key = q/displaystylefrac{partial^{,2}mbox{boldmathxi}}{partialt^{,2}}+2,mbox{boldmathOme2}mbox{boldmathxi}-nablaleft(frac{p}{rho}+{mitPhi}+chiright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4384.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\mbox{\boldmath$\xi$}}{\partial t^{ 2}}+2 \...
...^{ 2}\mbox{\boldmath$\xi$} -\nabla\left(\frac{p}{\rho}+{\mit\Phi}+\chi\right).$">|; 

$key = q/v_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6981.png"
 ALT="$ v_{ij}$">|; 

$key = q/displaystyleD,{mitPhi}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4769.png"
 ALT="$\displaystyle D {\mit\Phi}'$">|; 

$key = q/displaystyle=-left(frac{partialZ}{partialY}right)_Xleftslashleft(frac{partialZ}{partialX}right)_Yright.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5576.png"
 ALT="$\displaystyle = -\left(\frac{\partial Z}{\partial Y}\right)_X\left/\left(\frac{\partial Z}{\partial X}\right)_Y\right. $">|; 

$key = q/a_i,b_i=a_1,b_1+a_2,b_2+a_3,b_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6774.png"
 ALT="$ a_i b_i= a_1 b_1+a_2 b_2+a_3 b_3$">|; 

$key = q/(varpi,,z)=(0,,-a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2833.png"
 ALT="$ (\varpi, z) = (0, -a)$">|; 

$key = q/displaystyle{bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="$\displaystyle {\bf0}$">|; 

$key = q/displaystylefrac{dx}{dphi}=frac{dx}{df},frac{df}{dphi}=-frac{l,k,cos(2,phi)}{(1-k^{,2},sin^2phi)^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1172.png"
 ALT="$\displaystyle \frac{dx}{d\phi} = \frac{dx}{df} \frac{df}{d\phi} = -\frac{l k \cos(2 \phi)}{(1-k^{ 2} \sin^2\phi)^{1/2}},$">|; 

$key = q/z=d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4476.png"
 ALT="$ z=d$">|; 

$key = q/displaystylenabla^{,2}T=frac{c}{kappa}frac{partialT}{partialt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6620.png"
 ALT="$\displaystyle \nabla^{ 2} T = \frac{c}{\kappa} \frac{\partial T}{\partial t}.$">|; 

$key = q/T_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5250.png"
 ALT="$ T_c$">|; 

$key = q/DslashDoverline{t}equivpartialslashpartialoverline{t}+overline{bfv}cdotoverline{nabla};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="$ D/D\overline{t}\equiv \partial/\partial \overline{t} + \overline{\bf v}\cdot\overline{\nabla}$">|; 

$key = q/omega_z=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1793.png"
 ALT="$ \omega_z=0$">|; 

$key = q/displaystyle=frac{1}{varpi},frac{partialpsi}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2547.png"
 ALT="$\displaystyle =\frac{1}{\varpi} \frac{\partial \psi}{\partial z},$">|; 

$key = q/displaystyle2,a_{iljk}+(a_{ilkj}+a_{ijlk}+a_{ikjl});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6968.png"
 ALT="$\displaystyle 2 a_{iljk}+(a_{ilkj}+a_{ijlk}+a_{ikjl})$">|; 

$key = q/v';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5899.png"
 ALT="$ v'$">|; 

$key = q/displaystylebeta_{-n,,-n'}^{,-m,,-m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4954.png"
 ALT="$\displaystyle \beta_{-n, -n'}^{ -m, -m'}$">|; 

$key = q/displaystylefrac{d^{,2}u}{dt^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img758.png"
 ALT="$\displaystyle \frac{d^{ 2}u}{dt^{ 2}}$">|; 

$key = q/S,cosalpha_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6339.png"
 ALT="$ S \cos\alpha_y$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter15slashaspect.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img6086.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter15/aspect.eps}}
\end{figure}">|; 

$key = q/p(x,t)=p_0+p_1(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5354.png"
 ALT="$ p(x,t)=p_0+p_1(x,t)$">|; 

$key = q/{calE}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5448.png"
 ALT="$ {\cal E}_2$">|; 

$key = q/displaystyleA_{ij},x_j=lambda,x_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7002.png"
 ALT="$\displaystyle A_{ij} x_j = \lambda x_i
$">|; 

$key = q/displaystylelambda_rint_{mitOmega}{mitPhi}_r,{mitPhi}_s,d{mitOmega}=lambda_sint_{mitOmega}{mitPhi}_r,{mitPhi}_s,d{mitOmega}=delta_{r,s}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4818.png"
 ALT="$\displaystyle \lambda_r\int_{\mit\Omega} {\mit\Phi}_r {\mit\Phi}_s d{\mit\Ome...
...a_s\int_{\mit\Omega} {\mit\Phi}_r {\mit\Phi}_s d{\mit\Omega} = \delta_{r s}.$">|; 

$key = q/{bfe}_xtimes{bfe}_y={bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6370.png"
 ALT="$ {\bf e}_x\times
{\bf e}_y={\bf e}_z$">|; 

$key = q/lnr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1745.png"
 ALT="$ \ln r$">|; 

$key = q/p=p(r,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img815.png"
 ALT="$ p=p(r,z)$">|; 

$key = q/theta>theta_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5717.png"
 ALT="$ \theta&gt;\theta_{\rm max}$">|; 

$key = q/{rmRe}^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3097.png"
 ALT="$ {\rm Re}^{-1/2}$">|; 

$key = q/{bfe}_yequiv(0,,1,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6265.png"
 ALT="$ {\bf e}_y\equiv (0, 1, 0)$">|; 

$key = q/displaystylefrac{1}{V},frac{dbar{F}}{dz}=-{rmi},gamma^{,-1slash2}+{rmi},left(frac{1}{gamma}-1right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2280.png"
 ALT="$\displaystyle \frac{1}{V} \frac{d\bar{F}}{dz} = -{\rm i} \gamma^{ -1/2}+{\rm i} \left(\frac{1}{\gamma}-1\right)^{1/2},$">|; 

$key = q/beta_{-r,-s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4889.png"
 ALT="$ \beta_{-r,-s}$">|; 

$key = q/displaystyle=delta_{ij}-deltatheta_l,epsilon_{lij}-deltatheta_l,epsilon_{lji}=delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6831.png"
 ALT="$\displaystyle =\delta_{ij} -\delta\theta_l \epsilon_{lij}-\delta\theta_l \epsilon_{lji}=\delta_{ij},$">|; 

$key = q/-sinepsilon,(zeta_M+zeta_E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4559.png"
 ALT="$ -\sin\epsilon (\zeta_M+\zeta_E)$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashseparation.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="569" HEIGHT="341" BORDER="0"
 SRC="|."$dir".q|img3089.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/separation.eps}}
\end{figure}">|; 

$key = q/0.6627,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1286.png"
 ALT="$ 0.6627 c$">|; 

$key = q/displaystylenablaphi=sum_{i=1,3}frac{partialphi}{partialu_i},nablau_i=sum_{i=1,3}frac{1}{h_i},frac{partialphi}{partialu_i},{bfe}_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7065.png"
 ALT="$\displaystyle \nabla \phi = \sum_{i=1,3} \frac{\partial \phi}{\partial u_i} \nabla u_i=\sum_{i=1,3}\frac{1}{h_i} \frac{\partial \phi}{\partial u_i} {\bf e}_i.$">|; 

$key = q/displaystyley>0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6048.png"
 ALT="$\displaystyle y&gt;0,$">|; 

$key = q/displaystyleG(eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3169.png"
 ALT="$\displaystyle G(\eta)$">|; 

$key = q/displaystylecostheta,eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4776.png"
 ALT="$\displaystyle \cos\theta \eta$">|; 

$key = q/displaystylezeta(theta,phi)=sum_{n=1,infty}zeta_n(theta,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4511.png"
 ALT="$\displaystyle \zeta(\theta,\phi)=\sum_{n=1,\infty} \zeta_n(\theta,\phi),$">|; 

$key = q/m,B_1,A_2=U,epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5974.png"
 ALT="$ m B_1 A_2=U \epsilon$">|; 

$key = q/displaystyleF(z)={rmi},frac{mitGamma}{2pi},lnz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1984.png"
 ALT="$\displaystyle F(z) = {\rm i} \frac{\mit\Gamma}{2\pi} \ln z,$">|; 

$key = q/displaystyle{bfa}'simeq{bfa}+delta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6399.png"
 ALT="$\displaystyle {\bf a}' \simeq {\bf a} + \delta$">|; 

$key = q/p_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$ p_2$">|; 

$key = q/c_n^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4317.png"
 ALT="$ c_n^{ m}$">|; 

$key = q/r=a,xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5274.png"
 ALT="$ r=a \xi$">|; 

$key = q/sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5058.png"
 ALT="$ \sin\theta$">|; 

$key = q/displaystyle=left.left(frac{2,n+1}{2,n-2}right)rightslashleft[1+frac{(2,n+1),(2,n^{,2}+4,n+3)}{(6+n^{,2})},frac{mu}{rho,g,a}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4434.png"
 ALT="$\displaystyle = \left.\left(\frac{2 n+1}{2 n-2}\right)\right/\left[1+\frac{(2 n+1) (2 n^{ 2}+4 n+3)}{(6+n^{ 2})} \frac{\mu}{\rho g a}\right],$">|; 

$key = q/displaystyle[({bfA}cdotnabla),{bfA}]_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7137.png"
 ALT="$\displaystyle [({\bf A}\cdot\nabla) {\bf A}]_\theta$">|; 

$key = q/displaystylef_i=int_VF_i,dV+int_Vfrac{partialsigma_{ij}}{partialx_j},dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="$\displaystyle f_i = \int_V F_i dV + \int_V \frac{\partial \sigma_{ij}}{\partial x_j} dV.$">|; 

$key = q/displaystyle-frac{dF}{dz}=frac{4V_0}{n^2},frac{sin^2(zetaslash2)}{sin^2(zetaslashn)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2480.png"
 ALT="$\displaystyle -\frac{dF}{dz} = \frac{4V_0}{n^2} \frac{\sin^2(\zeta/2)}{\sin^2(\zeta/n)}.
$">|; 

$key = q/h=-kslash(rho,g);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7349.png"
 ALT="$ h=-k/(\rho g)$">|; 

$key = q/displaystyle{mitPhi}_{mitTheta}=oint_Stheta,{bfv}cdotd{bfS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="$\displaystyle {\mit\Phi}_{\mit\Theta} = \oint_S \theta {\bf v}\cdot d{\bf S},$">|; 

$key = q/a,cos[k,(x-V,t)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4206.png"
 ALT="$ a \cos[k (x-V t)]$">|; 

$key = q/displaystylelambda^{,3}-{rmtr}(A),lambda^{,2}+{mitPi}(A),lambda-{rmdet}(A)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7006.png"
 ALT="$\displaystyle \lambda^{ 3} - {\rm tr}(A)  \lambda^{ 2} + {\mit\Pi}(A) \lambda - {\rm det}(A) = 0,
$">|; 

$key = q/displaystylebar{z}=left.frac{1}{12},a^4rightslashfrac{1}{3},a^3=frac{1}{4},a.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6516.png"
 ALT="$\displaystyle \bar{z} = \left.\frac{1}{12} a^4\right/\frac{1}{3} a^3 = \frac{1}{4} a.$">|; 

$key = q/y=y_U(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5816.png"
 ALT="$ y=y_U(x)$">|; 

$key = q/V_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3537.png"
 ALT="$ V_s$">|; 

$key = q/B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$ B$">|; 

$key = q/displaystyle{bfF}=-6pi,a,mu,{bfV},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3894.png"
 ALT="$\displaystyle {\bf F} = -6\pi a \mu {\bf V},
$">|; 

$key = q/sinmu=1slash{rmMa}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5734.png"
 ALT="$ \sin\mu=1/{\rm Ma}_1$">|; 

$key = q/displaystyle=frac{partial}{partialvarpi}left(frac{1}{varpi},frac{partialpsi}{parpi}right)+frac{1}{varpi},frac{partial^{,2}psi}{partialz^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2568.png"
 ALT="$\displaystyle =\frac{\partial}{\partial\varpi}\left(\frac{1}{\varpi} \frac{\pa...
...l\varpi}\right) + \frac{1}{\varpi} \frac{\partial^{ 2}\psi}{\partial z^{ 2}}$">|; 

$key = q/displaystyleI=int_a^bF(y_1,y_2,cdots,y_{calF},y_1',y_2',cdots,y_{calF}',x),dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7362.png"
 ALT="$\displaystyle I = \int_a^b F(y_1,y_2,\cdots,y_{\cal F},y_1',y_2',\cdots,y_{\cal F}',x) dx.$">|; 

$key = q/hat{T}=TslashT_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5139.png"
 ALT="$ \hat{T}=T/T_0$">|; 

$key = q/displaystylephantom{=}-int_{mitOmega}frac{1}{sintheta}left(frac{partial{mitPsi}_ltheta},frac{partial{mitPhi}_s}{partialphi}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4868.png"
 ALT="$\displaystyle \phantom{=}-\int_{\mit\Omega}\frac{1}{\sin\theta}\left(\frac{\par...
...\partial\theta} \frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}.$">|; 

$key = q/|z|rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2481.png"
 ALT="$ \vert z\vert\rightarrow \infty$">|; 

$key = q/{mitOmega}_varphi={mitOmega}_varphi(varpi,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2554.png"
 ALT="$ {\mit\Omega}_\varphi={\mit\Omega}_\varphi(\varpi,z)$">|; 

$key = q/kappa^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img659.png"
 ALT="$ \kappa^{ 2}$">|; 

$key = q/displaystyle{bfA}=frac{{bfb}times{bfc}}{{bfa}cdot{bfb}times{bfc}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6735.png"
 ALT="$\displaystyle {\bf A} = \frac{{\bf b}\times {\bf c}}{{\bf a}\cdot{\bf b}\times {\bf c}},
$">|; 

$key = q/displaystylefrac{v_p}{v_0}simeqfrac{1}{(2,k,l)^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4158.png"
 ALT="$\displaystyle \frac{v_p}{v_0} \simeq \frac{1}{(2 k l)^{1/2}},$">|; 

$key = q/beta_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1748.png"
 ALT="$ \beta_m$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter09slashwing1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="620" HEIGHT="305" BORDER="0"
 SRC="|."$dir".q|img3403.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter09/wing1.eps}}
\end{figure}">|; 

$key = q/beta_1'-beta_1<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6172.png"
 ALT="$ \beta_1'-\beta_1&lt;0$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialx^{,2}}-frac{1}{lambda^{,2}},frac{partial^{,2}phi}{partialy^{,2}}simeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5986.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial x^{ 2}} -\frac{1}{\lambda^{ 2}} \frac{\partial^{ 2}\phi}{\partial y^{ 2}}\simeq 0$">|; 

$key = q/r=R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1113.png"
 ALT="$ r=R$">|; 

$key = q/rho_0,V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img702.png"
 ALT="$ \rho_0 V_0$">|; 

$key = q/A_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1600.png"
 ALT="$ A_1$">|; 

$key = q/displaystyle{bfB}({bfr})=frac{1}{4pi}intfrac{{bfj}({bfr}')times({bfr}-{bfr}')}{|{bfr}-{bfr}'|^{,3}},d^{,3}{bfr}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3386.png"
 ALT="$\displaystyle {\bf B}({\bf r}) = \frac{1}{4\pi}\int\frac{{\bf j}({\bf r}')\times ({\bf r}-{\bf r}')}{\vert{\bf r}-{\bf r}'\vert^{ 3}} d^{ 3}{\bf r}'.$">|; 

$key = q/displaystyleE=int_Vrho,{calE},dV+int_Vfrac{1}{2},rho,v_i,v_i,dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="$\displaystyle E = \int_V \rho {\cal E} dV + \int_V\frac{1}{2} \rho v_i v_i dV,$">|; 

$key = q/nabla^{,2}phi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2697.png"
 ALT="$ \nabla^{ 2}\phi=0$">|; 

$key = q/displaystylefrac{D{bfv}}{Dt}=-frac{nablap}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5437.png"
 ALT="$\displaystyle \frac{D{\bf v}}{Dt}= -\frac{\nabla p}{\rho},$">|; 

$key = q/v,{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6727.png"
 ALT="$ v {\bf b}$">|; 

$key = q/displaystyle{rmMa}_{,e}=sqrt{left(frac{2}{gamma-1}right)left[left(frac{p_0}{p_e}right)^{(gamma-1)slashgamma}-1right]},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="91" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5421.png"
 ALT="$\displaystyle {\rm Ma}_{ e} =\sqrt{ \left(\frac{2}{\gamma-1}\right) \left[\left(\frac{p_0}{p_e}\right)^{(\gamma-1)/\gamma}-1\right]},$">|; 

$key = q/9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4567.png"
 ALT="$ 9$">|; 

$key = q/displaystylep_0=p_0-rho,g,zeta+left.p_1right|_{z=0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3910.png"
 ALT="$\displaystyle p_0 = p_0 -\rho g \zeta + \left.p_1\right\vert _{z=0},$">|; 

$key = q/beta=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2152.png"
 ALT="$ \beta=\pi/2$">|; 

$key = q/kappa_x^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img660.png"
 ALT="$ \kappa_x^{ 2}$">|; 

$key = q/{bfa},%,{bfb}=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6299.png"
 ALT="$ {\bf a} \% {\bf b} = 1/2$">|; 

$key = q/displaystyle{calH}+frac{1}{2},v^{,2}+{mitPsi}={calC}_p,T+frac{1}{2},v^{,2}+{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5377.png"
 ALT="$\displaystyle {\cal H} +\frac{1}{2} v^{ 2}+{\mit\Psi} = {\cal C}_p T +\frac{1}{2} v^{ 2}+{\mit\Psi}$">|; 

$key = q/displaystyle=sum_{i=1,3}frac{h_i^{,2}}{h}nablau_icdotnablaleft(frac{h}{h_i},A_ir{1}{h},frac{partial}{partialu_i}!left(frac{h}{h_i},A_iright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7071.png"
 ALT="$\displaystyle =\sum_{i=1,3}\frac{h_i^{ 2}}{h}\nabla u_i\cdot\nabla\left(\frac{...
...3} \frac{1}{h} \frac{\partial}{\partial u_i}\!\left(\frac{h}{h_i} A_i\right),$">|; 

$key = q/displaystyle=frac{xi_1^{,3}}{4pi,y_1},frac{M_ast}{R^{,3}}=7.63times10^4,{rmkg,m^{-3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5243.png"
 ALT="$\displaystyle = \frac{\xi_1^{ 3}}{4\pi y_1} \frac{M_\ast}{R^{ 3}} = 7.63\times 10^4 {\rm kg m^{-3}},$">|; 

$key = q/beta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2635.png"
 ALT="$ \beta_1$">|; 

$key = q/displaystylefrac{d{rmMa}_2^{,2}}{d{rmMa}_1^{,2}}=-left(frac{1+gamma}{1-gamma+2,gamma,{rmMa}_1^{,2}}right)^{,2}<0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5479.png"
 ALT="$\displaystyle \frac{d{\rm Ma}_2^{ 2}}{d{\rm Ma}_1^{ 2}}=-\left(\frac{1+\gamma}{1-\gamma+2 \gamma {\rm Ma}_1^{ 2}}\right)^{ 2}&lt;0.$">|; 

$key = q/ds=h_eta,deta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2718.png"
 ALT="$ ds= h_\eta d\eta$">|; 

$key = q/displaystylesimeq6.7times10^4,L({rmm}),V_0({rmm,s^{-1}}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="$\displaystyle \simeq 6.7\times 10^4 L({\rm m}) V_0({\rm m s^{-1}}),$">|; 

$key = q/xi=xi_0+pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2468.png"
 ALT="$ \xi=\xi_0+\pi$">|; 

$key = q/displaystyle=-frac{1}{varpi},frac{partialpsi}{partialvarpi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2549.png"
 ALT="$\displaystyle = -\frac{1}{\varpi} \frac{\partial\psi}{\partial\varpi},$">|; 

$key = q/z=z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3450.png"
 ALT="$ z=z$">|; 

$key = q/+(1slash2),zeta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4536.png"
 ALT="$ +(1/2) \zeta_2$">|; 

$key = q/displaystyleG(x)=-frac{dP}{dx}=12,mu,Q,d^{,-3}(x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3667.png"
 ALT="$\displaystyle G(x)= -\frac{dP}{dx} = 12 \mu Q d^{ -3}(x).$">|; 

$key = q/displaystylesimeq{rmMa}_infty^{,2},(gamma+1),frac{u'}{U},frac{partialu'}{partiala}_infty^{,2},(gamma-1),frac{u'}{U},frac{partialv'}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5909.png"
 ALT="$\displaystyle \simeq {\rm Ma}_\infty^{ 2} (\gamma+1) \frac{u'}{U} \frac{\pa...
...{\rm Ma}_\infty^{ 2} (\gamma-1) \frac{u'}{U} \frac{\partial v'}{\partial y}$">|; 

$key = q/F(gamma,alpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1006.png"
 ALT="$ F(\gamma,\alpha)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter09slashcyl2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3349.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/cyl2.eps}}
\end{figure}">|; 

$key = q/displaystyleQ(theta,phi_pm,t)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4738.png"
 ALT="$\displaystyle Q(\theta,\phi_\pm,t) = 0.$">|; 

$key = q/a=6.37times10^6,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4572.png"
 ALT="$ a=6.37\times 10^6 {\rm m}$">|; 

$key = q/{bfe}_r=nablarslash|nablar|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4252.png"
 ALT="$ {\bf e}_r=\nabla r/\vert\nabla r\vert$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter06slashconf1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="528" HEIGHT="255" BORDER="0"
 SRC="|."$dir".q|img2081.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter06/conf1.eps}}
\end{figure}">|; 

$key = q/d(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1522.png"
 ALT="$ d(x)$">|; 

$key = q/displaystylenuleft.frac{partial^{,2}u}{partialy^{,2}}right|_{y=0}=-U,frac{dU}{dx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3151.png"
 ALT="$\displaystyle \nu\left.\frac{\partial^{ 2} u}{\partial y^{ 2}}\right\vert _{y=0} = -U \frac{dU}{dx}.$">|; 

$key = q/displaystyle=delta_{ji},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6792.png"
 ALT="$\displaystyle = \delta_{ji},$">|; 

$key = q/displaystyle{bfr}cdot{bfb}times{bfc}=alpha,{bfa}cdot{bfb}times{bfc},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6416.png"
 ALT="$\displaystyle {\bf r} \cdot {\bf b} \times{\bf c} = \alpha  {\bf a}\cdot{\bf b} \times{\bf c},$">|; 

$key = q/bar{F}slash(k,V)=skew{3}bar{phi}slash(k,V)<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2278.png"
 ALT="$ \bar{F}/(k V)=\skew{3}\bar{\phi}/(k V)&lt;1$">|; 

$key = q/-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2579.png"
 ALT="$ -1$">|; 

$key = q/displaystyle=epsilon_{ijk}left[delta_{li},delta_{mj},delta_{nk}-deltatheta_alefti},delta_{nk}+epsilon_{ank},delta_{li},delta_{mj}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6836.png"
 ALT="$\displaystyle =\epsilon_{ijk}\left[\delta_{li} \delta_{mj} \delta_{nk}-\delta...
...delta_{li} \delta_{nk} +\epsilon_{ank} \delta_{li} \delta_{mj}\right)\right]$">|; 

$key = q/dcslashdV'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4194.png"
 ALT="$ dc/dV'=0$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialx^{,2}}+frac{partial^{,2}phi}{partialy^{,2}}simeq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6024.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial x^{ 2}}+ \frac{\partial^{ 2}\phi}{\partial y^{ 2}}\simeq 0.$">|; 

$key = q/displaystyle=kleft(1-costhetaright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7381.png"
 ALT="$\displaystyle =k \left(1-\cos\theta\right),$">|; 

$key = q/displaystyleK=U+lambda,l=int_{-aslash2}^{aslash2}(-rho,g,y+lambda),(1+y'^{,2})^{1slash2},dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7345.png"
 ALT="$\displaystyle K = U + \lambda l = \int_{-a/2}^{a/2}(-\rho g y+\lambda) (1+y'^{ 2})^{1/2} dx,$">|; 

$key = q/displaystyleF=sum_{r=1,infty}a_r,{mitPsi}_r.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4833.png"
 ALT="$\displaystyle F= \sum_{r=1,\infty} a_r {\mit\Psi}_r.$">|; 

$key = q/h_1slashh_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1490.png"
 ALT="$ h_1/h_2$">|; 

$key = q/displaystyle=-sum_{r=1,infty}lambda_r,p_r,{mitPhi}_r.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4895.png"
 ALT="$\displaystyle = -\sum_{r=1,\infty} \lambda_r p_r {\mit\Phi}_r.$">|; 

$key = q/displaystylefrac{Dv_theta}{Dt}+frac{v_r,v_theta}{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="$\displaystyle \frac{Dv_\theta}{Dt}+\frac{v_r v_\theta}{r}$">|; 

$key = q/rho,S_1,v_1,delta,ttimes{bfv}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1389.png"
 ALT="$ \rho S_1 v_1 \delta t\times {\bf v}_1$">|; 

$key = q/deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="$ \delta t$">|; 

$key = q/displaystyle=int_0^inftyleft[(U-u),frac{dU}{dx}+u,frac{partial(U-u)}{partialx}+v,frac{partial(U-u)}{partialy}right]dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3081.png"
 ALT="$\displaystyle =\int_0^\infty\left[(U-u) \frac{dU}{dx} + u \frac{\partial(U-u)}{\partial x} + v \frac{\partial(U-u)}{\partial y}\right]dy$">|; 

$key = q/displaystyle=frac{1}{4},(c^{,2}-delta^{,2})^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3308.png"
 ALT="$\displaystyle =\frac{1}{4} (c^{ 2}-\delta^{ 2})^{1/2}.$">|; 

$key = q/displaystyle=rho_2,u_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5463.png"
 ALT="$\displaystyle = \rho_2 u_2,$">|; 

$key = q/displaystyle{mitGamma}_C=oint_C{bfv}cdotd{bfr}=int_S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1551.png"
 ALT="$\displaystyle {\mit\Gamma}_C= \oint_C {\bf v}\cdot d{\bf r} = \int_S$">|; 

$key = q/psi_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1621.png"
 ALT="$ \psi_2$">|; 

$key = q/n>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3876.png"
 ALT="$ n&gt;1$">|; 

$key = q/h_2=|nablau_2|^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7031.png"
 ALT="$ h_2=\vert\nabla u_2\vert^{-1}$">|; 

$key = q/{rmMa}_inftysimeq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5912.png"
 ALT="$ {\rm Ma}_\infty \simeq 1$">|; 

$key = q/displaystylemu'=frac{M}{M'},frac{{a_0'}^{,3}}{R^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1015.png"
 ALT="$\displaystyle \mu'= \frac{M}{M'} \frac{{a_0'}^{ 3}}{R^{ 3}},$">|; 

$key = q/c_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3486.png"
 ALT="$ c_0$">|; 

$key = q/displaystyleQ=frac{G,d^{,3}}{12,mu}+frac{1}{2},U,d.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3587.png"
 ALT="$\displaystyle Q = \frac{G d^{ 3}}{12 \mu}+ \frac{1}{2} U d.$">|; 

$key = q/a_0=(a_1,a_2,a_3)^{1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1014.png"
 ALT="$ a_0=(a_1 a_2 a_3)^{1/3}$">|; 

$key = q/phi_-=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4932.png"
 ALT="$ \phi_-=0$">|; 

$key = q/B_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5969.png"
 ALT="$ B_2$">|; 

$key = q/displaystyle=-frac{1}{sintheta},frac{partialP}{partialphi}+frac{partialQ}{partialtheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4740.png"
 ALT="$\displaystyle =-\frac{1}{\sin\theta} \frac{\partial P}{\partial\phi} + \frac{\partial Q}{\partial\theta}.$">|; 

$key = q/displaystyleint_{-infty}^inftyu,dy=Q,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3028.png"
 ALT="$\displaystyle \int_{-\infty}^\infty u dy = Q,$">|; 

$key = q/displaystylesimeq-sqrt{frac{g}{h}},w,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3566.png"
 ALT="$\displaystyle \simeq - \sqrt{\frac{g}{h}} w,$">|; 

$key = q/displaystyle=a_j,frac{partialb_i}{partialx_j},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="$\displaystyle = a_j \frac{\partial b_i}{\partial x_j},$">|; 

$key = q/z=x_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6761.png"
 ALT="$ z= x_3$">|; 

$key = q/displaystyleint_P^Q{bfA}cdotd{bfr}=int_P^Q(A_x,dx+A_y,dy+A_z,dz).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6475.png"
 ALT="$\displaystyle \int_P^Q {\bf A}\cdot d{\bf r} = \int_P^Q (A_x dx+A_y dy + A_z dz).$">|; 

$key = q/displaystyleleft.frac{partial}{partialt}right|_{x,y,z}=left.frac{partial}{partialt}right|_{r,theta}-{bfV}cdotnabla.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2667.png"
 ALT="$\displaystyle \left.\frac{\partial}{\partial t}\right\vert _{x,y,z} =\left. \frac{\partial}{\partial t}\right\vert _{r,\theta} -{\bf V}\cdot\nabla.$">|; 

$key = q/|zeta|slash{muskipz@5mudividemuskipz@tw@mkernmuskipz@mathaccentV{tilde}07E{mkern-muskipz@{zeta}mkernmuskipz@}mkern-muskipz@}{}_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$ \vert\zeta \vert/{\muskip \z@ 5mu\divide \muskip \z@ \tw@ \mkern \muskip \z@ \...
...e}07E{\mkern -\muskip \z@ {\zeta }\mkern \muskip \z@ }\mkern -\muskip \z@ }{}_3$">|; 

$key = q/V(Q);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6554.png"
 ALT="$ V(Q)$">|; 

$key = q/displaystyleZ_2(x-c,t,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5098.png"
 ALT="$\displaystyle Z_2(x-c t,y)$">|; 

$key = q/2,alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2237.png"
 ALT="$ 2 \alpha$">|; 

$key = q/displaystyle=oint_C(phi,dx-psi,dy),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2345.png"
 ALT="$\displaystyle =\oint_C (\phi dx-\psi dy),$">|; 

$key = q/v_r(a,theta)=v_theta(a,theta)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3800.png"
 ALT="$ v_r(a,\theta)=v_\theta(a,\theta)=0$">|; 

$key = q/displaystylefrac{p}{rho}=frac{R,T}{M}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5120.png"
 ALT="$\displaystyle \frac{p}{\rho} = \frac{R T}{M}.$">|; 

$key = q/3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4217.png"
 ALT="$ 3/2$">|; 

$key = q/{bfL}=-k,v,v_y,{bfe}_x+k,v,v_x,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3555.png"
 ALT="$ {\bf L} = -k v v_y {\bf e}_x + k v v_x {\bf e}_y$">|; 

$key = q/zeta=r,{rme}^{,{rmi},theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2092.png"
 ALT="$ \zeta=r {\rm e}^{ {\rm i} \theta}$">|; 

$key = q/V_0leftrightarrowV_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img604.png"
 ALT="$ V_0\leftrightarrow V_1$">|; 

$key = q/displaystyley;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1941.png"
 ALT="$\displaystyle y$">|; 

$key = q/alpha_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1747.png"
 ALT="$ \alpha_m$">|; 

$key = q/dW={bfF}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6478.png"
 ALT="$ dW={\bf F}\cdot d{\bf r}$">|; 

$key = q/{rmFr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="$ {\rm Fr}$">|; 

$key = q/displaystylembox{boldmath{{xi{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4074.png"
 ALT="$\displaystyle \mbox{\boldmath$\xi$}$">|; 

$key = q/displaystylesimeq-Uleft(frac{dy}{dx}right)_{rmbody},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5949.png"
 ALT="$\displaystyle \simeq -U\left(\frac{dy}{dx}\right)_{\rm body},$">|; 

$key = q/displaystyled=cleft[frac{cos(pi,nslash2)+1}{|sin(pi,nslash2)|}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2486.png"
 ALT="$\displaystyle d = c\left[\frac{\cos(\pi n/2)+1}{\vert\sin(\pi n/2)\vert}\right].
$">|; 

$key = q/displaystyle=alphaleft[E(phi,k)+(1-k^{,2})^{1slash2},F(phi,k)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1321.png"
 ALT="$\displaystyle = \alpha\left[E(\phi,k) +(1-k^{ 2})^{1/2} F(\phi,k)\right].$">|; 

$key = q/G({rmFr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1447.png"
 ALT="$ G({\rm Fr})$">|; 

$key = q/displaystyle=U(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3077.png"
 ALT="$\displaystyle = U(x),$">|; 

$key = q/displaystyleint!int_Sf(x,y,z),dS=lim_{deltaS_irightarrow0}sum_if(x,y,z),deltaS_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6485.png"
 ALT="$\displaystyle \int\!\int_S f(x,y,z)  dS = \lim_{\delta S_i\rightarrow 0}\sum_i f(x,y,z)  \delta S_i$">|; 

$key = q/displaystyle=frac{{mitOmega}^{,2},a^{,2}}{3,g},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4541.png"
 ALT="$\displaystyle = \frac{{\mit\Omega}^{ 2} a^{ 2}}{3 g},$">|; 

$key = q/displaystyleequivnablatimes{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img531.png"
 ALT="$\displaystyle \equiv \nabla\times {\bf v}
$">|; 

$key = q/theta<pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1194.png"
 ALT="$ \theta &lt;\pi/2$">|; 

$key = q/V_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img688.png"
 ALT="$ V_1$">|; 

$key = q/displaystylepsi(xirightarrowinfty,eta)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2706.png"
 ALT="$\displaystyle \psi(\xi\rightarrow\infty,\eta)= 0.$">|; 

$key = q/Q>Q'>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2826.png"
 ALT="$ Q&gt;Q'&gt;0$">|; 

$key = q/displaystyle=a,omega,frac{cosh[k,(z+d)]}{sinh(k,d)},sin(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3970.png"
 ALT="$\displaystyle =a \omega \frac{\cosh[k (z+d)]}{\sinh(k d)} \sin(\omega t-k x),$">|; 

$key = q/{bfv}=nablaztimesnablapsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1654.png"
 ALT="$ {\bf v} = \nabla z\times \nabla\psi$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter06slashconf.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img2143.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter06/conf.eps}}
\end{figure}">|; 

$key = q/Wsim{calO}(alpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3527.png"
 ALT="$ W\sim{\cal O}(\alpha)$">|; 

$key = q/displaystylevarphi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4301.png"
 ALT="$\displaystyle \varphi'$">|; 

$key = q/h_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1465.png"
 ALT="$ h_1$">|; 

$key = q/displaystyle=frac{w}{2}left[coshleft(pi,frac{x}{w}right)cosleft(pi,frac{y}{w}right)-1right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2163.png"
 ALT="$\displaystyle =\frac{w}{2}\left[\cosh\left(\pi \frac{x}{w}\right)\cos\left(\pi \frac{y}{w}\right)-1\right],$">|; 

$key = q/displaystylepsi(xi,eta)=frac{1}{2},c^{,2},V,coseta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2853.png"
 ALT="$\displaystyle \psi(\xi,\eta) = \frac{1}{2} c^{ 2} V \cos\eta
$">|; 

$key = q/{bfb}={bfr}+mu,{bfp};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6731.png"
 ALT="$ {\bf b} = {\bf r}+\mu {\bf p}$">|; 

$key = q/U_0,deltaslashnu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2997.png"
 ALT="$ U_0 \delta/\nu$">|; 

$key = q/|{bfr}-{bfk}|^{-n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6751.png"
 ALT="$ \vert{\bf r}-{\bf k}\vert^{-n}$">|; 

$key = q/z=R,{rme}^{,{rmi},theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3279.png"
 ALT="$ z=R {\rm e}^{ {\rm i} \theta}$">|; 

$key = q/displaystylez=-frac{x}{h}+c,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7352.png"
 ALT="$\displaystyle z =-\frac{x}{h} + c,$">|; 

$key = q/displaystylevarpi^{,2}=b^{,2},(costheta_1-costheta_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2841.png"
 ALT="$\displaystyle \varpi^{ 2} = b^{ 2} (\cos\theta_1-\cos\theta_2),
$">|; 

$key = q/displaystyle=-D-2,rho,nu,frac{d}{dl}!int_{-h}^hu(y),dy,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3059.png"
 ALT="$\displaystyle = -D -2 \rho \nu \frac{d}{dl}\!\int_{-h}^h u(y)  dy,$">|; 

$key = q/{mitLambda}=lambda=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3202.png"
 ALT="$ {\mit\Lambda}=\lambda=0$">|; 

$key = q/_S({bfr}')times;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3462.png"
 ALT="$ _S({\bf r}')\times$">|; 

$key = q/rrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4361.png"
 ALT="$ r\rightarrow \infty$">|; 

$key = q/displaystyle({bfA}cdotnabla),{bfB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7081.png"
 ALT="$\displaystyle ({\bf A}\cdot\nabla)  {\bf B}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashvortex1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="335" BORDER="0"
 SRC="|."$dir".q|img1707.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/vortex1.eps}}
\end{figure}">|; 

$key = q/displaystyleU(theta)=V,(c+delta),frac{f(theta)}{h(theta)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3358.png"
 ALT="$\displaystyle U(\theta) = V (c+\delta) \frac{f(\theta)}{h(\theta)},$">|; 

$key = q/{bfa}={bfa}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6432.png"
 ALT="$ {\bf a} = {\bf a} (t)$">|; 

$key = q/dxslashdt=u-c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5663.png"
 ALT="$ dx/dt=u-c$">|; 

$key = q/displaystyle{calS}_n(theta,varphi)=sum_{m=0,n}left[c_n^{,m},P_n^{,m}(costheta),cos(m,varphi)+d_n^{,m},P_n^{,m}(costheta),sin(m,varphi)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="552" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4316.png"
 ALT="$\displaystyle {\cal S}_n(\theta,\varphi) = \sum_{m=0,n} \left[c_n^{ m} P_n^{\...
... \cos(m \varphi) + d_n^{ m} P_n^{ m}(\cos\theta) \sin(m \varphi)\right],$">|; 

$key = q/displaystyleintx_1,x_2,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img850.png"
 ALT="$\displaystyle \int x_1 x_2 dV$">|; 

$key = q/displaystyle=left[1+left(frac{gamma-1}{gamma+1}right)left({rmMa}^{,2}-1right)right]^{-1slash(gamma-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5406.png"
 ALT="$\displaystyle =\left[1+\left(\frac{\gamma-1}{\gamma+1}\right)\left({\rm Ma}^{ 2}-1\right)\right]^{-1/(\gamma-1)}.$">|; 

$key = q/c=-w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2159.png"
 ALT="$ c=-w$">|; 

$key = q/displaystyleint_0^inftyleft[frac{a_1^{,2},a_2^{,2}}{(a_1^{,2}+u),(a_2^{,2}+u)}-frac{a_3^{,2}}{(a_3^{,2}+u)}right]frac{du}{mitDelta}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="366" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img924.png"
 ALT="$\displaystyle \int_0^\infty\left[\frac{a_1^{ 2} a_2^{ 2}}{(a_1^{ 2}+u) (a_2^{ 2}+u)}- \frac{a_3^{ 2}}{(a_3^{ 2}+u)}\right]\frac{du}{\mit\Delta} = 0.$">|; 

$key = q/displaystylenablap=c^{,2},nablarho,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5882.png"
 ALT="$\displaystyle \nabla p= c^{ 2} \nabla\rho,$">|; 

$key = q/U=U_0,x^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3225.png"
 ALT="$ U=U_0 x^{ m}$">|; 

$key = q/displaystylev_y=frac{{mitGamma}}{4pi,z}int_0^{pislash2}cosphi,dphi=frac{{mitGamma}}{4pi,z}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3478.png"
 ALT="$\displaystyle v_y = \frac{{\mit\Gamma}}{4\pi z}\int_0^{\pi/2}\cos\phi d\phi = \frac{{\mit\Gamma}}{4\pi z}.$">|; 

$key = q/partialsigma_{ij}slashpartialx_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="$ \partial \sigma_{ij}/\partial x_j$">|; 

$key = q/displaystylephi(x,z,t)=A,{rme}^{,k,z},cos(omega,t-k,x)+B,{rme}^{-k,z},cos(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="458" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3919.png"
 ALT="$\displaystyle \phi(x,z,t) = A {\rm e}^{ k z} \cos(\omega t-k x) + B {\rm e}^{-k z} \cos(\omega t-k x),$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter15slashcone.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="341" BORDER="0"
 SRC="|."$dir".q|img5719.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter15/cone.eps}}
\end{figure}">|; 

$key = q/displaystyle=frac{1}{sintheta},frac{partialP'}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4749.png"
 ALT="$\displaystyle =\frac{1}{\sin\theta} \frac{\partial P'}{\partial\phi},$">|; 

$key = q/phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1201.png"
 ALT="$ \phi$">|; 

$key = q/displaystyledzeta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2070.png"
 ALT="$\displaystyle d\zeta'$">|; 

$key = q/A_1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5972.png"
 ALT="$ A_1=0$">|; 

$key = q/Z_2(x,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5101.png"
 ALT="$ Z_2(x,0)$">|; 

$key = q/(x,,y)rightarrow(xi,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6032.png"
 ALT="$ (x, y)\rightarrow (\xi,\eta)$">|; 

$key = q/X=costheta,eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4765.png"
 ALT="$ X=\cos\theta \eta$">|; 

$key = q/S_1,(p_1+rho,v_1^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1395.png"
 ALT="$ S_1 (p_1+\rho v_1^{ 2})$">|; 

$key = q/displaystyleint_0^1x,dxint_0^{1-x}y^{,2},dy=int_0^1frac{x}{3},(1-x)^3,dx=frac{1}{60}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6495.png"
 ALT="$\displaystyle \int_0^1 x dx \int_0^{1-x} y^{ 2} dy = \int_0^1\frac{x}{3}  (1-x)^3 dx = \frac{1}{60}.$">|; 

$key = q/displaystyledeltaI=int_a^bleft(frac{partialF}{partialy},deltay+frac{partialF}{partialy'},deltay'right)dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7309.png"
 ALT="$\displaystyle \delta I = \int_a^b\left(\frac{\partial F}{\partial y} \delta y+ \frac{\partial F}{\partial y'} \delta y'\right)dx,$">|; 

$key = q/z<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img568.png"
 ALT="$ z&lt;0$">|; 

$key = q/{bfv}=d{bfr}slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6333.png"
 ALT="$ {\bf v} = d{\bf r}/dt$">|; 

$key = q/displaystylefrac{rho}{rho_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5390.png"
 ALT="$\displaystyle \frac{\rho}{\rho_0}$">|; 

$key = q/6000,{rmkm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5119.png"
 ALT="$ 6000 {\rm km}$">|; 

$key = q/z=pm0.65tmspace+thinmuskip{.1667em}c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$ z=\pm 0.65\tmspace +\thinmuskip {.1667em}c$">|; 

$key = q/displaystyle=F(pislash2,k)-F(phi,k),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1179.png"
 ALT="$\displaystyle = F(\pi/2,k)- F(\phi,k),$">|; 

$key = q/displaystylesimeqmu_1+epsilon,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6118.png"
 ALT="$\displaystyle \simeq \mu_1+\epsilon,$">|; 

$key = q/displaystyleomega=nabla^{,2}psi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img539.png"
 ALT="$\displaystyle \omega = \nabla^{ 2}\psi.
$">|; 

$key = q/x=pmd;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1193.png"
 ALT="$ x=\pm d$">|; 

$key = q/-S,p_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1415.png"
 ALT="$ -S p_0$">|; 

$key = q/{mitPsi}rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5269.png"
 ALT="$ {\mit\Psi}\rightarrow 0$">|; 

$key = q/u=-V_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5558.png"
 ALT="$ u=-V_p$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter07slashsphere.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img2651.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter07/sphere.eps}}
\end{figure}">|; 

$key = q/displaystyleW=int_{infty}^0frac{-y,dy}{(a^{,2}+y^{,2})^{3slash2}}=left[frac{1}{(y^{,2}+a^{,2})^{1slash2}}right]^0_infty=frac{1}{a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6482.png"
 ALT="$\displaystyle W = \int_{\infty}^0 \frac{-y dy}{(a^{ 2} + y^{ 2})^{3/2}} = \left[\frac{1}{(y^{ 2}+a^{ 2})^{1/2}} \right]^0_\infty = \frac{1}{a}.$">|; 

$key = q/e_{13}>0.93858;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img965.png"
 ALT="$ e_{13}&gt;0.93858$">|; 

$key = q/0leqmuleqpislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5720.png"
 ALT="$ 0\leq\mu\leq \pi/2$">|; 

$key = q/displaystyle=a_i',b_{jk}'={calR}_{il},a_l,{calR}_{jm},{calR}_{kn},b_{mn}={calR}_{il},{calR}_{jm},{calR}_{kn},a_l,b_{mn};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="416" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6853.png"
 ALT="$\displaystyle = a_i' b_{jk}' = {\cal R}_{il} a_l {\cal R}_{jm} {\cal R}_{kn} b_{mn} = {\cal R}_{il} {\cal R}_{jm} {\cal R}_{kn} a_l b_{mn}$">|; 

$key = q/displaystyleH_L=H_1-H_2=h_1-h_2+frac{v_1^{,2}}{2,g}-frac{v_2^{,2}}{2,g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1503.png"
 ALT="$\displaystyle H_L= H_1-H_2 = h_1-h_2 +\frac{v_1^{ 2}}{2 g} -\frac{v_2^{ 2}}{2 g}$">|; 

$key = q/displaystyle=frac{left[1-gamma+2,gamma,{rmMa}_1^{,2},sin^2betaright]left[2+(gamm,2},sin^2betaright]}{(gamma+1)^{,2},{rmMa}_1^{,2},sin^2beta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="431" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5689.png"
 ALT="$\displaystyle = \frac{\left[1-\gamma+2 \gamma {\rm Ma}_1^{ 2} \sin^2\beta\r...
..._1^{ 2} \sin^2\beta\right]}{(\gamma+1)^{ 2} {\rm Ma}_1^{ 2} \sin^2\beta},$">|; 

$key = q/y'equivdyslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7299.png"
 ALT="$ y'\equiv dy/dx$">|; 

$key = q/{mitDelta}p=p-p_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6165.png"
 ALT="$ {\mit\Delta}p=p-p_1$">|; 

$key = q/displaystyleoint_Sa_{ij},dS_j=int_Vfrac{partiala_{ij}}{partialx_j},dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6894.png"
 ALT="$\displaystyle \oint_S a_{ij} dS_j = \int_V \frac{\partial a_{ij}}{\partial x_j} dV,$">|; 

$key = q/displaystylekappa_y=left(frac{int_{A_0}x^2,dx,dy}{A_0}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img656.png"
 ALT="$\displaystyle \kappa_y = \left(\frac{\int_{A_0}x^2 dx dy}{A_0}\right)^{1/2},$">|; 

$key = q/0leqthetaleqpislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3091.png"
 ALT="$ 0\leq\theta\leq \pi/2$">|; 

$key = q/displaystylefrac{1}{sintheta}left[frac{partial}{partialtheta},(sintheta,xi)+frac{partialeta}{partialphi}right]+zeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4654.png"
 ALT="$\displaystyle \frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta} (\sin\theta \xi)+ \frac{\partial\eta}{\partial\phi}\right] + \zeta$">|; 

$key = q/{bfa}times{bfc}={bfb}times{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6717.png"
 ALT="$ {\bf a}\times {\bf c} = {\bf b}\times {\bf c}$">|; 

$key = q/displaystyle=-left(frac{partial{calF}}{partialv}right)_{calT},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5588.png"
 ALT="$\displaystyle =- \left(\frac{\partial {\cal F}}{\partial v}\right)_{\cal T},$">|; 

$key = q/S_x={bfe}_xcdot{bfS}={bfe}_xcdot{bfn},S=cosalpha_x,S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6337.png"
 ALT="$ S_x={\bf e}_x\cdot {\bf S}={\bf e}_x\cdot{\bf n} S=\cos\alpha_x S$">|; 

$key = q/displaystylesimeqleft(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right)(theta_1'+theta_2'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6199.png"
 ALT="$\displaystyle \simeq \left(\frac{\gamma {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right)(\theta_1'+\theta_2'),$">|; 

$key = q/displaystyleskew{3}hat{beta}_{-n,,n'}^{,-m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5002.png"
 ALT="$\displaystyle \skew{3}\hat{\beta}_{-n, n'}^{ -m, m'}$">|; 

$key = q/C_{p0}slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6074.png"
 ALT="$ C_{p0}/2$">|; 

$key = q/displaystyleoverline{p}_0-p_0=frac{2,gamma}{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3863.png"
 ALT="$\displaystyle \overline{p}_0 - p_0 = \frac{2 \gamma}{a},$">|; 

$key = q/displaystylef''(infty);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2944.png"
 ALT="$\displaystyle f''(\infty)$">|; 

$key = q/{calR};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5291.png"
 ALT="$ {\cal R}$">|; 

$key = q/displaystyles_pm=frac{1pmsqrt{1-c_0^{,2}slashc^{,2}}}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img682.png"
 ALT="$\displaystyle s_\pm = \frac{1\pm \sqrt{1-c_0^{ 2}/c^{ 2}}}{2},$">|; 

$key = q/Dsimeq(2,L_0slashA),alpha^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3523.png"
 ALT="$ D\simeq (2 L_0/A) \alpha^{ 2}$">|; 

$key = q/{rmMa}_1geq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5501.png"
 ALT="$ {\rm Ma}_1\geq 0$">|; 

$key = q/displaystylef_i=int_VF_i,dV+oint_Ssigma_{ij},dS_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$\displaystyle f_i = \int_V F_i dV + \oint_S \sigma_{ij} dS_j,$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter11slashdeep.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="257" BORDER="0"
 SRC="|."$dir".q|img3947.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter11/deep.eps}}
\end{figure}">|; 

$key = q/h_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4444.png"
 ALT="$ h_n$">|; 

$key = q/phi(x,y,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1651.png"
 ALT="$ \phi(x,y,t)$">|; 

$key = q/0leqthetaleq2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3485.png"
 ALT="$ 0\leq \theta\leq 2\pi$">|; 

$key = q/displaystylembox{boldmath{{theta{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6400.png"
 ALT="$\displaystyle \mbox{\boldmath$\theta$}$">|; 

$key = q/displaystyle=7.32times10^3,{rmm},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4579.png"
 ALT="$\displaystyle = 7.32\times 10^3 {\rm m},$">|; 

$key = q/|V|<|c|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4131.png"
 ALT="$ \vert V\vert&lt;\vert c\vert$">|; 

$key = q/Tsim{calO}(alpha^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3526.png"
 ALT="$ T\sim{\cal O}(\alpha^{ 2})$">|; 

$key = q/displaystyle=frac{G,m',a^{2}}{R^{,3}}left[frac{1}{2}left(1+3,e,cosleft[(omega_1-n^2delta,cos[2,omega_1,t]right){calR}_2^{,(0)}(r,theta)right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="576" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4303.png"
 ALT="$\displaystyle =\frac{G m' a^{2}}{R^{ 3}}\left[\frac{1}{2}\left(1+3 e \cos\...
...} \sin^2\delta \cos[2 \omega_1 t]\right){\cal R}_2^{ (0)}(r,\theta)\right.$">|; 

$key = q/{bfb}equiv(1slash!sqrt{2},,-1slash!sqrt{2},,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6298.png"
 ALT="$ {\bf b} \equiv (1/\!\sqrt{2}, -
1/\!\sqrt{2},  0)$">|; 

$key = q/lambda_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4198.png"
 ALT="$ \lambda_c$">|; 

$key = q/BP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2830.png"
 ALT="$ BP$">|; 

$key = q/beta_1'-beta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6167.png"
 ALT="$ \beta_1'-\beta_1$">|; 

$key = q/displaystyle=a,(2,s+tantheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img743.png"
 ALT="$\displaystyle = a (2 s+\tan\theta).$">|; 

$key = q/displaystylefrac{d}{dx}left(rho,u,{calE}+p,u+frac{1}{2},rho,u^{,3}right)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5458.png"
 ALT="$\displaystyle \frac{d}{dx}\left(\rho u {\cal E} + p u+\frac{1}{2} \rho u^{ 3}\right)=0,$">|; 

$key = q/rhoslashskew{3}bar{rho}=5.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4713.png"
 ALT="$ \rho/\skew{3}\bar{\rho}=5.5$">|; 

$key = q/displaystyleU(theta)=2,U_0,sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3090.png"
 ALT="$\displaystyle U(\theta) = 2 U_0 \sin\theta,$">|; 

$key = q/displaystylel=left(frac{gamma}{rho,g}right)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1120.png"
 ALT="$\displaystyle l = \left(\frac{\gamma}{\rho g}\right)^{1/2}$">|; 

$key = q/displaystyleX-{rmi},Y=frac{1}{2},{rmi},rhooint_{C'}left(frac{dF}{dz}right)^{,2},Q}{pi}right)^2oint_{C'}frac{z^{,2}}{(z^{,2}+a^{,2})^{,2}},dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2415.png"
 ALT="$\displaystyle X - {\rm i} Y = \frac{1}{2} {\rm i} \rho\oint_{C'}\left(\frac{...
...ft(\frac{Q}{\pi}\right)^2\oint_{C'}\frac{z^{ 2}}{(z^{ 2}+a^{ 2})^{ 2}} dz,$">|; 

$key = q/displaystyle=f(x-lambda,y)=frac{U,epsilon}{lambda},sin[alpha,(x-lambda,y)],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5994.png"
 ALT="$\displaystyle = f(x-\lambda y) = \frac{U \epsilon}{\lambda} \sin[\alpha (x-\lambda y)],$">|; 

$key = q/{rmtr}(A)=A_{ii};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7007.png"
 ALT="$ {\rm tr}(A) = A_{ii}$">|; 

$key = q/displaystyle=(4,nu-lambda-mu)slash10.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6976.png"
 ALT="$\displaystyle = (4 \nu-\lambda-\mu)/10.$">|; 

$key = q/{mitPhi}_{r'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4816.png"
 ALT="$ {\mit\Phi}_{r'}$">|; 

$key = q/displaystyle=frac{1}{3},c-frac{1}{9},b^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6094.png"
 ALT="$\displaystyle = \frac{1}{3} c - \frac{1}{9} b^{ 2},$">|; 

$key = q/lambda_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4823.png"
 ALT="$ \lambda_1$">|; 

$key = q/displaystyler-r_0=left.frac{dr}{dp}right|_{H_0},(p-p_0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img720.png"
 ALT="$\displaystyle r-r_0=\left. \frac{dr}{dp}\right\vert _{H_0} (p-p_0),$">|; 

$key = q/displaystylezeta(x)proptocosleft(pi,frac{x}{l}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3997.png"
 ALT="$\displaystyle \zeta(x)\propto \cos\left(\pi \frac{x}{l}\right).$">|; 

$key = q/displaystyle=v',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5897.png"
 ALT="$\displaystyle = v',$">|; 

$key = q/c=sqrt{gamma,pslashrho};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5364.png"
 ALT="$ c=\sqrt{\gamma p/\rho}$">|; 

$key = q/|delta_1|lldelta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img612.png"
 ALT="$ \vert\delta_1\vert\ll \delta_0$">|; 

$key = q/displaystylefrac{rho}{rho_c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5205.png"
 ALT="$\displaystyle \frac{\rho}{\rho_c}$">|; 

$key = q/displaystyle=-frac{Q}{2pi},lnr,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1978.png"
 ALT="$\displaystyle =-\frac{Q}{2\pi} \ln r,$">|; 

$key = q/F_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3054.png"
 ALT="$ F_x$">|; 

$key = q/displaystylefrac{alpha}{u}+frac{beta}{v}+frac{gamma}{w}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6729.png"
 ALT="$\displaystyle \frac{\alpha}{u} +\frac{\beta}{v} + \frac{\gamma}{w} = 0.
$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{AppendixAslashfigA.26.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="282" BORDER="0"
 SRC="|."$dir".q|img6648.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{AppendixA/figA.26.eps}}
\end{figure}">|; 

$key = q/Lslash(q_1,c);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5856.png"
 ALT="$ L/(q_1 c)$">|; 

$key = q/deltallx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3018.png"
 ALT="$ \delta\ll x$">|; 

$key = q/pm1.3,X_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4224.png"
 ALT="$ \pm 1.3 X_1)$">|; 

$key = q/displaystyle=frac{1}{12},a^{,4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6515.png"
 ALT="$\displaystyle = \frac{1}{12} a^{ 4}.$">|; 

$key = q/u_2slashc_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5698.png"
 ALT="$ u_2/c_2$">|; 

$key = q/displaystyle{bfA}=A_r,{bfe}_r+A_theta,{bfe}_theta+A_z,{bfe}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7109.png"
 ALT="$\displaystyle {\bf A} = A_r {\bf e}_r+ A_\theta {\bf e}_\theta + A_z {\bf e}_z,$">|; 

$key = q/D=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3778.png"
 ALT="$ D=0$">|; 

$key = q/l=2pislashalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5951.png"
 ALT="$ l=2\pi/\alpha$">|; 

$key = q/displaystylephantom{=}left.+frac{1}{2}left(frac{partialv_x}{partialz}+frac{partirtialv_y}{partialz}+frac{partialv_z}{partialy}right)^2right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="$\displaystyle \phantom{=}\left.+\frac{1}{2}\left(\frac{\partial v_x}{\partial z...
...frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right)^2\right].$">|; 

$key = q/m^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1736.png"
 ALT="$ m^{ 2}$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi}-frac{partialphi}{partialt}=C,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1593.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} + {\mit\Psi} - \frac{\partial\phi}{\partial t} = C,$">|; 

$key = q/r_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1347.png"
 ALT="$ r_-$">|; 

$key = q/{mitGamma},{mitGamma}'<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2440.png"
 ALT="$ {\mit\Gamma} {\mit\Gamma}'&lt;0$">|; 

$key = q/phi({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3904.png"
 ALT="$ \phi({\bf r},t)$">|; 

$key = q/displaystylep_1=rho,frac{partialphi}{partialt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3905.png"
 ALT="$\displaystyle p_1 = \rho \frac{\partial \phi}{\partial t}.$">|; 

$key = q/pislash4leqphileq3pislash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$ \pi /4\leq \phi \leq 3\pi /4$">|; 

$key = q/p(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5441.png"
 ALT="$ p(x)$">|; 

$key = q/displaystylefrac{partial(rho,v_i)}{partialt}+frac{partial(rho,v_i,v_j)}{partialx_j}=F_i+frac{partialsigma_{ij}}{partialx_j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="$\displaystyle \frac{\partial (\rho v_i)}{\partial t} + \frac{\partial(\rho v_i v_j)}{\partial x_j} = F_i + \frac{\partial\sigma_{ij}}{\partial x_j}$">|; 

$key = q/displaystylebeta=mu+epsilon,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5731.png"
 ALT="$\displaystyle \beta=\mu+\epsilon,$">|; 

$key = q/r^{,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2592.png"
 ALT="$ r^{ l}$">|; 

$key = q/displaystylea_i={calR}_{ji},a_j'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6828.png"
 ALT="$\displaystyle a_i = {\cal R}_{ji} a_j'.$">|; 

$key = q/displaystyle=sum_{n=1,infty}^{n,{rmeven}}sum_{m=0,n}zeta_n^{,m},{mitPhi}_n^m(theta,phi),cos(n,pislash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5024.png"
 ALT="$\displaystyle = \sum_{n=1,\infty}^{n {\rm even}}\sum_{m=0,n}\zeta_n^{ m} {\mit\Phi}_n^m(\theta,\phi) \cos(n \pi/2),$">|; 

$key = q/displaystyle=(1-e_{13}^{,2})^{1slash2},e_{13}^{,2}int_0^inftyfrac{lambda,dlambda}{(1+lambda)^2,(1+lambda-e_{13}^{,2})^{3slash2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img879.png"
 ALT="$\displaystyle =(1-e_{13}^{ 2})^{1/2} e_{13}^{ 2}\int_0^\infty \frac{\lambda d\lambda}{(1+\lambda)^2 (1+\lambda -e_{13}^{ 2})^{3/2}}$">|; 

$key = q/displaystylez^{,2}+varpi^{,2}=r^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2728.png"
 ALT="$\displaystyle z^{ 2}+\varpi^{ 2} = r^{ 2},$">|; 

$key = q/b=0.6416,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1267.png"
 ALT="$ b=0.6416 c$">|; 

$key = q/displaystyle{mitGamma}_{n,n'}^{,m,m'}=(a_n^{,m})^{-1}left(delta_{n,n'}^{,m,m'}+{rmi},kappa_{n,n'}^{,m,m'}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4981.png"
 ALT="$\displaystyle {\mit\Gamma}_{n,n'}^{ m,m'}=(a_n^{ m})^{-1}\left(\delta_{n,n'}^{ m,m'}+{\rm i} \kappa_{n,n'}^{ m,m'}\right),$">|; 

$key = q/displaystylezeta_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4580.png"
 ALT="$\displaystyle \zeta_M$">|; 

$key = q/displaystylenabla^{,2}phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1968.png"
 ALT="$\displaystyle \nabla^{ 2}\phi$">|; 

$key = q/tanh^2alpha=2slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3241.png"
 ALT="$ \tanh^2\alpha =2/3$">|; 

$key = q/J=J';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2373.png"
 ALT="$ J=J'$">|; 

$key = q/displaystyle=-int_Vrho,g,dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img583.png"
 ALT="$\displaystyle =-\int_V \rho g dV.$">|; 

$key = q/V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$ V$">|; 

$key = q/displaystyletheta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6200.png"
 ALT="$\displaystyle \theta_2$">|; 

$key = q/displaystylefrac{v_x(x,y)}{U_0}=1-frac{Q}{U_0,delta},frac{1}{sqrt{pi}},exp(-y^{,2}slashdelta^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3032.png"
 ALT="$\displaystyle \frac{v_x(x,y)}{U_0} = 1 - \frac{Q}{U_0 \delta} \frac{1}{\sqrt{\pi}} \exp(-y^{ 2}/\delta^{ 2}),$">|; 

$key = q/displaystyleD=rho,U_0,Q.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3060.png"
 ALT="$\displaystyle D = \rho U_0 Q.$">|; 

$key = q/nabla^2phi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3273.png"
 ALT="$ \nabla^2\phi=0$">|; 

$key = q/displaystyleepsilonsimeqfrac{5}{4},m.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img911.png"
 ALT="$\displaystyle \epsilon \simeq \frac{5}{4} m.$">|; 

$key = q/displaystylezeta=a,sin(k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1343.png"
 ALT="$\displaystyle \zeta = a \sin(k x),
$">|; 

$key = q/displaystyledeltap(x,0)=-frac{rho}{2},left|frac{dF}{dz}right|^{,2}_{y=0}=-frac{ri,a}right)^2frac{x^{,2}slasha^{,2}}{(1+x^{,2}slasha^{,2})^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2029.png"
 ALT="$\displaystyle \delta p(x,0) = -\frac{\rho}{2} \left\vert\frac{dF}{dz}\right\ve...
...2}\left(\frac{Q}{\pi a}\right)^2\frac{x^{ 2}/a^{ 2}}{(1+x^{ 2}/a^{ 2})^2}.$">|; 

$key = q/zeta_{b,n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4342.png"
 ALT="$ \zeta_{b n}$">|; 

$key = q/displaystylefrac{dF}{dz}=frac{dF}{dzeta},frac{dzeta}{dz}=left[V,{rme}^{,{rmi},alzeta^{,2}}right]left(frac{zeta^{,2}}{zeta^{,2}-l^{,2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="478" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3311.png"
 ALT="$\displaystyle \frac{dF}{dz} = \frac{dF}{d\zeta} \frac{d\zeta}{dz} = \left[V {...
...lpha}}{\zeta^{ 2}}\right]\left(\frac{\zeta^{ 2}}{\zeta^{ 2}-l^{ 2}}\right).$">|; 

$key = q/y_1=y(xi_1)=2.018;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5223.png"
 ALT="$ y_1=y(\xi_1) = 2.018$">|; 

$key = q/displaystylefrac{d{mitGamma}_C}{dt}=-oint_Cnabla!left(frac{p}{rho}-frac{1}{2},v^{,2}+{mitPsi}right)cdotd{bfr}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1558.png"
 ALT="$\displaystyle \frac{d{\mit\Gamma}_C}{dt} =- \oint_C\nabla\!\left(\frac{p}{\rho}- \frac{1}{2} v^{ 2}+{\mit\Psi}\right)\cdot d{\bf r}= 0,$">|; 

$key = q/displaystyleu_1=sqrt{{calR},T_0left(frac{2}{gamma+1}right)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5431.png"
 ALT="$\displaystyle u_1=\sqrt{{\cal R} T_0\left(\frac{2}{\gamma+1}\right)}.$">|; 

$key = q/mu=7.50times10^{-6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1029.png"
 ALT="$ \mu=7.50\times 10^{-6}$">|; 

$key = q/displaystyleeta=left[frac{(m+1),U_0,x^{,m-1}}{2,nu}right]^{1slash2}y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2919.png"
 ALT="$\displaystyle \eta =\left[\frac{(m+1) U_0 x^{ m-1}}{2 \nu}\right]^{1/2}y.$">|; 

$key = q/displaystyle=-frac{1}{2},rhooint_Cleft|frac{dF}{dz}right|^{,2}(x,dx+y,dy).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2396.png"
 ALT="$\displaystyle =-\frac{1}{2} \rho\oint_C \left\vert\frac{dF}{dz}\right\vert^{ 2}(x dx+y dy).$">|; 

$key = q/displaystyledeltar=frac{R}{r},deltap,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img711.png"
 ALT="$\displaystyle \delta r = \frac{R}{r} \delta p,$">|; 

$key = q/displaystyler'^{,2}+r''^{,2}=(r'+r'')^2-2,r',r''=frac{B^{,2}}{A^2}-2,frac{C}{A},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7248.png"
 ALT="$\displaystyle r'^{ 2}+r''^{ 2} = (r'+r'')^2- 2 r' r'' = \frac{B^{ 2}}{A^2} - 2 \frac{C}{A},$">|; 

$key = q/displaystyleskew{5}ddot{xi}-2,{mitOmega},costheta,skew{3}dot{eta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4655.png"
 ALT="$\displaystyle \skew{5}\ddot{\xi} -2 {\mit\Omega} \cos\theta \skew{3}\dot{\eta}$">|; 

$key = q/displaystyle2,(2,n+1),A_n,mu+2,B_n,mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4425.png"
 ALT="$\displaystyle 2 (2 n+1) A_n \mu + 2 B_n \mu$">|; 

$key = q/displaystyle{bfa}cdot{bfb}equiva_x,b_x+a_y,b_y+a_z,b_z={rmscalar~number}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6300.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b} \equiv a_x  b_x + a_y  b_y + a_z  b_z = {\rm scalar&nbsp;number}.$">|; 

$key = q/rho,S_1,v_1^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1385.png"
 ALT="$ \rho S_1 v_1^{ 2}$">|; 

$key = q/displaystyle{calG}({rmFr}_1),{calG}({rmFr}_2)=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1491.png"
 ALT="$\displaystyle {\cal G}({\rm Fr}_1) {\cal G}({\rm Fr}_2)= 1,$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.08.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="196" BORDER="0"
 SRC="|."$dir".q|img6371.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.08.eps}}
\end{figure}">|; 

$key = q/displaystyle=frac{1}{r},frac{partial(r,v_r)}{partialr}+frac{1}{r},frac{partialv_theta}{partialtheta}+frac{partialv_z}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="$\displaystyle =\frac{1}{r} \frac{\partial (r v_r)}{\partial r} +\frac{1}{r}  \frac{\partial v_\theta}{\partial \theta} + \frac{\partial v_z}{\partial z},$">|; 

$key = q/1.4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$ 1.4$">|; 

$key = q/{mitDelta}u=left.uright|_{z=d}-left.uright|_{z=0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4501.png"
 ALT="$ {\mit\Delta} u = \left.u\right\vert _{z=d}-\left.u\right\vert _{z=0}$">|; 

$key = q/Vrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$ V\rightarrow 0$">|; 

$key = q/B=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2201.png"
 ALT="$ B=1$">|; 

$key = q/displaystylea_{ikj}=a_{jik}=a_{kji}=-a_{ijk}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6933.png"
 ALT="$\displaystyle a_{ikj} = a_{jik}=a_{kji} = -a_{ijk}.$">|; 

$key = q/x=x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3669.png"
 ALT="$ x=x_1$">|; 

$key = q/x_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7003.png"
 ALT="$ x_j$">|; 

$key = q/r=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1714.png"
 ALT="$ r=a$">|; 

$key = q/beta=2,mslash(m+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2959.png"
 ALT="$ \beta=2 m/(m+1)$">|; 

$key = q/displaystylea_n^{,m}=left{{array}{lll}(4slashlambda_n^{,m})^{1slash2}&mbox{hspac&m>0[0.5ex](2slashlambda_n^{,m})^{1slash2}&&m=0{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4939.png"
 ALT="$\displaystyle a_n^{ m} =\left\{\begin{array}{lll} (4/\lambda_n^{ m})^{1/2}&amp;\m...
...space{0.5cm}}&amp; m&gt;0\ [0.5ex] (2/\lambda_n^{ m})^{1/2}&amp;&amp;m=0 \end{array}\right..$">|; 

$key = q/H>(3slash2),(Q^{,2}slashg)^{1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1434.png"
 ALT="$ H&gt;(3/2) (Q^{ 2}/g)^{1/3}$">|; 

$key = q/d{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1550.png"
 ALT="$ d{\bf r}$">|; 

$key = q/displaystyleF=-frac{Q}{pi},lnzeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2166.png"
 ALT="$\displaystyle F= -\frac{Q}{\pi} \ln \zeta$">|; 

$key = q/displaystylec_n^{,m}=a_n^{,m},b_n^{,m}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4965.png"
 ALT="$\displaystyle c_n^{ m}= a_n^{ m} b_n^{ m}.$">|; 

$key = q/aslashr'll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4250.png"
 ALT="$ a/r' \ll 1$">|; 

$key = q/AA';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4011.png"
 ALT="$ AA'$">|; 

$key = q/displaystyleF_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1781.png"
 ALT="$\displaystyle F_x$">|; 

$key = q/displaystyle2,a_{sis}+a_{iss}+a_{ssi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6928.png"
 ALT="$\displaystyle 2 a_{sis} + a_{iss} + a_{ssi}$">|; 

$key = q/displaystyleQ=1.33,(nu,U_0,L)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3061.png"
 ALT="$\displaystyle Q = 1.33 (\nu U_0 L)^{1/2},$">|; 

$key = q/displaystylephi(x.y)=f(x-lambda,y)+g(x+lambda,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6046.png"
 ALT="$\displaystyle \phi(x.y) = f(x-\lambda y)+g(x+\lambda x),$">|; 

$key = q/displaystyle={bfe}_varphicdotnabla^{,2}(psi,nablavarphi)=nabla^{,2}left(frac{psi}{varpi}right)-frac{psi}{varpi^3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2567.png"
 ALT="$\displaystyle = {\bf e}_\varphi\cdot\nabla^{ 2}(\psi \nabla\varphi)= \nabla^{ 2}\left(\frac{\psi}{\varpi}\right)-\frac{\psi}{\varpi^3}$">|; 

$key = q/displaystyle=epsilon_{ijk},(delta_{li}-deltatheta_a,epsilon_{ali}),(delta_{mj}-dheta_b,epsilon_{bmj}),(delta_{nk}-deltatheta_c,epsilon_{cnk});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6835.png"
 ALT="$\displaystyle = \epsilon_{ijk} (\delta_{li}-\delta\theta_a \epsilon_{ali}) (...
...}-\delta\theta_b \epsilon_{bmj}) (\delta_{nk}-\delta\theta_c \epsilon_{cnk})$">|; 

$key = q/displaystyle{rmMa}_1^{,2},sin^2beta-1simeqleft[left(frac{gamma+1}{2}right){rmMa}_1^{,2},tanbetaright]theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5715.png"
 ALT="$\displaystyle {\rm Ma}_1^{ 2} \sin^2\beta-1 \simeq \left[\left(\frac{\gamma+1}{2}\right){\rm Ma}_1^{ 2} \tan\beta\right]\theta.$">|; 

$key = q/displaystyle=chi+frac{kappa,M}R,nabla^{,2}!left(frac{p}{rho}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="$\displaystyle = \chi + \frac{\kappa M}R \nabla^{ 2}\!\left(\frac{p}{\rho}\right),$">|; 

$key = q/c=sqrt{gamma,{calR},T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5381.png"
 ALT="$ c=\sqrt{\gamma {\cal R} T}$">|; 

$key = q/displaystylezeta(x,y,t)=Z_1(x+c,t,y)+Z_2(x-c,t,y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5095.png"
 ALT="$\displaystyle \zeta(x,y,t)= Z_1(x+c t,y)+Z_2(x-c t,y),
$">|; 

$key = q/displaystyle=Vleft(1-frac{a^{,3}}{r^{,3}}right)costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2646.png"
 ALT="$\displaystyle = V\left(1-\frac{a^{ 3}}{r^{ 3}}\right)\cos\theta,$">|; 

$key = q/h+deltah;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3602.png"
 ALT="$ h+\delta h$">|; 

$key = q/displaystylec=frac{V',s}{1+s}pmleft[c_1^{,2}-frac{s,V'^{,2}}{(1+s)^2}right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4199.png"
 ALT="$\displaystyle c = \frac{V' s}{1+s} \pm\left[c_1^{ 2}- \frac{s V'^{ 2}}{(1+s)^2}\right]^{1/2}.$">|; 

$key = q/{calC}_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5325.png"
 ALT="$ {\cal C}_p$">|; 

$key = q/displaystyle=int_{S}left(frac{partialphi}{partialx}-frac{partialpsi}{partialy}right)dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2371.png"
 ALT="$\displaystyle =\int_{S}\left(\frac{\partial\phi}{\partial x} - \frac{\partial\psi}{\partial y}\right)dS,$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partialf(theta,phi_pm)}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4799.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial f(\theta,\phi_\pm)}{\partial \phi}$">|; 

$key = q/displaystyleC_psimeq-frac{2,u'}{U}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5933.png"
 ALT="$\displaystyle C_p\simeq -\frac{2 u'}{U}.$">|; 

$key = q/displaystylepsi(r,theta)=-frac{Q}{2pi},theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1674.png"
 ALT="$\displaystyle \psi(r,\theta) = -\frac{Q}{2\pi} \theta.$">|; 

$key = q/displaystyle=a_{ssk},delta_{ij}+a_{sjs},delta_{ik},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6916.png"
 ALT="$\displaystyle =a_{ssk} \delta_{ij} + a_{sjs} \delta_{ik},$">|; 

$key = q/=omega_z,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1648.png"
 ALT="$ = \omega_z {\bf e}_z$">|; 

$key = q/displaystylephi(x,z,t)=A,{rme}^{,k,z},cos(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3936.png"
 ALT="$\displaystyle \phi(x,z,t) = A {\rm e}^{ k z} \cos(\omega t-k x).$">|; 

$key = q/displaystyleC_f=frac{f_x}{f_y}=frac{f_x'}{f_y'}=frac{4}{3},frac{d_0}{l},H(k),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3720.png"
 ALT="$\displaystyle C_f = \frac{f_x}{f_y}=\frac{f_x'}{f_y'} = \frac{4}{3} \frac{d_0}{l} H(k),$">|; 

$key = q/displaystyle=V_x(t),frac{a^{,2}}{r^{,2}},costheta+V_y(t),frac{a^{,.2}}{r^{,2}},sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1831.png"
 ALT="$\displaystyle = V_x(t) \frac{a^{ 2}}{r^{ 2}} \cos\theta+V_y(t) \frac{a^{ .2}}{r^{ 2}} \sin\theta,$">|; 

$key = q/displaystyle{rme}^{,{rmi},theta}=costheta+{rmi},sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1938.png"
 ALT="$\displaystyle {\rm e}^{ {\rm i} \theta}= \cos\theta+{\rm i} \sin\theta,$">|; 

$key = q/F(zeta)=F[f(z)]=G(z)=widetilde{phi}(x,y)+{rmi},widetilde{psi}(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2086.png"
 ALT="$ F(\zeta)=F[f(z)] = G(z) = \widetilde{\phi}(x,y) + {\rm i} \widetilde{\psi}(x,y)$">|; 

$key = q/displaystyle=frac{1}{2},frac{m'}{m},frac{a^{,4}}{R^{,3}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4543.png"
 ALT="$\displaystyle =\frac{1}{2} \frac{m'}{m} \frac{a^{ 4}}{R^{ 3}}.$">|; 

$key = q/v(x,0,t)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5094.png"
 ALT="$ v(x,0,t)=0$">|; 

$key = q/sigmall|f_0|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5112.png"
 ALT="$ \sigma\ll \vert f_0\vert$">|; 

$key = q/h=delta_0^{,2},A_0slash2,V_0=s,cslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img667.png"
 ALT="$ h=\delta_0^{ 2} A_0/2 V_0=s c/2$">|; 

$key = q/displaystyle=muleft(frac{1}{r,sintheta},frac{partialv_r}{partialvarphi}+frac{partialv_varphi}{partialr}-frac{v_varphi}{r}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img499.png"
 ALT="$\displaystyle =\mu\left(\frac{1}{r \sin\theta} \frac{\partial v_r}{\partial \varphi} + \frac{\partial v_\varphi}{\partial r}-\frac{v_\varphi}{r}\right),$">|; 

$key = q/displaystyle{mitDelta}v_zequivv_z(y=0_+)-v_z(y=0_-)={mitOmega}_x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3381.png"
 ALT="$\displaystyle {\mit\Delta} v_z \equiv v_z(y=0_+)-v_z(y=0_-)={\mit\Omega}_x.$">|; 

$key = q/B=rho_0,V_0,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img607.png"
 ALT="$ B = \rho_0 V_0 g$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashwalz.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img3205.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/walz.eps}}
\end{figure}">|; 

$key = q/displaystyleV'=c_m=(1+s)^{1slash2},c_1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4195.png"
 ALT="$\displaystyle V' = c_m = (1+s)^{1/2} c_1.$">|; 

$key = q/dbar{F}slashdz=-{rmi},V,partialskew{3}bar{phi}slashpartialy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2297.png"
 ALT="$ d\bar{F}/dz=-{\rm i} V \partial\skew{3}\bar{\phi}/\partial y$">|; 

$key = q/displaystyleoint_C({bfg}times{bfb})cdot{bft},dr=int_Snablatimes({bfg}times{bfb})cdot{bfn},dS.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1097.png"
 ALT="$\displaystyle \oint_C ({\bf g}\times {\bf b})\cdot {\bf t} dr= \int_S \nabla\times({\bf g}\times{\bf b})\cdot {\bf n} dS.$">|; 

$key = q/x_Lleqxleqx_R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5824.png"
 ALT="$ x_L\leq x\leq x_R$">|; 

$key = q/deltay=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7314.png"
 ALT="$ \delta y=0$">|; 

$key = q/hslashHll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1461.png"
 ALT="$ h/H\ll 1$">|; 

$key = q/v_p=c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4110.png"
 ALT="$ v_p=c$">|; 

$key = q/delta=epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4582.png"
 ALT="$ \delta= \epsilon$">|; 

$key = q/M_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5232.png"
 ALT="$ M_\ast$">|; 

$key = q/(dslash2)(-cosalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2447.png"
 ALT="$ (d/2)(-\cos\alpha$">|; 

$key = q/{bfr}=(x,,y,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6748.png"
 ALT="$ {\bf r}=(x, y, z)$">|; 

$key = q/-12leq{mitLambda}leq12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3172.png"
 ALT="$ -12\leq {\mit\Lambda}\leq 12$">|; 

$key = q/M_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$ M_f$">|; 

$key = q/displaystyle=left(frac{3}{2,n+1}right)frac{skew{3}bar{rho}}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4432.png"
 ALT="$\displaystyle = \left(\frac{3}{2 n+1}\right)\frac{\skew{3}\bar{\rho}}{\rho},$">|; 

$key = q/c_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5635.png"
 ALT="$ c_4$">|; 

$key = q/displaystyle(nabla^{,2}{bfv})_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="$\displaystyle (\nabla^{ 2}{\bf v})_i$">|; 

$key = q/displaystyle=int_S(nablatimes{bfA})_z,dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2357.png"
 ALT="$\displaystyle =\int_S (\nabla\times {\bf A})_z dS,$">|; 

$key = q/|V|>|c|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4130.png"
 ALT="$ \vert V\vert&gt; \vert c\vert$">|; 

$key = q/displaystylesimeqleft(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right)(theta_1+theta_2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6195.png"
 ALT="$\displaystyle \simeq \left(\frac{\gamma {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right)(\theta_1+\theta_2),$">|; 

$key = q/displaystyle=frac{p}{p_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2884.png"
 ALT="$\displaystyle = \frac{p}{p_0},$">|; 

$key = q/-M_0,omega^{,2},r_H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img826.png"
 ALT="$ -M_0 \omega^{ 2} r_H$">|; 

$key = q/1-s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img606.png"
 ALT="$ 1-s$">|; 

$key = q/displaystyle=(pislash2-1),alpha^{,-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3265.png"
 ALT="$\displaystyle =(\pi/2-1) \alpha^{ -1},$">|; 

$key = q/displaystyleleft[(2,A_n+B_n),r^{,2},p_n+n,phi_nright]_{r=a}=a,zeta_{b,n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4418.png"
 ALT="$\displaystyle \left[(2 A_n+B_n) r^{ 2} p_n + n \phi_n\right]_{r=a} = a \zeta_{b n},$">|; 

$key = q/displaystyle{bfA}=sum_{i=1,3}A_i,{bfe}_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7040.png"
 ALT="$\displaystyle {\bf A} = \sum_{i=1,3} A_i {\bf e}_i,$">|; 

$key = q/exp(-{rmi},m,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1738.png"
 ALT="$ \exp(-{\rm i} m \theta)$">|; 

$key = q/displaystylephi(r,theta,varphi)=-V,r,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2638.png"
 ALT="$\displaystyle \phi(r,\theta,\varphi)= -V r \cos\theta$">|; 

$key = q/displaystyleF(theta,phi_pm)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4832.png"
 ALT="$\displaystyle F(\theta,\phi_\pm) =0.$">|; 

$key = q/oint{bfv}cdotd{bfr}slashdS;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6653.png"
 ALT="$ \oint {\bf v}\cdot d{\bf r}/ dS$">|; 

$key = q/S_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7143.png"
 ALT="$ S_{ij}$">|; 

$key = q/displaystylenurightarrownu_{rmmax}equivfrac{pi}{2}left[left(frac{gamma+1}{gamma-1}right)^{1slash2}-1right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6150.png"
 ALT="$\displaystyle \nu\rightarrow\nu_{\rm max} \equiv \frac{\pi}{2}\left[\left(\frac{\gamma+1}{\gamma-1}\right)^{1/2}-1\right],
$">|; 

$key = q/displaystylefrac{1}{rho},frac{Drho}{Dt}=frac{Dlnrho}{Dt}=-nablacdot{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="$\displaystyle \frac{1}{\rho} \frac{D\rho}{Dt}= \frac{D\ln\rho}{Dt}=-\nabla\cdot{\bf v},$">|; 

$key = q/displaystylef^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1297.png"
 ALT="$\displaystyle f^{ 2}$">|; 

$key = q/exp(,{rmi},sigma_j,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4924.png"
 ALT="$ \exp( {\rm i} \sigma_j t)$">|; 

$key = q/{bfa},%,{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6293.png"
 ALT="$ {\bf a} \% {\bf b}$">|; 

$key = q/displaystylelambda=-sintheta=expleft(-frac{pi}{2},frac{V}{skew{3}bar{psi}_1},sright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2222.png"
 ALT="$\displaystyle \lambda=-\sin\theta = \exp\left(-\frac{\pi}{2} \frac{V}{\skew{3}\bar{\psi}_1} s\right).$">|; 

$key = q/0.1,{rmmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1151.png"
 ALT="$ 0.1 {\rm mm}$">|; 

$key = q/displaystyledelta_2(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3183.png"
 ALT="$\displaystyle \delta_2(x)$">|; 

$key = q/displaystylef=left(frac{2pi,sinalpha}{pi,sinalpha+4}right)rho,V^{,2},b.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2336.png"
 ALT="$\displaystyle f =\left(\frac{2\pi \sin\alpha}{\pi \sin\alpha+4}\right)\rho V^{ 2} b.$">|; 

$key = q/displaystylephi(xi_0,eta)=V,aleft[frac{e-(1-e^{,2})^{1slash2},sin^{-1}e}{sin^{-1}e-e,(1-e^{,2})^{1slash2}}right]coseta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2779.png"
 ALT="$\displaystyle \phi(\xi_0,\eta) = V a\left[\frac{e-(1-e^{ 2})^{1/2} \sin^{-1} e}{\sin^{-1} e-e (1-e^{ 2})^{1/2}}\right]\cos\eta.$">|; 

$key = q/V=V'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4112.png"
 ALT="$ V=V'=0$">|; 

$key = q/sindeltall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4299.png"
 ALT="$ \sin\delta \ll 1$">|; 

$key = q/displaystylesimn,{mitDelta}thetasimtheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5754.png"
 ALT="$\displaystyle \sim n {\mit\Delta}\theta\sim \theta,$">|; 

$key = q/displaystyleepsilon=frac{rho,omega^{,2},a^{,3}}{8,gamma},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1345.png"
 ALT="$\displaystyle \epsilon = \frac{\rho \omega^{ 2} a^{ 3}}{8 \gamma},
$">|; 

$key = q/{bfh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6604.png"
 ALT="$ {\bf h}$">|; 

$key = q/rho>rho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5373.png"
 ALT="$ \rho&gt;\rho_0$">|; 

$key = q/6.960times10^8,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5246.png"
 ALT="$ 6.960\times 10^8 {\rm m}$">|; 

$key = q/hggdelta(l);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3040.png"
 ALT="$ h\gg \delta(l)$">|; 

$key = q/displaystylefrac{1}{(1+f_x^{,2})^{1slash2}}=C-frac{f^{,2}}{2,l^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1163.png"
 ALT="$\displaystyle \frac{1}{(1+f_x^{ 2})^{1/2}} = C-\frac{f^{ 2}}{2 l^{ 2}},$">|; 

$key = q/{bfV}=V,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2702.png"
 ALT="$ {\bf V} =V {\bf e}_z$">|; 

$key = q/displaystyle{bfF}=-rho,nabla{mitPsi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="$\displaystyle {\bf F} = - \rho \nabla{\mit\Psi},$">|; 

$key = q/delta_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5012.png"
 ALT="$ \delta_{n, n'}^{ m, m'}$">|; 

$key = q/displaystyleS_{x'}=S,cos,(alpha-theta)=S,cosalpha,costheta+S,sinalpha,sintheta=S_x,costheta+S_y,sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="586" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6344.png"
 ALT="$\displaystyle S_{x'} = S \cos (\alpha-\theta) = S \cos\alpha \cos\theta + S \sin\alpha \sin\theta = S_x \cos\theta + S_y \sin\theta,$">|; 

$key = q/displaystyle{bfF}=-m',frac{d{bfV}}{dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1817.png"
 ALT="$\displaystyle {\bf F} =- m' \frac{d{\bf V}}{dt},$">|; 

$key = q/displaystylefrac{dM}{dt}+{mitPhi}_M=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="$\displaystyle \frac{dM}{dt} +{\mit\Phi}_M=0$">|; 

$key = q/displaystyleu_{e,{rmmax}}=left(frac{2,gamma}{gamma-1}right)^{1slash2}c_0=sqrt{2,{calC}_p,T_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5433.png"
 ALT="$\displaystyle u_{e {\rm max}}= \left(\frac{2 \gamma}{\gamma-1}\right)^{1/2}c_0=\sqrt{2 {\cal C}_p T_0},$">|; 

$key = q/displaystylef;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1303.png"
 ALT="$\displaystyle f$">|; 

$key = q/displaystylefrac{1}{2},rholeft[left(frac{3}{2},G,M,alpha_1-omega^{,2}right)x_1^{{,2}right)x_2^{,2}+frac{3}{2},G,M,alpha_3,x_3^{,2}right]=p_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="593" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img864.png"
 ALT="$\displaystyle \frac{1}{2} \rho\left[\left(\frac{3}{2} G M \alpha_1-\omega^{...
...ega^{ 2}\right)x_2^{ 2}+ \frac{3}{2} G M \alpha_3 x_3^{ 2}\right] = p_0,$">|; 

$key = q/zleq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img556.png"
 ALT="$ z\leq 0$">|; 

$key = q/sigma_{13};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="$ \sigma_{13}$">|; 

$key = q/xi_z(x,0,t)=zeta(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4090.png"
 ALT="$ \xi_z(x,0,t)=\zeta(x,t)$">|; 

$key = q/7.1,{rmkm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4652.png"
 ALT="$ 7.1 {\rm km}$">|; 

$key = q/oint{bfA}cdotd{bfr}neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6561.png"
 ALT="$ \oint {\bf A}\cdot d{\bf r} \neq 0$">|; 

$key = q/displaystylealpha,{bfa}+beta,{bfb}+gamma,{bfc}={bf0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6725.png"
 ALT="$\displaystyle \alpha {\bf a} + \beta {\bf b} + \gamma {\bf c} = {\bf0},
$">|; 

$key = q/displaystylefrac{delta_2^{,2}}{nu}=frac{0.47}{U^{,6}}int_0^xU^{,5}(x'),dx',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3207.png"
 ALT="$\displaystyle \frac{\delta_2^{ 2}}{\nu} = \frac{0.47}{U^{ 6}}\int_0^x U^{ 5}(x') dx',$">|; 

$key = q/displaystylenablatimes({bfA}times{bfB});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6697.png"
 ALT="$\displaystyle \nabla\times({\bf A}\times{\bf B})$">|; 

$key = q/h<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1153.png"
 ALT="$ h&lt;0$">|; 

$key = q/{rmi},dskew{2}bar{z}=dy+{rmi},dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2400.png"
 ALT="$ {\rm i} d\skew{2}\bar{z} = dy+{\rm i} dx$">|; 

$key = q/e_{13};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$ e_{13}$">|; 

$key = q/displaystyle{bfq}=-kappa,nablaT.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="$\displaystyle {\bf q} = -\kappa \nabla T.$">|; 

$key = q/zeta_j(mu,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4672.png"
 ALT="$ \zeta_j(\mu,\phi,t)$">|; 

$key = q/displaystyle=(8,sslashtantheta)^{1slash2},a,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img776.png"
 ALT="$\displaystyle = (8 s/\tan\theta)^{1/2} a,$">|; 

$key = q/nu=1.5times10^{-5},{rmm^2,s^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="$ \nu= 1.5\times 10^{-5} {\rm m^2 s^{-1}}$">|; 

$key = q/f(x,y)=x,y^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6445.png"
 ALT="$ f(x,y)= x y^{ 2}$">|; 

$key = q/displaystylea_r=mu_rint_{mitOmega}F,{mitPsi}_r,d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4834.png"
 ALT="$\displaystyle a_r = \mu_r\int_{\mit\Omega} F {\mit\Psi}_r d{\mit\Omega}.$">|; 

$key = q/displaystylepsi=-frac{1}{2},V,r^{,2},sin^2thetaleft(1-frac{a^{,3}}{r^{,3}}right){,2}left[1-frac{a^{,3}}{(varpi^{,2}+z^{,2})^{3slash2}}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2649.png"
 ALT="$\displaystyle \psi = -\frac{1}{2} V r^{ 2} \sin^2\theta\left(1-\frac{a^{ 3...
...2} V \varpi^{ 2}\left[1-\frac{a^{ 3}}{(\varpi^{ 2}+z^{ 2})^{3/2}}\right].$">|; 

$key = q/x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$ x_1$">|; 

$key = q/displaystylesinbeta=frac{v_p}{V}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4015.png"
 ALT="$\displaystyle \sin\beta =\frac{v_p}{V}.$">|; 

$key = q/psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$ \psi $">|; 

$key = q/p_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$ p_1$">|; 

$key = q/displaystyleQ=rho(x),A(x),u(x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5410.png"
 ALT="$\displaystyle Q = \rho(x) A(x) u(x).$">|; 

$key = q/mu<0.06757;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1036.png"
 ALT="$ \mu&lt;0.06757$">|; 

$key = q/displaystylenabla^{,2}phi=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img546.png"
 ALT="$\displaystyle \nabla^{ 2}\phi = 0,
$">|; 

$key = q/1,{rmmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1149.png"
 ALT="$ 1 {\rm mm}$">|; 

$key = q/beta(y)=(1slash2),A,u_0(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3130.png"
 ALT="$ \beta(y) = (1/2) A u_0(y)$">|; 

$key = q/-psi_Q({bfr}-{bfr}_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1687.png"
 ALT="$ -\psi_Q({\bf r}-{\bf r}_0)$">|; 

$key = q/displaystyleu(x)=left[-1+left(frac{x-x_2}{x_1-x_2}right)right]V_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5568.png"
 ALT="$\displaystyle u(x) = \left[-1+\left(\frac{x-x_2}{x_1-x_2}\right)\right]V_p$">|; 

$key = q/displaystylerholeft(frac{partialv_i}{partialt}+v_j,frac{partialv_i}{partialx_j}right)=F_i+frac{partialsigma_{ij}}{partialx_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="$\displaystyle \rho\left(\frac{\partial v_i}{\partial t} + v_j \frac{\partial v_i}{\partial x_j}\right)= F_i + \frac{\partial\sigma_{ij}}{\partial x_j}.$">|; 

$key = q/displaystylefrac{T}{T_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5616.png"
 ALT="$\displaystyle \frac{T}{T_0}$">|; 

$key = q/displaystyleD,{mitPsi}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4771.png"
 ALT="$\displaystyle D {\mit\Psi}'$">|; 

$key = q/A_{ijkl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="$ A_{ijkl}$">|; 

$key = q/displaystyle=-overline{rho},overline{nabla}cdotoverline{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="$\displaystyle = -\overline{\rho} \overline{\nabla}\cdot\overline{\bf v},$">|; 

$key = q/displaystyle(a_{ik}'-{calR}_{il},{calR}_{km},a_{lm}),b_{jk}'=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6864.png"
 ALT="$\displaystyle (a_{ik}'- {\cal R}_{il} {\cal R}_{km} a_{lm}) b_{jk}' = 0.$">|; 

$key = q/displaystylefrac{dF}{dz}=V,{rme}^{,{rmi},alpha}+{rmi},frac{{mitGamma}}{2pi,z}+frac{B}{z^{,2}}+cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3282.png"
 ALT="$\displaystyle \frac{dF}{dz} = V {\rm e}^{ {\rm i} \alpha} + {\rm i} \frac{{\mit\Gamma}}{2\pi z}+ \frac{B}{z^{ 2}}+\cdots$">|; 

$key = q/z_{GH}=c,(1-s)slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img672.png"
 ALT="$ z_{GH} = c (1-s)/2$">|; 

$key = q/PQRS;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6231.png"
 ALT="$ PQRS$">|; 

$key = q/displaystylev_z(r,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3677.png"
 ALT="$\displaystyle v_z(r,z)$">|; 

$key = q/gamma<2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1771.png"
 ALT="$ \gamma&lt;2$">|; 

$key = q/{bfa}times({bfb}times{bfc})=({bfa}times{bfb})times{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6713.png"
 ALT="$ {\bf a}\times ({\bf b}\times {\bf c})= ({\bf a}\times {\bf b})\times {\bf c}$">|; 

$key = q/deltar,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1127.png"
 ALT="$ \delta r \cos\theta$">|; 

$key = q/W,z_G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img701.png"
 ALT="$ W z_G$">|; 

$key = q/delta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2854.png"
 ALT="$ \delta$">|; 

$key = q/F=x^{,3},y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6469.png"
 ALT="$ F=x^{ 3} y$">|; 

$key = q/displaystylesigma_{rr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="$\displaystyle \sigma_{rr}$">|; 

$key = q/Gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3390.png"
 ALT="$ \Gamma$">|; 

$key = q/2,p-q=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3250.png"
 ALT="$ 2 p-q=0$">|; 

$key = q/displaystylerightarrow-V,costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3756.png"
 ALT="$\displaystyle \rightarrow -V \cos\theta,$">|; 

$key = q/rho,A,u=rho_1,A_1,u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5411.png"
 ALT="$ \rho A u=\rho_1 A_1 u_1$">|; 

$key = q/displaystylez_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6128.png"
 ALT="$\displaystyle z_1$">|; 

$key = q/displaystyleint_{P}^Qf(x,y),dl={rmArea~under~the~curve},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6442.png"
 ALT="$\displaystyle \int_{P}^Q f(x,y) dl = {\rm Area&nbsp;under&nbsp;the&nbsp;curve},$">|; 

$key = q/nablacdot{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="$ \nabla\cdot{\bf v}$">|; 

$key = q/displaystyleC_psimeqfrac{2}{U}left.frac{partialphi}{partialx}right|_{y=0}=-frac{2,epsilon,alpha}{sqrt{1-{rmMa}_infty^{,2}}},sin(alpha,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5978.png"
 ALT="$\displaystyle C_p \simeq \frac{2}{U}\left.\frac{\partial\phi}{\partial x}\right...
...= -\frac{2 \epsilon \alpha}{\sqrt{1-{\rm Ma}_\infty^{ 2}}} \sin(\alpha x).$">|; 

$key = q/psi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1620.png"
 ALT="$ \psi_1$">|; 

$key = q/{rmMa}_2simeq{rmMa}_1,(1-delta_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6185.png"
 ALT="$ {\rm Ma}_2\simeq {\rm Ma}_1 (1-\delta_1)$">|; 

$key = q/p_n({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4395.png"
 ALT="$ p_n({\bf r})$">|; 

$key = q/displaystyle=a_1,b_2-a_2,b_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6785.png"
 ALT="$\displaystyle =a_1 b_2-a_2 b_1$">|; 

$key = q/displaystyle=nabla^{,2}A_theta+frac{2}{r^{,2}},frac{partialA_r}{partialtheta}-freta}-frac{2}{r^{,2},sintheta},frac{partialA_phi}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="386" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7196.png"
 ALT="$\displaystyle = \nabla^{ 2} A_\theta +\frac{2}{r^{ 2}} \frac{\partial A_r}{\...
...^2\theta} -\frac{2}{r^{ 2} \sin\theta} \frac{\partial A_\phi}{\partial\phi},$">|; 

$key = q/pislash2leqphileqpi-phi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1332.png"
 ALT="$ \pi/2\leq \phi\leq \pi-\phi_0$">|; 

$key = q/eta<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3175.png"
 ALT="$ \eta&lt;1$">|; 

$key = q/nabla^{,2}psi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2025.png"
 ALT="$ \nabla^{ 2}\psi=0$">|; 

$key = q/displaystyle{bfe}_icdot{bfe}_jtimes{bfe}_k=epsilon_{ijk}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7039.png"
 ALT="$\displaystyle {\bf e}_i\cdot{\bf e}_j\times {\bf e}_k = \epsilon_{ijk}.$">|; 

$key = q/theta=30^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$ \theta =30^\circ $">|; 

$key = q/displaystylefrac{partiala_i}{partialx_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6874.png"
 ALT="$\displaystyle \frac{\partial a_i}{\partial x_j}.$">|; 

$key = q/displaystylefrac{rho_1(x,t)}{rho_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5358.png"
 ALT="$\displaystyle \frac{\rho_1(x,t)}{\rho_0}$">|; 

$key = q/displaystylev_perp=frac{psi_2-psi_1}{deltas}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1632.png"
 ALT="$\displaystyle v_\perp = \frac{\psi_2-\psi_1}{\delta s}.$">|; 

$key = q/x_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$ x_3$">|; 

$key = q/S,cosalpha_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6336.png"
 ALT="$ S \cos\alpha_x$">|; 

$key = q/I(z),dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3467.png"
 ALT="$ I(z) dz$">|; 

$key = q/DslashDoverline{t}=partialslashpartialoverline{t}+overline{bfv}cdotoverline{nabla};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="$ D/D\overline{t}= \partial/\partial \overline{t} + \overline{\bf v}\cdot\overline{\nabla}$">|; 

$key = q/delta<c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3305.png"
 ALT="$ \delta&lt;c$">|; 

$key = q/{rmMa},{rmPr},overline{p}_0sim{calO}(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="$ {\rm Ma}, {\rm Pr}, \overline{p}_0\sim {\cal O}(1)$">|; 

$key = q/rho_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5447.png"
 ALT="$ \rho_2$">|; 

$key = q/displaystylefrac{a}{l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3686.png"
 ALT="$\displaystyle \frac{a}{l}$">|; 

$key = q/displaystyle=int_{mitOmega}left(frac{partial{mitPsi}_r'}{partialtheta},frac{partialphi},frac{partial{mitPsi}_s}{partialphi}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4883.png"
 ALT="$\displaystyle =\int_{\mit\Omega} \left(\frac{\partial {\mit\Psi}_r'}{\partial\t...
...}{\partial\phi} \frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega},$">|; 

$key = q/displaystyle=lambda_1,lambda_2,lambda_3.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7013.png"
 ALT="$\displaystyle =\lambda_1 \lambda_2 \lambda_3.$">|; 

$key = q/CC''=[M'slash(M+M')],R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img971.png"
 ALT="$ CC'' = [M'/(M+M')] R$">|; 

$key = q/sigma_{23};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$ \sigma_{23}$">|; 

$key = q/{rmMa}_1geq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5721.png"
 ALT="$ {\rm Ma}_1\geq 1$">|; 

$key = q/{mitGamma}={mitGamma}_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$ {\mit \Gamma }={\mit \Gamma }_\infty $">|; 

$key = q/displaystyle{mitDeltavarphi}(theta,phi)=m_j,frac{pi}{2}+tan^{-1}left[frac{zeta_s(theta,phi)}{zeta_c(theta,phi)}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5030.png"
 ALT="$\displaystyle {\mit\Delta\varphi}(\theta,\phi) =m_j \frac{\pi}{2} + \tan^{-1}\left[\frac{\zeta_s(\theta,\phi)}{\zeta_c(\theta,\phi)}\right].$">|; 

$key = q/displaystyle=(4,mu-nu-lambda)slash10,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6974.png"
 ALT="$\displaystyle = (4 \mu-\nu-\lambda)/10,$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi}=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1375.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} + {\mit\Psi} =$">|; 

$key = q/displaystyleD,P_n^{,m}(mu),{rme}^{,{rmi},m,phi}=-n,(n+1),P_n^{,m}(mu),{rme}^{,{rmi},m,phi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4681.png"
 ALT="$\displaystyle D  P_n^{ m}(\mu) {\rm e}^{ {\rm i} m \phi} = -n (n+1) P_n^{ m}(\mu) {\rm e}^{ {\rm i} m \phi}.$">|; 

$key = q/{bfk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6753.png"
 ALT="$ {\bf k}$">|; 

$key = q/displaystylefrac{beta_{-n,,-n'}^{,-m,,-m'}}{c_n^{,m},c_{n'}^{,m'}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4962.png"
 ALT="$\displaystyle \frac{\beta_{-n, -n'}^{ -m, -m'}}{c_n^{ m} c_{n'}^{ m'}}$">|; 

$key = q/d{mitOmega}=sintheta,dtheta,dphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4753.png"
 ALT="$ d{\mit\Omega}=\sin\theta d\theta d\phi$">|; 

$key = q/displaystyle{bfE}({bfr})=frac{1}{4pi,epsilon_0}left[frac{3,({bfe}_rcdot{bfp}),{bfe}_r-{bfp}}{r^3}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7208.png"
 ALT="$\displaystyle {\bf E}({\bf r})= \frac{1}{4\pi \epsilon_0} \left[\frac{3 ({\bf e}_r\cdot {\bf p}) {\bf e}_r-{\bf p}}{r^3}\right],
$">|; 

$key = q/2pi,kappa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1909.png"
 ALT="$ 2\pi \kappa$">|; 

$key = q/displaystylefrac{Dv_varphi}{Dt}+frac{v_r,v_varphi+cottheta,v_theta,v_varphi}{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img508.png"
 ALT="$\displaystyle \frac{Dv_\varphi}{Dt}+\frac{v_r v_\varphi + \cot\theta v_\theta v_\varphi}{r}$">|; 

$key = q/sigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5185.png"
 ALT="$ \sigma$">|; 

$key = q/gamma_{13}<gamma_{23}-gamma_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1133.png"
 ALT="$ \gamma_{13}&lt; \gamma_{23}-\gamma_{12}$">|; 

$key = q/y(0)=y(1)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7376.png"
 ALT="$ y(0)=y(1)=1$">|; 

$key = q/y(xi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$ y(\xi )$">|; 

$key = q/b>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6217.png"
 ALT="$ b&gt;0$">|; 

$key = q/displaystylenablacdot{bfA}=lim_{dVrightarrow0}frac{oint{bfA}cdotd{bfS}}{dV}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6583.png"
 ALT="$\displaystyle \nabla\cdot{\bf A} = \lim_{dV\rightarrow 0} \frac{\oint {\bf A}\cdot d{\bf S}} {dV}.$">|; 

$key = q/displaystylefrac{partial}{partialx'}=left(frac{partialx}{partialx'}right)_{y',z'ac{partialz}{partialx'}right)_{y',z'}frac{partial}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6570.png"
 ALT="$\displaystyle \frac{\partial}{\partial x'} = \left(\frac{\partial x}{\partial x...
...eft(\frac{\partial z}{\partial x'} \right)_{y',z'} \frac{\partial}{\partial z},$">|; 

$key = q/displaystylefrac{[rho,({bfv}cdotnabla),{bfv}]_r}{(mu,nabla^2{bfv})_r}simfrac{rho,V,r}{mu}sim{rmRe}left(frac{r}{a}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3831.png"
 ALT="$\displaystyle \frac{[\rho ({\bf v}\cdot\nabla) {\bf v}]_r }{(\mu \nabla^2{\bf v})_r} \sim \frac{\rho V r}{\mu}\sim {\rm Re}\left(\frac{r}{a}\right),$">|; 

$key = q/displaystyle=c,coshxi_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2755.png"
 ALT="$\displaystyle =c \cosh\xi_0,$">|; 

$key = q/displaystylez=zeta+frac{l^{,2}}{zeta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2121.png"
 ALT="$\displaystyle z = \zeta+ \frac{l^{ 2}}{\zeta},$">|; 

$key = q/displaystylec'=c_0left{1+left(frac{gamma+1}{gamma-1}right)left[left(frac{rho}{rho_0}right)^{,(gamma-1)slash2}-1right]right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5372.png"
 ALT="$\displaystyle c' =c_0\left\{1+\left(\frac{\gamma+1}{\gamma-1}\right)\left[\left(\frac{\rho}{\rho_0}\right)^{ (\gamma-1)/2}-1\right]\right\}.$">|; 

$key = q/displaystyle=-frac{1}{rho,r,sintheta},frac{partialp}{partialvarphi}-frac{1}{r,sift(nabla^{,2}v_varphi-frac{v_varphi}{r^{,2},sin^2theta}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="440" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img509.png"
 ALT="$\displaystyle = - \frac{1}{\rho r \sin\theta} \frac{\partial p}{\partial \va...
...ho}\left(\nabla^{ 2} v_\varphi-\frac{v_\varphi}{r^{ 2} \sin^2\theta} \right.$">|; 

$key = q/displaystyleint_a^bleft[frac{partialF}{partialy}-frac{d}{dx}!left(frac{partialF}{partialy'}right)right]deltay,dx=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7316.png"
 ALT="$\displaystyle \int_a^b\left[\frac{\partial F}{\partial y}- \frac{d}{dx}\!\left(\frac{\partial F}{\partial y'}\right)\right]\delta y dx =0.$">|; 

$key = q/{bfv}=v_y,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3477.png"
 ALT="$ {\bf v}= v_y {\bf e}_y$">|; 

$key = q/dbar{F}slashdzsimeq1slashV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2244.png"
 ALT="$ d\bar{F}/dz\simeq 1/V$">|; 

$key = q/displaystylefrac{{rmMa}_infty^{,2},(gamma+1),epsilon,alpha}{(1-{rmMa}_infty^{,2})^{,3slash2}}ll1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5982.png"
 ALT="$\displaystyle \frac{{\rm Ma}_\infty^{ 2} (\gamma+1) \epsilon \alpha}{(1-{\rm Ma}_\infty^{ 2})^{ 3/2}}\ll 1.$">|; 

$key = q/displaystyle=frac{partialvarpi}{partialeta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2685.png"
 ALT="$\displaystyle =\frac{\partial\varpi}{\partial\eta},$">|; 

$key = q/eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1579.png"
 ALT="$ \eta$">|; 

$key = q/x,V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$ x V$">|; 

$key = q/z=z_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2058.png"
 ALT="$ z=z_0$">|; 

$key = q/skew{5}bar{zeta}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4522.png"
 ALT="$ \skew{5}\bar{\zeta}_2$">|; 

$key = q/displaystyle=left(frac{p_2slashp_1}{p_4slashp_1}right)^{,1slashgamma_4},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5650.png"
 ALT="$\displaystyle = \left(\frac{p_2/p_1}{p_4/p_1}\right)^{ 1/\gamma_4},$">|; 

$key = q/_x|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$ _x\vert$">|; 

$key = q/T_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5638.png"
 ALT="$ T_3$">|; 

$key = q/displaystylenablatimes({bfg}times{bfb})=-(nablacdot{bfg}),{bfb}+({bfb}cdotnabla),{bfg},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1098.png"
 ALT="$\displaystyle \nabla\times ({\bf g}\times {\bf b}) = - (\nabla\cdot{\bf g}) {\bf b} + ({\bf b}\cdot\nabla) {\bf g},$">|; 

$key = q/displaystylefrac{omega^{,2}}{2pi,G,rho}=left(frac{3-2,e_{13}^{,2}}{e_{13}^{,3}}rin^{-1}e_{13}-left(frac{3}{e_{13}^{,2}}right)(1-e_{13}^{,2}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="510" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img881.png"
 ALT="$\displaystyle \frac{\omega^{ 2}}{2\pi G \rho} = \left(\frac{3-2 e_{13}^{ 2...
...^{1/2} \sin^{-1} e_{13} - \left(\frac{3}{e_{13}^{ 2}}\right)(1-e_{13}^{ 2}).$">|; 

$key = q/displaystylefrac{dp}{dz}=-rho,g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img565.png"
 ALT="$\displaystyle \frac{dp}{dz} = -\rho g.$">|; 

$key = q/displaystyle=p_0left(1-frac{gamma-1}{gamma},frac{z}{H}right)^{gammaslash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5151.png"
 ALT="$\displaystyle = p_0\left(1-\frac{\gamma-1}{\gamma} \frac{z}{H}\right)^{\gamma/(\gamma-1)},$">|; 

$key = q/rho=rho(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img562.png"
 ALT="$ \rho=\rho(z)$">|; 

$key = q/0^circ,{rmC};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5126.png"
 ALT="$ 0^\circ  {\rm C}$">|; 

$key = q/f=f_0-beta,x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5104.png"
 ALT="$ f= f_0-\beta x$">|; 

$key = q/dFslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7323.png"
 ALT="$ dF/dx$">|; 

$key = q/|y|ggdelta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3049.png"
 ALT="$ \vert y\vert\gg\delta$">|; 

$key = q/v_i({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="$ v_i({\bf r}, t)$">|; 

$key = q/{bfv}=u(x,y),{bfe}_x+v(x,y),{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3072.png"
 ALT="$ {\bf v} = u(x,y) {\bf e}_x+v(x,y) {\bf e}_y$">|; 

$key = q/D(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6486.png"
 ALT="$ D(x,y)$">|; 

$key = q/displaystylefrac{dwidehat{P}}{dX}=-widehat{U},frac{dwidehat{U}}{dX},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2896.png"
 ALT="$\displaystyle \frac{d\widehat{P}}{d X} = - \widehat{U} \frac{d\widehat{U}}{dX},$">|; 

$key = q/displaystyle=theta^{,3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5206.png"
 ALT="$\displaystyle = \theta^{ 3},$">|; 

$key = q/C_1=C_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1381.png"
 ALT="$ C_1=C_2$">|; 

$key = q/k,lgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4161.png"
 ALT="$ k l\gg 1$">|; 

$key = q/displaystyleleft.frac{partialpsi}{partialr}right|_{r=a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3761.png"
 ALT="$\displaystyle \left.\frac{\partial\psi}{\partial r}\right\vert _{r=a}$">|; 

$key = q/displaystyle=frac{{mitOmega}^{,2},a^{,2}}{3}left(frac{r}{a}right)^2P_2^{,0}(costheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4382.png"
 ALT="$\displaystyle =\frac{{\mit\Omega}^{ 2} a^{ 2}}{3}\left(\frac{r}{a}\right)^2 P_2^{ 0}(\cos\theta).$">|; 

$key = q/displaystyleQ=h,v.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1427.png"
 ALT="$\displaystyle Q = h v.$">|; 

$key = q/kappa_H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="$ \kappa_H$">|; 

$key = q/displaystyleleft(frac{partial{calS}}{partialv}right)_{T}=left(frac{partialp}{partialT}right)_v.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5589.png"
 ALT="$\displaystyle \left(\frac{\partial {\cal S}}{\partial v}\right)_{T} = \left(\frac{\partial p}{\partial T}\right)_v.
$">|; 

$key = q/displaystyleoint{bfA}cdotd{bfr}=left(frac{partialA_z}{partialy}-frac{partialA_y}{partialz}right),dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6633.png"
 ALT="$\displaystyle \oint {\bf A}\cdot d{\bf r} = \left(\frac{\partial A_z}{\partial y}- \frac{\partial A_y}{\partial z}\right) dS,$">|; 

$key = q/displaystyledeltaQ=V,deltah,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3606.png"
 ALT="$\displaystyle \delta Q = V \delta h,$">|; 

$key = q/{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$ {\bf r}$">|; 

$key = q/displaystylefrac{{mitDelta}p}{p}simeqleft(frac{gamma,{rmMa}^{,2}}{sqrt{{rmMa}^{,2}-1}}right){mitDelta}theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5785.png"
 ALT="$\displaystyle \frac{{\mit\Delta}p}{p}\simeq \left(\frac{\gamma {\rm Ma}^{ 2}}{\sqrt{{\rm Ma}^{ 2}-1}}\right){\mit\Delta}\theta,$">|; 

$key = q/displaystylefrac{d{mitGamma}}{dt}=-k,V,U_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3332.png"
 ALT="$\displaystyle \frac{d{\mit\Gamma}}{dt} = - k V U_0.$">|; 

$key = q/displaystyleoint_C{bfF}cdotd{bfr}=int_Snablatimes{bfF}cdotd{bfS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1092.png"
 ALT="$\displaystyle \oint_C {\bf F}\cdot d{\bf r} = \int_S \nabla\times {\bf F}\cdot d{\bf S},$">|; 

$key = q/displaystylec_p-c_v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5602.png"
 ALT="$\displaystyle c_p - c_v$">|; 

$key = q/displaystyle=frac{partialzeta_b}{partialt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4471.png"
 ALT="$\displaystyle =\frac{\partial\zeta_b}{\partial t},$">|; 

$key = q/displaystylelambda=-sintheta=-left(frac{k}{k+s}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2271.png"
 ALT="$\displaystyle \lambda=-\sin\theta =-\left(\frac{k}{k+s}\right)^{1/2}.$">|; 

$key = q/gamma_{r,,s}rightarrowgamma_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4985.png"
 ALT="$ \gamma_{r, s}\rightarrow \gamma_{n, n'}^{ m, m'}$">|; 

$key = q/displaystyle=-a,(k,d)^{-1},cos(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3957.png"
 ALT="$\displaystyle = -a (k d)^{-1} \cos(\omega t-k x),$">|; 

$key = q/displaystyleP_2(cosgamma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4260.png"
 ALT="$\displaystyle P_2(\cos\gamma)$">|; 

$key = q/displaystylefrac{omega(x,y)}{U_0slashdelta}=-0.749left(frac{L}{x}right)^{1slash2}frac{y}{delta},exp(-y^{,2}slashdelta^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3064.png"
 ALT="$\displaystyle \frac{\omega(x,y)}{U_0/\delta} = -0.749 \left(\frac{L}{x}\right)^{1/2}\frac{y}{\delta} \exp(-y^{ 2}/\delta^{ 2}),$">|; 

$key = q/-p,delta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="$ -p \delta_{ij}$">|; 

$key = q/displaystyle=a_{ssk},delta_{ij}+a_{iss},delta_{jk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6918.png"
 ALT="$\displaystyle = a_{ssk} \delta_{ij} + a_{iss} \delta_{jk},$">|; 

$key = q/p_4>p_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5633.png"
 ALT="$ p_4&gt;p_1$">|; 

$key = q/{bfe}_xi=nablaxislash|nablaxi|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2708.png"
 ALT="$ {\bf e}_\xi=\nabla\xi/\vert\nabla\xi\vert$">|; 

$key = q/displaystylefrac{p}{rho^{,gamma}}={rmconstant}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5118.png"
 ALT="$\displaystyle \frac{p}{\rho^{ \gamma}} = {\rm constant}.$">|; 

$key = q/displaystyle=-frac{1}{2},B,cosh^2xileft[frac{sinhxi'}{cosh^2xi'}-tan^{-1}left(frac{1}{sinhxi'}right)right]^{infty}_{xi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2766.png"
 ALT="$\displaystyle =-\frac{1}{2}  B \cosh^2\xi\left[ \frac{\sinh\xi'}{\cosh^2\xi'}-\tan^{-1}\left(\frac{1}{\sinh\xi'}\right)\right]^{\infty}_{\xi}$">|; 

$key = q/displaystylenuleft.frac{partial^{,2}u}{partialy^{,2}}right|_{y=0}=-frac{1}{U},frac{dU}{dx}=frac{1}{rho},frac{dP}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3142.png"
 ALT="$\displaystyle \nu\left.\frac{\partial^{ 2} u}{\partial y^{ 2}}\right\vert _{y=0} = -\frac{1}{U} \frac{dU}{dx} = \frac{1}{\rho} \frac{dP}{dx},$">|; 

$key = q/displaystylefrac{dx}{dt}=c_2-V_p=c_1-frac{1}{2},(gamma+1),V_p.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5559.png"
 ALT="$\displaystyle \frac{dx}{dt}=c_2-V_p = c_1-\frac{1}{2} (\gamma+1) V_p.$">|; 

$key = q/displaystylefrac{delta_2(x)}{x}=frac{0.69}{{rmRe}^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3211.png"
 ALT="$\displaystyle \frac{\delta_2(x)}{x} = \frac{0.69}{{\rm Re}^{1/2}},$">|; 

$key = q/epsilon=23.44^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4586.png"
 ALT="$ \epsilon=23.44^\circ$">|; 

$key = q/v'slashV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5908.png"
 ALT="$ v'/V$">|; 

$key = q/displaystyle{mitDelta}(delta{mitPhi})simd,frac{partialdelta{mitPhi}}{partialz}simfrac{d}{a},delta{mitPhi}lldelta{mitPhi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4481.png"
 ALT="$\displaystyle {\mit\Delta}(\delta{\mit\Phi})\sim d \frac{\partial \delta{\mit\Phi}}{\partial z} \sim \frac{d}{a} \delta{\mit\Phi}\ll \delta{\mit\Phi},$">|; 

$key = q/displaystylephantom{=}left.-frac{A_i}{h_i,h_j^{,2}}left[frac{2}{h_i}left(frac{pau_j}right)^2-frac{partial^2h_i}{partialu_j^{,2}}right]right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7102.png"
 ALT="$\displaystyle \phantom{=}\left.-\frac{A_i}{h_i h_j^{ 2}}\left[\frac{2}{h_i}\l...
...partial u_j}\right)^2-\frac{\partial^2 h_i}{\partial u_j^{ 2}}\right]\right\}.$">|; 

$key = q/B=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2202.png"
 ALT="$ B=-1$">|; 

$key = q/0.125;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$ 0.125$">|; 

$key = q/displaystyleF(z)=-frac{a,pi,V_0}{tanh(a,pislashz)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2498.png"
 ALT="$\displaystyle F(z)= - \frac{a \pi V_0}{\tanh(a \pi/z)}
$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partialA(theta,phi_pm,t)}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4786.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial A(\theta,\phi_\pm,t)}{\partial \phi}$">|; 

$key = q/24.066;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4624.png"
 ALT="$ 24.066$">|; 

$key = q/displaystyleF''-F^{,2}=-1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3237.png"
 ALT="$\displaystyle F'' - F^{ 2} = -1,
$">|; 

$key = q/displaystyle={calR}_{il},{calR}_{qk},a_{lk},b_{jq}={calR}_{il},{calR}_{km},a_{lm},b_{jk}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6862.png"
 ALT="$\displaystyle = {\cal R}_{il} {\cal R}_{qk} a_{lk} b_{jq} = {\cal R}_{il} {\cal R}_{km} a_{lm} b_{jk}',$">|; 

$key = q/displaystyleU(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3333.png"
 ALT="$\displaystyle U (\theta)$">|; 

$key = q/displaystyle=W_y=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img581.png"
 ALT="$\displaystyle = W_y = 0,$">|; 

$key = q/displaystyleoverline{nabla}cdotoverline{bfv}ll1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="$\displaystyle \overline{\nabla}\cdot\overline{\bf v} \ll 1.$">|; 

$key = q/(partialuslashpartialx)_{x=x_0}=infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3109.png"
 ALT="$ (\partial u/\partial x)_{x=x_0}=\infty$">|; 

$key = q/(1-m),(1-alphaslash2)=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1880.png"
 ALT="$ (1-m) (1-\alpha/2) = 1/2$">|; 

$key = q/e_{12}=sinbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1024.png"
 ALT="$ e_{12}=\sin\beta$">|; 

$key = q/{rmMa}_infty^{,2}-1equivlambda^{,2}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5985.png"
 ALT="$ {\rm Ma}_\infty^{ 2}-1\equiv \lambda^{ 2} &gt; 0$">|; 

$key = q/displaystylefrac{1}{rho},frac{dp}{dr}=frac{kappa^{,2},r}{a^{,4}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1907.png"
 ALT="$\displaystyle \frac{1}{\rho} \frac{dp}{dr} = \frac{\kappa^{ 2} r}{a^{ 4}},
$">|; 

$key = q/displaystylev_r(rrightarrowinfty,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3755.png"
 ALT="$\displaystyle v_r(r\rightarrow \infty,\theta)$">|; 

$key = q/displaystylefrac{x_1^{,2}}{a_1^{,2}}+frac{x_2^{,2}}{a_2^{,2}}=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img835.png"
 ALT="$\displaystyle \frac{x_1^{ 2}}{a_1^{ 2}}+ \frac{x_2^{ 2}}{a_2^{ 2}}=1,$">|; 

$key = q/displaystyletheta=n,{mitDelta}theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5752.png"
 ALT="$\displaystyle \theta = n {\mit\Delta}\theta,$">|; 

$key = q/pslashrho^{,gamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="$ p/\rho^{ \gamma}$">|; 

$key = q/displaystyleF(z)=frac{D,{rme}^{,{rmi},alpha}}{2pi,z}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2451.png"
 ALT="$\displaystyle F(z) = \frac{D {\rm e}^{ {\rm i} \alpha}}{2\pi z}.
$">|; 

$key = q/displaystyle8,s^{,2}-6,s+1>0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img769.png"
 ALT="$\displaystyle 8 s^{ 2}-6 s +1&gt;0,$">|; 

$key = q/{bfv}=nablavarphitimesnablapsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2604.png"
 ALT="$ {\bf v} = \nabla\varphi\times\nabla\psi$">|; 

$key = q/{bfr}={bfd}slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1681.png"
 ALT="$ {\bf r}={\bf d}/2$">|; 

$key = q/displaystyle_Ssimeq{mitOmega}_z,{bfe}_z:;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3446.png"
 ALT="$\displaystyle _S \simeq {\mit\Omega}_z {\bf e}_z:$">|; 

$key = q/displaystyle{bfv}=nablavarphitimesnablapsi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3731.png"
 ALT="$\displaystyle {\bf v} =\nabla\varphi\times\nabla\psi,$">|; 

$key = q/psi=psi(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1711.png"
 ALT="$ \psi=\psi(r)$">|; 

$key = q/displaystyle|z|=|alpha|left[E(phi,k)+{rmsgn}(alpha,beta),(1-k^{,2})^{1slash2},F(phi,k)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1308.png"
 ALT="$\displaystyle \vert z\vert= \vert\alpha\vert\left[E(\phi,k) + {\rm sgn}(\alpha \beta) (1-k^{ 2})^{1/2} F(\phi,k)\right],$">|; 

$key = q/{bfgrad},T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6539.png"
 ALT="$ {\bf grad} T$">|; 

$key = q/displaystylenablacdot{bfh}=-c,frac{partialT}{partialt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6608.png"
 ALT="$\displaystyle \nabla\cdot{\bf h} = -c \frac{\partial T}{\partial t}.$">|; 

$key = q/displaystyleG(e)=frac{e-(1-e^{,2})^{1slash2},sin^{-1}e}{sin^{-1}e-e,(1-e^{,2})^{1slash2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2782.png"
 ALT="$\displaystyle G(e)= \frac{e-(1-e^{ 2})^{1/2} \sin^{-1} e}{\sin^{-1} e-e (1-e^{ 2})^{1/2}}$">|; 

$key = q/displaystylea,x_1^{,2}+b,x_2^{,2}+c,x_3^{,2}+2,f,x_1,x_2+2,g,x_1,x_3+2,h,x_2,x_3=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="454" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6994.png"
 ALT="$\displaystyle a x_1^{ 2} + b x_2^{ 2}+c x_3^{ 2} + 2 f x_1 x_2+2 g x_1 x_3+2 h x_2 x_3=1.
$">|; 

$key = q/displaystylefrac{U_0}{L},frac{partialV_x}{partialX}+frac{U_1}{delta},frac{partialV_y}{partialY}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2887.png"
 ALT="$\displaystyle \frac{U_0}{L} \frac{\partial V_x}{\partial X} + \frac{U_1}{\delta} \frac{\partial V_y}{\partial Y} = 0.$">|; 

$key = q/displaystyle{bfa}times({bfb}times{bfc})neq({bfa}times{bfb})times{bfc}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6366.png"
 ALT="$\displaystyle {\bf a}\times({\bf b} \times{\bf c})\neq ({\bf a}\times{\bf b}) \times{\bf c}.$">|; 

$key = q/displaystyleF_2(lambda);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3193.png"
 ALT="$\displaystyle F_2(\lambda)$">|; 

$key = q/(1-alpha'),pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2957.png"
 ALT="$ (1-\alpha') \pi$">|; 

$key = q/alpha=1slashT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5607.png"
 ALT="$ \alpha=1/T$">|; 

$key = q/displaystyle=-alpha(x)+frac{dh}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5842.png"
 ALT="$\displaystyle = -\alpha(x)+\frac{dh}{dx},$">|; 

$key = q/{rmMa}rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6149.png"
 ALT="$ {\rm Ma}\rightarrow \infty$">|; 

$key = q/displaystylefrac{dw}{w}=-frac{dtheta}{sqrt{{rmMa}^{,2}-1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5759.png"
 ALT="$\displaystyle \frac{dw}{w}= -\frac{d\theta}{\sqrt{{\rm Ma}^{ 2}-1}}.$">|; 

$key = q/alpha+delta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3517.png"
 ALT="$ \alpha+\delta$">|; 

$key = q/displaystylesimeqskew{5}tilde{phi}-frac{k^{,2}}{4},(skew{5}tilde{phi}+sinskew{5}tilde{phi},cosskew{5}tilde{phi}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1209.png"
 ALT="$\displaystyle \simeq \skew{5}\tilde{\phi} - \frac{k^{ 2}}{4} (\skew{5}\tilde{\phi}+\sin\skew{5}\tilde{\phi} \cos\skew{5}\tilde{\phi}),$">|; 

$key = q/displaystylefrac{T}{T_0}=1-frac{z}{c},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5259.png"
 ALT="$\displaystyle \frac{T}{T_0} = 1 - \frac{z}{c},
$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashturn1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="283" BORDER="0"
 SRC="|."$dir".q|img5760.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/turn1.eps}}
\end{figure}">|; 

$key = q/displaystyledelta(x)=1.72left(frac{nu,x}{U_0}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2989.png"
 ALT="$\displaystyle \delta(x) = 1.72\left(\frac{\nu x}{U_0}\right)^{1/2}.$">|; 

$key = q/(c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2039.png"
 ALT="$ (c$">|; 

$key = q/pislash2geqphigeqphi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1331.png"
 ALT="$ \pi/2\geq \phi\geq \phi_0$">|; 

$key = q/skew{5}hat{zeta}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4717.png"
 ALT="$ \skew{5}\hat{\zeta}_j$">|; 

$key = q/v_x(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3593.png"
 ALT="$ v_x(0)=0$">|; 

$key = q/m<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2956.png"
 ALT="$ m&lt;0$">|; 

$key = q/displaystyle=alpha+beta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1290.png"
 ALT="$\displaystyle =\alpha+\beta,$">|; 

$key = q/phi(u_1,u_2,u_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7063.png"
 ALT="$ \phi(u_1,u_2,u_3)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter07slashstokes.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="335" BORDER="0"
 SRC="|."$dir".q|img2535.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter07/stokes.eps}}
\end{figure}">|; 

$key = q/displaystyle=left(frac{m-m'}{m+m'}right),frac{g}{omega^{,2}}left[sin(omega,t)-omega,tright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1854.png"
 ALT="$\displaystyle = \left(\frac{m-m'}{m+m'}\right) \frac{g}{\omega^{ 2}}\left[\sin(\omega t)-\omega t\right],$">|; 

$key = q/displaystyleint_S{bfv}cdotd{bfS}=4pi,r^{,2},v_r(r)=Q,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2606.png"
 ALT="$\displaystyle \int_S {\bf v} \cdot d{\bf S} = 4\pi r^{ 2} v_r(r) = Q,$">|; 

$key = q/displaystyle=f(z_0+dz'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2061.png"
 ALT="$\displaystyle =f(z_0+dz'),$">|; 

$key = q/u_c,tanhu_c=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1279.png"
 ALT="$ u_c \tanh u_c=1$">|; 

$key = q/displaystylefrac{Dv_r}{Dt}-frac{v_theta^{,2}+v_varphi^{,2}}{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img502.png"
 ALT="$\displaystyle \frac{Dv_r}{Dt}-\frac{v_\theta^{ 2}+v_\varphi^{ 2}}{r}$">|; 

$key = q/rhorightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6148.png"
 ALT="$ \rho\rightarrow 0$">|; 

$key = q/displaystylea_{sskl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6949.png"
 ALT="$\displaystyle a_{sskl}$">|; 

$key = q/displaystyle={rmi}^{,n+n'+1},kappa_{n,,n'}^{,m,,m'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5011.png"
 ALT="$\displaystyle = {\rm i}^{ n+n'+1}  \kappa_{n, n'}^{ m, m'}.$">|; 

$key = q/displaystylesin^2mu+left(frac{gamma-1}{gamma+1}right)cos^2mu=frac{c_1^{,2}}{w^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6206.png"
 ALT="$\displaystyle \sin^2\mu +\left(\frac{\gamma-1}{\gamma+1}\right)\cos^2\mu=\frac{c_1^{ 2}}{w^{ 2}},
$">|; 

$key = q/{figure}{epsfysize{=2.in{centerline{epsffile{AppendixAslashfigA.20.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img6547.png"
 ALT="\begin{figure}
\epsfysize =2.in
\centerline{\epsffile{AppendixA/figA.20.eps}}
\end{figure}">|; 

$key = q/displaystyle=phi,nablacdot{bfA}+{bfA}cdotnablaphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6689.png"
 ALT="$\displaystyle =\phi  \nabla\cdot {\bf A} + {\bf A}\cdot \nabla\phi,$">|; 

$key = q/theta(xi_1)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5213.png"
 ALT="$ \theta(\xi_1)=0$">|; 

$key = q/h(x,z,t)=xi_z(x,z,t)-z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3948.png"
 ALT="$ h(x,z,t) = \xi_z(x,z,t)-z$">|; 

$key = q/displaystylev_x(y)=frac{G}{2,mu},y,(2,h-y).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3596.png"
 ALT="$\displaystyle v_x(y) = \frac{G}{2 \mu} y (2 h-y).$">|; 

$key = q/displaystyle{bfv}=v(r),{bfe}_r,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2605.png"
 ALT="$\displaystyle {\bf v} = v(r) {\bf e}_r,$">|; 

$key = q/v_xrightarrowV,sinalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2293.png"
 ALT="$ v_x\rightarrow V \sin\alpha$">|; 

$key = q/T_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5393.png"
 ALT="$ T_1$">|; 

$key = q/(pmc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2467.png"
 ALT="$ (\pm c$">|; 

$key = q/displaystylefrac{F''}{F};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5962.png"
 ALT="$\displaystyle \frac{F''}{F}$">|; 

$key = q/displaystylenablaphi=left(frac{partialphi}{partialx},,frac{partialphi}{partialy},,frac{partialphi}{partialz}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6600.png"
 ALT="$\displaystyle \nabla\phi = \left(\frac{\partial \phi}{\partial x},  \frac{\partial \phi} {\partial y},  \frac{\partial \phi}{\partial z}\right),$">|; 

$key = q/displaystylesum_{i=1,3}alpha_i,a_i^{,2}=int_0^inftysum_{i=1,3}frac{a_i^{,2},du}{ft(frac{u}{mitDelta}right)+frac{1}{mitDelta}right]du=alpha_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="545" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7277.png"
 ALT="$\displaystyle \sum_{i=1,3}\alpha_i a_i^{ 2} = \int_0^\infty \sum_{i=1,3}\frac...
...u}\!\left(\frac{u}{\mit\Delta}\right)+\frac{1}{\mit\Delta}\right]du = \alpha_0.$">|; 

$key = q/displaystyleC_p=frac{2}{gamma,{rmMa}_infty^{,2}}left(left[1-frac{1}{2},(gamma-1)+v'^{,2}}{U^{,2}}right)right]^{,gammaslash(gamma-1)}-1right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="565" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5931.png"
 ALT="$\displaystyle C_p = \frac{2}{\gamma {\rm Ma}_\infty^{ 2}}\left(\left[1-\frac{...
...\frac{u'^{ 2}+v'^{ 2}}{U^{ 2}}\right)\right]^{ \gamma/(\gamma-1)}-1\right).$">|; 

$key = q/theta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1664.png"
 ALT="$ \theta_0$">|; 

$key = q/displaystylefrac{d}{dx}!left(y',frac{partialF}{partialy'}right)=frac{dF}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7324.png"
 ALT="$\displaystyle \frac{d}{dx}\!\left(y' \frac{\partial F}{\partial y'}\right) = \frac{dF}{dx},$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$ a$">|; 

$key = q/displaystylerho,g,frac{partialzeta}{partialt}=-rho,gleft.frac{partialphi}{partialz}right|_{z=0}=left.frac{partialp_1}{partialt}right|_{z=0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3912.png"
 ALT="$\displaystyle \rho g \frac{\partial \zeta}{\partial t} = -\rho g\left.\frac{...
...l z}\right\vert _{z=0}=\left.\frac{\partial p_1}{\partial t}\right\vert _{z=0},$">|; 

$key = q/displaystylek^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1299.png"
 ALT="$\displaystyle k^{ 2}$">|; 

$key = q/_y({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$ _y({\bf r},t)$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter03slashfig2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="251" BORDER="0"
 SRC="|."$dir".q|img1122.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter03/fig2.eps}}
\end{figure}">|; 

$key = q/displaystylelambda'+cosh(mpaslash2,h+c)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7356.png"
 ALT="$\displaystyle \lambda' + \cosh(\mp a/2 h + c) = 0.$">|; 

$key = q/{mitPsi}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5344.png"
 ALT="$ {\mit\Psi}=0$">|; 

$key = q/B'C';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2235.png"
 ALT="$ B'C'$">|; 

$key = q/{rmMa}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$ {\rm Ma}_1$">|; 

$key = q/displaystyleG(e)=-left{frac{1+(1slash2,e)lnleft[(1-e)slash(1+e)right]}{(1-e^{,2})^{,-1}+(1slash2,e)lnleft[(1-e)slash(1+e)right]}right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2813.png"
 ALT="$\displaystyle G(e)=-\left\{\frac{1 +(1/2 e)\ln\left[(1-e)/(1+e)\right]} {(1-e^{ 2})^{ -1}+(1/2 e)\ln\left[(1-e)/(1+e)\right]}\right\}.$">|; 

$key = q/displaystylez_M=frac{kappa_x^{,2},A_0}{V_0}+z_H,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img650.png"
 ALT="$\displaystyle z_M = \frac{\kappa_x^{ 2} A_0}{V_0}+ z_H,$">|; 

$key = q/displaystyle=frac{2,gamma+(gamma+1),beta}{2,gamma+(gamma-1),beta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5533.png"
 ALT="$\displaystyle =\frac{2 \gamma+(\gamma+1) \beta}{2 \gamma+(\gamma-1) \beta},$">|; 

$key = q/overline{alpha_C}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5854.png"
 ALT="$ \overline{\alpha_C}=0$">|; 

$key = q/displaystyleF_y(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1814.png"
 ALT="$\displaystyle F_y (t)$">|; 

$key = q/displaystyle{bfv}=v_r(r,theta,t),{bfe}_r+v_theta(r,theta,t),{bfe}_theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1643.png"
 ALT="$\displaystyle {\bf v} = v_r(r,\theta,t) {\bf e}_r + v_\theta(r,\theta,t) {\bf e}_\theta,$">|; 

$key = q/{mitGamma}(z+dz);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3466.png"
 ALT="$ {\mit\Gamma}(z+dz)$">|; 

$key = q/u(x)=v_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5443.png"
 ALT="$ u(x)=v_x$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter07slashdipole.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="465" HEIGHT="277" BORDER="0"
 SRC="|."$dir".q|img2621.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter07/dipole.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{partial^{,2}psi}{partialx,partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2932.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\psi}{\partial x \partial y}$">|; 

$key = q/displaystyle({bfa}times{bfb})_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6784.png"
 ALT="$\displaystyle ({\bf a}\times {\bf b})_3$">|; 

$key = q/dx=dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6458.png"
 ALT="$ dx=dy$">|; 

$key = q/displaystylefrac{partialp}{partialz}+rho,frac{partial({mitPsi}+{mitPsi}')}{partialz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img817.png"
 ALT="$\displaystyle \frac{\partial p}{\partial z} + \rho \frac{\partial ({\mit\Psi}+{\mit\Psi}')}{\partial z}$">|; 

$key = q/{rmFr}<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1443.png"
 ALT="$ {\rm Fr}&lt;1$">|; 

$key = q/theta_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5709.png"
 ALT="$ \theta_{\rm max}$">|; 

$key = q/displaystyleF(z)=-frac{2,c,{rmi},V_0}{n},cot(zetaslashn),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2477.png"
 ALT="$\displaystyle F(z) = -\frac{2 c {\rm i} V_0}{n} \cot(\zeta/n),
$">|; 

$key = q/displaystylefrac{c_p}{c_v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5610.png"
 ALT="$\displaystyle \frac{c_p}{c_v}$">|; 

$key = q/v_z(y=0_+);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3378.png"
 ALT="$ v_z(y=0_+)$">|; 

$key = q/eta=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2705.png"
 ALT="$ \eta=\pi$">|; 

$key = q/-p,{bfn};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="$ -p {\bf n}$">|; 

$key = q/displaystylexi_r=frac{x_i,xi_i}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4402.png"
 ALT="$\displaystyle \xi_r = \frac{x_i \xi_i}{r},$">|; 

$key = q/displaystyle=(1-pislash4),alpha^{,-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3267.png"
 ALT="$\displaystyle =(1-\pi/4) \alpha^{ -1},$">|; 

$key = q/displaystylepsi(varpi,z)=-frac{1}{4},Vleft[a^{,2}+z^{,2}+varpi^{,2}-sqrt{(a^{,2}^{,2})^2-4,a^{,2},varpi^{,2}},right]+frac{1}{2},V,varpi^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="618" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2794.png"
 ALT="$\displaystyle \psi(\varpi,z)=- \frac{1}{4} V\left[a^{ 2}+z^{ 2}+\varpi^{ 2}...
...rpi^{ 2})^2-4 a^{ 2} \varpi^{ 2}} \right] + \frac{1}{2} V \varpi^{ 2}.$">|; 

$key = q/p_2slashp_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5643.png"
 ALT="$ p_2/p_1$">|; 

$key = q/zeta(0)>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3998.png"
 ALT="$ \zeta(0)&gt;0$">|; 

$key = q/displaystylelambda(theta)=0.47,frac{h^{,4}(theta),g(theta)}{f^{,5}(theta)}int_{theta_0}^thetafrac{f^{,5}(theta')}{h^{,4}(theta')},dtheta'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3364.png"
 ALT="$\displaystyle \lambda(\theta) = 0.47 \frac{h^{ 4}(\theta) g(\theta)}{f^{ 5}...
...ta)}\int_{\theta_0}^\theta \frac{f^{ 5}(\theta')}{h^{ 4}(\theta')} d\theta'.$">|; 

$key = q/b=-c=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2197.png"
 ALT="$ b=-c=1$">|; 

$key = q/displaystylev({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5082.png"
 ALT="$\displaystyle v({\bf r},t)$">|; 

$key = q/delta_0+delta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img611.png"
 ALT="$ \delta_0+\delta_1$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter06slashschwarz2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="256" BORDER="0"
 SRC="|."$dir".q|img2178.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter06/schwarz2.eps}}
\end{figure}">|; 

$key = q/displaystylecpmsqrt{c^{,2}-a^{,2}slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1051.png"
 ALT="$\displaystyle c\pm \sqrt{c^{ 2}-a^{ 2}/2}.
$">|; 

$key = q/displaystylep_n-p_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5753.png"
 ALT="$\displaystyle p_n-p_1$">|; 

$key = q/(xi,,eta,,varpi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2689.png"
 ALT="$ (\xi, \eta, \varpi)$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi}=C_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1377.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} + {\mit\Psi} = C_1,$">|; 

$key = q/y=dslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3583.png"
 ALT="$ y=d/2$">|; 

$key = q/{rmIm}(z)rightarrowb;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2250.png"
 ALT="$ {\rm Im}(z)\rightarrow b$">|; 

$key = q/displaystyle=frac{dyslashdbeta}{dxslashdbeta}=frac{(1slash2),dXslashdbeta,sinbeteta)}{dXslashdbeta,[1-(1slash2),sin^2beta]-X,sinbeta,cosbeta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4042.png"
 ALT="$\displaystyle =\frac{dy/d\beta}{dx/d\beta} = \frac{(1/2) dX/d\beta \sin\beta\...
...\beta-\sin^2\beta)} {dX/d\beta [1-(1/2) \sin^2\beta]-X \sin\beta \cos\beta}$">|; 

$key = q/C_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1380.png"
 ALT="$ C_2$">|; 

$key = q/{rmFr}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1607.png"
 ALT="$ {\rm Fr}_2$">|; 

$key = q/displaystyle(widetilde{nabla{bfA}})_{thetatheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7149.png"
 ALT="$\displaystyle (\widetilde{\nabla {\bf A}})_{\theta\theta}$">|; 

$key = q/displaystyle=v(x,y),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3012.png"
 ALT="$\displaystyle = v(x,y),$">|; 

$key = q/displaystylerightarrowP(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2868.png"
 ALT="$\displaystyle \rightarrow P(x)$">|; 

$key = q/displaystyleD=2int_0^Lleft.sigma_{xy}right|_{y=0},dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2993.png"
 ALT="$\displaystyle D = 2\int_0^L\left.\sigma_{xy}\right\vert _{y=0} dx,$">|; 

$key = q/H-d(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1527.png"
 ALT="$ H-d(x)$">|; 

$key = q/displaystyle{calG}(x)equivfrac{1}{2}left(-1+sqrt{1+8,x^{,2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1492.png"
 ALT="$\displaystyle {\cal G}(x)\equiv \frac{1}{2}\left(-1+\sqrt{1+8 x^{ 2}}\right).$">|; 

$key = q/displaystylefrac{partialu}{partialt}+u,frac{partialu}{partialx}+frac{2,a}{gamma-1},frac{partialc}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5657.png"
 ALT="$\displaystyle \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} +\frac{2 a}{\gamma-1} \frac{\partial c}{\partial x}$">|; 

$key = q/{bfd}=d,(costheta_0,{bfe}_x+sintheta_0,{bfe}_y)=d,[cos(theta-theta_0),{bfe}_r-sin(theta-theta_0),{bfe}_theta];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="509" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1690.png"
 ALT="$ {\bf d} = d (\cos\theta_0 {\bf e}_x+ \sin\theta_0 {\bf e}_y)=d [\cos(\theta-\theta_0) {\bf e}_r
-\sin(\theta-\theta_0) {\bf e}_\theta]$">|; 

$key = q/displaystylefrac{Doverline{rho}}{Doverline{t}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="$\displaystyle \frac{D\overline{\rho}}{D\overline{t}}$">|; 

$key = q/3,alpha=-2,beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="$ 3 \alpha=-2 \beta$">|; 

$key = q/displaystylea_{ijk}={calR}_{li},{calR}_{mj},{calR}_{nk},a_{lmn}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6844.png"
 ALT="$\displaystyle a_{ijk} = {\cal R}_{li} {\cal R}_{mj} {\cal R}_{nk} a_{lmn}'.$">|; 

$key = q/overline{bfv}={bfv}slashV_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="$ \overline{\bf v} = {\bf v}/V_0$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter09slashellipse.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="338" BORDER="0"
 SRC="|."$dir".q|img3304.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter09/ellipse.eps}}
\end{figure}">|; 

$key = q/displaystylev(x,0)=left|frac{dF}{dz}right|(x,0)=frac{Q}{pi,a},frac{|x|slasha}{1+x^{,2}slasha^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2028.png"
 ALT="$\displaystyle v(x,0)= \left\vert\frac{dF}{dz}\right\vert(x, 0) = \frac{Q}{\pi a} \frac{\vert x\vert/a}{1+x^{ 2}/a^{ 2}}.$">|; 

$key = q/=omega,{bfe}_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img974.png"
 ALT="$ =\omega {\bf e}_3$">|; 

$key = q/tanh(xi_0slash2)=[(1-e)slash(1+e)]^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2802.png"
 ALT="$ \tanh(\xi_0/2)=[(1-e)/(1+e)]^{1/2}$">|; 

$key = q/dz=r-a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5061.png"
 ALT="$ dz = r-a$">|; 

$key = q/rho({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="$ \rho({\bf r},t)$">|; 

$key = q/displaystylefrac{dF}{dz}=V,{rme}^{,{rmi},alpha}+frac{A}{z}+frac{B}{z^{,2}}+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3277.png"
 ALT="$\displaystyle \frac{dF}{dz} =V {\rm e}^{ {\rm i} \alpha} + \frac{A}{z}+ \frac{B}{z^{ 2}}+\cdots.$">|; 

$key = q/delta_1slashdelta_2=2.55;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3215.png"
 ALT="$ \delta_1/\delta_2=2.55$">|; 

$key = q/P=P(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3637.png"
 ALT="$ P=P(r)$">|; 

$key = q/theta(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5204.png"
 ALT="$ \theta(r)$">|; 

$key = q/displaystylesimeqfrac{2}{k}-frac{k}{2},[1-cos(2,skew{5}tilde{phi})].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1213.png"
 ALT="$\displaystyle \simeq \frac{2}{k}-\frac{k}{2} [1-\cos(2 \skew{5}\tilde{\phi})].$">|; 

$key = q/{bfD};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3445.png"
 ALT="$ {\bf D}$">|; 

$key = q/displaystyle=frac{v_y}{U_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2882.png"
 ALT="$\displaystyle = \frac{v_y}{U_1},$">|; 

$key = q/displaystylefrac{beta}{(1-beta)^4}=zeta^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5227.png"
 ALT="$\displaystyle \frac{\beta}{(1-\beta)^4} = \zeta^{ 2},$">|; 

$key = q/displaystyleG(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3679.png"
 ALT="$\displaystyle G(z)$">|; 

$key = q/displaystylev_theta(r)=left{{array}{lll}({mitGamma}slash2pi),(rslasha^{,2})&mboxrleqa[0.5ex]({mitGamma}slash2pi),(1slashr)&&r>a{array}right.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1717.png"
 ALT="$\displaystyle v_\theta(r)= \left\{\begin{array}{lll} ({\mit\Gamma}/2\pi) (r/a^...
...ace{1cm}}&amp;r\leq a\ [0.5ex] ({\mit\Gamma}/2\pi) (1/r)&amp;&amp;r&gt;a \end{array}\right.,$">|; 

$key = q/mu=0.68;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5236.png"
 ALT="$ \mu=0.68$">|; 

$key = q/displaystyle=z^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1945.png"
 ALT="$\displaystyle = z^{ 2},$">|; 

$key = q/H_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img717.png"
 ALT="$ H_0$">|; 

$key = q/displaystyle(rho-rho'),g=frac{k,rho,(V-c)^2}{tanh(k,d)}+frac{k,rho',(V'-c)^2}{tanh(k,d')},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4107.png"
 ALT="$\displaystyle (\rho-\rho') g = \frac{k \rho (V-c)^2}{\tanh(k d)}+ \frac{k \rho' (V'-c)^2}{\tanh(k d')},$">|; 

$key = q/displaystylefrac{d{bfa}}{dt}=frac{dphi}{dt},{bfb}+phi,frac{d{bfb}}{dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6438.png"
 ALT="$\displaystyle \frac{d {\bf a}}{dt} = \frac{d\phi}{dt}  {\bf b} + \phi  \frac{d {\bf b}}{dt}.$">|; 

$key = q/displaystylefrac{c_p}{c_v}=frac{kappa_T}{kappa_{calS}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5606.png"
 ALT="$\displaystyle \frac{c_p}{c_v} = \frac{\kappa_T}{\kappa_{\cal S}}.
$">|; 

$key = q/displaystyle=-frac{partialphi}{partialx}=-frac{U,epsilon,alpha}{lambda},cos[alpha,(x-lambda,y)],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5995.png"
 ALT="$\displaystyle =-\frac{\partial\phi}{\partial x} = -\frac{U \epsilon \alpha}{\lambda} \cos[\alpha (x-\lambda y)],$">|; 

$key = q/1slashr^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1699.png"
 ALT="$ 1/r^{ 2}$">|; 

$key = q/{rmFr},{rmPr}sim{calO}(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="$ {\rm Fr}, {\rm Pr} \sim {\cal O}(1)$">|; 

$key = q/{rmMa}_3simeq{rmMa}_1,(1-2,delta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6175.png"
 ALT="$ {\rm Ma}_3\simeq {\rm Ma}_1 (1-2 \delta)$">|; 

$key = q/ABCA'B'C';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2203.png"
 ALT="$ ABCA'B'C'$">|; 

$key = q/displaystyleh'=frac{int_{A_0}int_0^{delta'}z,dx,dy,dz}{V_0}=frac{1}{2,V_0}int_{A2}(x,y)-2,theta,y,delta(x,y)+{calO}(theta^{,2})right]dx,dy=h,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="627" HEIGHT="81" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img638.png"
 ALT="$\displaystyle h' = \frac{\int_{A_0}\int_0^{\delta'} z dx dy dz}{V_0}=\frac{1...
... 2}(x,y) -2 \theta y \delta(x,y) + {\cal O}(\theta^{ 2})\right]dx dy = h,$">|; 

$key = q/widehat{mitGamma}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3340.png"
 ALT="$ \widehat{\mit\Gamma}=0$">|; 

$key = q/beta=pmtan^{-1}(1slashsqrt{8})=pm19.47^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4055.png"
 ALT="$ \beta=\pm \tan^{-1}(1/\sqrt{8}) = \pm 19.47^\circ$">|; 

$key = q/z=r-a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4455.png"
 ALT="$ z=r-a$">|; 

$key = q/stackrel{displaystylerightarrow}{BC};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="30" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6703.png"
 ALT="$ \stackrel{\displaystyle \rightarrow}{BC}$">|; 

$key = q/xrightarrowx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1189.png"
 ALT="$ x\rightarrow x$">|; 

$key = q/displaystylepsirightarrowfrac{Q}{4pi}left(frac{d,cos^2theta}{r}-frac{d}{r}right)frac{D}{4pi},frac{varpi^{,2}}{(varpi^{,2}+z^{,2})^{3slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="500" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2632.png"
 ALT="$\displaystyle \psi\rightarrow \frac{Q}{4\pi}\left(\frac{d \cos^2\theta}{r}-\fr...
...\theta}{r} =-\frac{D}{4\pi} \frac{\varpi^{ 2}}{(\varpi^{ 2}+z^{ 2})^{3/2}},$">|; 

$key = q/displaystylephantom{=}left.-frac{v_theta}{r^{,2},sin^2theta}-frac{2,cottheta}{r^hi}+frac{1}{3,r},frac{partial{mitDelta}}{partialtheta}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img507.png"
 ALT="$\displaystyle \phantom{=}\left.-\frac{v_\theta}{r^{ 2} \sin^2\theta}-\frac{2\...
...l\varphi}+ \frac{1}{3 r} \frac{\partial{\mit\Delta}}{\partial \theta}\right),$">|; 

$key = q/{rmMa}_3simeq{rmMa}_4simeq{rmMa}_1,(1-delta_1-delta_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6186.png"
 ALT="$ {\rm Ma}_3\simeq {\rm Ma}_4 \simeq {\rm Ma}_1 (1-\delta_1-\delta_2)$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img755.png"
 ALT="$ u$">|; 

$key = q/displaystylec_V=M,{calC}_V.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5306.png"
 ALT="$\displaystyle c_V = M {\cal C}_V.$">|; 

$key = q/displaystylec^{,2}+frac{1}{2},(gamma-1),q^{,2}=c_infty^{,2}+frac{1}{2},(gamma-1),U^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5925.png"
 ALT="$\displaystyle c^{ 2} + \frac{1}{2} (\gamma-1) q^{ 2} = c_\infty^{ 2}+\frac{1}{2} (\gamma-1) U^{ 2},$">|; 

$key = q/displaystyle=a,frac{sinh[k,(z+d)]}{sinh(k,d)},sin(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3969.png"
 ALT="$\displaystyle =a \frac{\sinh[k (z+d)]}{\sinh(k d)} \sin(\omega t-k x),$">|; 

$key = q/S_1=S_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1401.png"
 ALT="$ S_1=S_2$">|; 

$key = q/theta=theta_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3362.png"
 ALT="$ \theta=\theta_\pm$">|; 

$key = q/displaystyle={rmi}^{,n+n'+1},beta_{-n,,-n'}^{,-m,,-m'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5005.png"
 ALT="$\displaystyle = {\rm i}^{ n+n'+1} \beta_{-n, -n'}^{ -m, -m'},$">|; 

$key = q/e_{13}<0.81267;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img956.png"
 ALT="$ e_{13}&lt; 0.81267$">|; 

$key = q/displaystylefrac{v}{u}=frac{beta(y)}{u_0(y)sqrt{x_0-x}}+{calO}(1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3129.png"
 ALT="$\displaystyle \frac{v}{u} = \frac{\beta(y)}{u_0(y) \sqrt{x_0-x}}+ {\cal O}(1).$">|; 

$key = q/displaystylepsi(x,y)=V_0left(sintheta_0,x-costheta_0,yright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1665.png"
 ALT="$\displaystyle \psi(x,y) = V_0\left(\sin\theta_0 x-\cos\theta_0 y\right).$">|; 

$key = q/displaystyler_1-r_2=frac{d,cos[(theta_2+theta_1)slash2]}{cos[(theta_2-theta_1)slash2]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2626.png"
 ALT="$\displaystyle r_1 - r_2= \frac{d \cos[(\theta_2+\theta_1)/2]}{\cos[(\theta_2-\theta_1)/2]}.$">|; 

$key = q/displaystylep;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2140.png"
 ALT="$\displaystyle p$">|; 

$key = q/displaystyle=a_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6798.png"
 ALT="$\displaystyle = a_i,$">|; 

$key = q/displaystylefrac{Dv_z}{Dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="$\displaystyle \frac{Dv_z}{Dt}$">|; 

$key = q/displaystylephantom{=}-tan^{-1}left(left[{rmMa}^{,2}-1right]^{1slash2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6212.png"
 ALT="$\displaystyle \phantom{=}-\tan^{-1}\left(\left[{\rm Ma}^{ 2}-1\right]^{1/2}\right).$">|; 

$key = q/rho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="$ \rho_0$">|; 

$key = q/{figure}{epsfysize{=2.4in{centerline{epsffile{Chapter12slashdiurnal2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="547" HEIGHT="274" BORDER="0"
 SRC="|."$dir".q|img4720.png"
 ALT="\begin{figure}
\epsfysize =2.4in
\centerline{\epsffile{Chapter12/diurnal2.eps}}
\end{figure}">|; 

$key = q/displaystyle=pmfrac{2}{k},(1-k^{,2},sin^2phi)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1187.png"
 ALT="$\displaystyle = \pm\frac{2}{k} (1-k^{ 2} \sin^2\phi)^{1/2},$">|; 

$key = q/P';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3404.png"
 ALT="$ P'$">|; 

$key = q/displaystylec^{,2},frac{drho}{rho}+u,du=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5623.png"
 ALT="$\displaystyle c^{ 2} \frac{d\rho}{\rho}+u du=0,
$">|; 

$key = q/displaystyledV=r^{,2},sintheta,dr,dtheta,dphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7177.png"
 ALT="$\displaystyle dV = r^{ 2} \sin\theta dr d\theta d\phi.$">|; 

$key = q/displaystylefrac{partial^{,2}phi}{partialx^{,2}}=frac{partial^{,2}psi}{partialx,si}{partialy,partialx}=-frac{partial^{,2}phi}{partialy^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1659.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\phi}{\partial x^{ 2}} = \frac{\partial^{ 2...
...2}\psi}{\partial y \partial x} = -\frac{\partial^{ 2}\phi}{\partial y^{ 2}},$">|; 

$key = q/H';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img621.png"
 ALT="$ H'$">|; 

$key = q/displaystylefrac{1}{gamma-1},left(rho_2^{,gamma-1}-rho_1^{,gamma-1}right)left(frac{p_1}{rho_1^{,gamma}}-frac{p_2}{rho_2^{,gamma}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5266.png"
 ALT="$\displaystyle \frac{1}{\gamma-1} \left(\rho_2^{ \gamma-1}-\rho_1^{ \gamma-1}\right)\left(\frac{p_1}{\rho_1^{ \gamma}}-\frac{p_2}{\rho_2^{ \gamma}}\right),
$">|; 

$key = q/displaystyle=-frac{g,a^{,2}}{r}left(1+3,frac{d}{a},frac{skew{3}bar{rho}}{rho}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4365.png"
 ALT="$\displaystyle =- \frac{g a^{ 2}}{r}\left(1 + 3 \frac{d}{a} \frac{\skew{3}\bar{\rho}}{\rho}\right),$">|; 

$key = q/{bfV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2428.png"
 ALT="$ {\bf V}$">|; 

$key = q/displaystylegamma_{23}+gamma_{12}>gamma_{13}>gamma_{23}-gamma_{12}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1131.png"
 ALT="$\displaystyle \gamma_{23}+\gamma_{12}&gt; \gamma_{13} &gt; \gamma_{23}-\gamma_{12}.$">|; 

$key = q/displaystylefrac{{mitDelta}t(theta,phi)}{T}=frac{{mitDelta}varphi(theta,phi)}{2pi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5029.png"
 ALT="$\displaystyle \frac{{\mit\Delta}t(\theta,\phi)}{T} = \frac{{\mit\Delta}\varphi(\theta,\phi)}{2\pi},$">|; 

$key = q/x=x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3670.png"
 ALT="$ x=x_2$">|; 

$key = q/displaystylefrac{partialp}{partialr}+rho,frac{partial({mitPsi}+{mitPsi}')}{partialr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img816.png"
 ALT="$\displaystyle \frac{\partial p}{\partial r} + \rho \frac{\partial ({\mit\Psi}+{\mit\Psi}')}{\partial r}$">|; 

$key = q/displaystyle-int_{-h}^hrho,U_0,dy+int_{-h}^{h}rho,[U_0-u(y)],dy+2int_{-l}^{l}rho,v(x),dx=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3046.png"
 ALT="$\displaystyle -\int_{-h}^h\rho U_0 dy + \int_{-h}^{h}\rho [U_0-u(y)] dy+2\int_{-l}^{l}\rho v(x) dx = 0,$">|; 

$key = q/displaystyle=frac{partialphi}{partialx}+{rmi},frac{partialpsi}{partialx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1963.png"
 ALT="$\displaystyle = \frac{\partial \phi}{\partial x} + {\rm i}  \frac{\partial \psi} {\partial x}.$">|; 

$key = q/alpha=2,mslash(m+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2955.png"
 ALT="$ \alpha=2 m/(m+1)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashsource2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img2050.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/source2.eps}}
\end{figure}">|; 

$key = q/displaystyle=u_0,{rme}^{,{rmi},(sigma,t-{bfk}cdot{bfr})},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5081.png"
 ALT="$\displaystyle = u_0 {\rm e}^{ {\rm i} (\sigma t-{\bf k}\cdot{\bf r})},$">|; 

$key = q/displaystyle=-rho,{mitGamma},V_y-m',frac{dV_x}{dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1840.png"
 ALT="$\displaystyle = -\rho {\mit\Gamma} V_y - m' \frac{dV_x}{dt},$">|; 

$key = q/nablacdot{bfn}neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1111.png"
 ALT="$ \nabla\cdot{\bf n}\neq 0$">|; 

$key = q/rho=rho(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5342.png"
 ALT="$ \rho=\rho(x,t)$">|; 

$key = q/|alpha|>|beta|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1295.png"
 ALT="$ \vert\alpha\vert&gt;\vert\beta\vert$">|; 

$key = q/displaystyle=-frac{1}{h_eta},frac{partialphi}{partialeta}=frac{1}{varpi,h_xi},frac{partialpsi}{partialxi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2696.png"
 ALT="$\displaystyle = -\frac{1}{h_\eta} \frac{\partial\phi}{\partial\eta} =\frac{1}{\varpi h_\xi} \frac{\partial\psi}{\partial \xi}.$">|; 

$key = q/displaystylefrac{W}{g},frac{d^{,2}delta_1}{dt^{,2}}=W-B'=-frac{W}{delta_0},delta_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img615.png"
 ALT="$\displaystyle \frac{W}{g} \frac{d^{ 2} \delta_1}{dt^{ 2}} = W- B' = -\frac{W}{\delta_0} \delta_1,$">|; 

$key = q/displaystylephantom{=}+int_{mitOmega}frac{1}{sintheta}left(frac{partial{mitPhi}'ltheta},frac{partial{mitPsi}_r}{partialphi}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4907.png"
 ALT="$\displaystyle \phantom{=}+\int_{\mit\Omega}\frac{1}{\sin\theta}\left(\frac{\par...
...\partial\theta} \frac{\partial{\mit\Psi}_r}{\partial\phi}\right)d{\mit\Omega}.$">|; 

$key = q/overline{{mitPsi}}={mitPsi}slash(g,L);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="$ \overline{{\mit\Psi}} = {\mit\Psi}/(g L)$">|; 

$key = q/psi=psi(r,theta,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1728.png"
 ALT="$ \psi=\psi(r,\theta,t)$">|; 

$key = q/thetageq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img529.png"
 ALT="$ \theta\geq 0$">|; 

$key = q/widetilde{phi}(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2089.png"
 ALT="$ \widetilde{\phi}(x,y)$">|; 

$key = q/BCD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4051.png"
 ALT="$ BCD$">|; 

$key = q/displaystylet_astsimeqfrac{2}{(gamma+1),k,u_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5668.png"
 ALT="$\displaystyle t_\ast \simeq \frac{2}{(\gamma+1) k u_0}
$">|; 

$key = q/displaystylefrac{partialpsi}{partialeta}=frac{partialpsi_0}{partialeta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2711.png"
 ALT="$\displaystyle \frac{\partial\psi}{\partial\eta} = \frac{\partial\psi_0}{\partial\eta}$">|; 

$key = q/displaystylea_{ij}'=b_{ij}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6846.png"
 ALT="$\displaystyle a_{ij}'=b_{ij}'$">|; 

$key = q/1geqk>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1317.png"
 ALT="$ 1\geq k&gt; 0$">|; 

$key = q/displaystyleK;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2809.png"
 ALT="$\displaystyle K$">|; 

$key = q/v_x(0)=v_x(d)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3581.png"
 ALT="$ v_x(0)=v_x(d)=0$">|; 

$key = q/displaystylefrac{partialzeta}{partialt}=-frac{d}{a,sintheta}left[frac{partial}{partialtheta},(sintheta,u)+frac{partialv}{partialphi}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4477.png"
 ALT="$\displaystyle \frac{\partial \zeta}{\partial t}= - \frac{d}{a \sin\theta}\left...
...rtial}{\partial\theta} (\sin\theta u)+\frac{\partial v}{\partial\phi}\right].$">|; 

$key = q/displaystylebeta_{r,,s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4864.png"
 ALT="$\displaystyle \beta_{r, s}$">|; 

$key = q/{mitDelta}P=P(z_2)-P(z_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3683.png"
 ALT="$ {\mit\Delta} P = P(z_2)-P(z_1)$">|; 

$key = q/eta=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3161.png"
 ALT="$ \eta=1$">|; 

$key = q/displaystyle{calL}-f_z,frac{partial{calL}}{partialf_z}=C',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1251.png"
 ALT="$\displaystyle {\cal L}-f_z \frac{\partial {\cal L}}{\partial f_z} = C',$">|; 

$key = q/q^{,2}=(U+u')^{,2}+v'^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5926.png"
 ALT="$ q^{ 2}= (U+u')^{ 2} + v'^{ 2}$">|; 

$key = q/displaystylel=left(frac{gamma}{rho,g}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4149.png"
 ALT="$\displaystyle l = \left(\frac{\gamma}{\rho g}\right)^{1/2}.$">|; 

$key = q/omega_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4298.png"
 ALT="$ \omega_2$">|; 

$key = q/displaystyle{rmMa}_1lesseqgtr1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5499.png"
 ALT="$\displaystyle {\rm Ma}_1\lesseqgtr 1.$">|; 

$key = q/displaystyle-g^{,-1},delta{mitPhi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4505.png"
 ALT="$\displaystyle - g^{ -1} \delta{\mit\Phi}$">|; 

$key = q/displaystyle=f(z_0+dz'').;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2063.png"
 ALT="$\displaystyle =f(z_0+dz'').$">|; 

$key = q/e_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4591.png"
 ALT="$ e_M$">|; 

$key = q/M=V,rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7271.png"
 ALT="$ M= V \rho$">|; 

$key = q/displaystylefrac{1}{2},rho,g,a^{,2},w,V=frac{1}{2},rho,g,a^{,2},w,v_g+D,V.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3989.png"
 ALT="$\displaystyle \frac{1}{2} \rho g a^{ 2} w V= \frac{1}{2} \rho g a^{ 2} w v_g + D V.$">|; 

$key = q/0.25<s<0.28125;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img804.png"
 ALT="$ 0.25&lt;s&lt;0.28125$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="192" BORDER="0"
 SRC="|."$dir".q|img6255.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.04.eps}}
\end{figure}">|; 

$key = q/displaystyle=nablacdot({calR}_n,nablax_i+x_i,nabla{calR}_n)=2,nablax_icdotnabla{calR}_n=2,frac{partial{calR}_n}{partialx_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4333.png"
 ALT="$\displaystyle = \nabla\cdot({\cal R}_n \nabla x_i + x_i \nabla{\cal R}_n) = 2 \nabla x_i\cdot\nabla{\cal R}_n = 2 \frac{\partial {\cal R}_n}{\partial x_i},$">|; 

$key = q/displaystyle=-V_0,r,sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1974.png"
 ALT="$\displaystyle =-V_0 r \sin\theta.$">|; 

$key = q/+4.25times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4616.png"
 ALT="$ +4.25\times 10^{-2}$">|; 

$key = q/c(phi)=c_0,sinphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3488.png"
 ALT="$ c(\phi)=c_0 \sin\phi$">|; 

$key = q/xi_0<xi<infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2703.png"
 ALT="$ \xi_0&lt;\xi&lt;\infty$">|; 

$key = q/v_p=(gslashk)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4025.png"
 ALT="$ v_p=(g/k)^{1/2}$">|; 

$key = q/displaystylefrac{u}{rho},frac{partialrho}{partialx}+frac{v}{rho},frac{partialrho}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5883.png"
 ALT="$\displaystyle \frac{u}{\rho} \frac{\partial\rho}{\partial x} + \frac{v}{\rho} \frac{\partial \rho}{\partial y}$">|; 

$key = q/(x,,y,,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6275.png"
 ALT="$ (x, y, z)$">|; 

$key = q/displaystyle=oint_C(psi,dx+phi,dy).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2347.png"
 ALT="$\displaystyle =\oint_C (\psi dx+\phi dy).$">|; 

$key = q/displaystylefrac{d{mitGamma}}{dt}=-ointomega_z,{bfv}cdotd{bfS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3327.png"
 ALT="$\displaystyle \frac{d{\mit\Gamma}}{dt} = -\oint \omega_z {\bf v}\cdot d{\bf S}.$">|; 

$key = q/displaystylesimeqfrac{gamma,(gamma+1),{rmMa}_p^{,2}}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5545.png"
 ALT="$\displaystyle \simeq \frac{\gamma (\gamma+1) {\rm Ma}_p^{ 2}}{2},$">|; 

$key = q/displaystyle{mitOmega}(r)=frac{1}{r^{,2}}left(frac{{mitOmega}_1-{mitOmega}_2}{r_r_1^{,-2}-{mitOmega}_1,r_2^{,-2}}{r_1^{,-2}-r_2^{,-2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3638.png"
 ALT="$\displaystyle {\mit\Omega}(r) = \frac{1}{r^{ 2}}\left(\frac{{\mit\Omega}_1-{\m...
...mega}_2 r_1^{ -2}- {\mit\Omega}_1 r_2^{ -2}}{r_1^{ -2}-r_2^{ -2}}\right).$">|; 

$key = q/displaystyle=-frac{v,v_y}{h},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3557.png"
 ALT="$\displaystyle = - \frac{v v_y}{h},$">|; 

$key = q/displaystyle{mitPsi}=-frac{3}{4},G,Mleft(alpha_0-sum_{i=1,3}alpha_i,x_i^{,2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img857.png"
 ALT="$\displaystyle {\mit\Psi} = - \frac{3}{4} G M\left(\alpha_0-\sum_{i=1,3}\alpha_i x_i^{ 2}\right),$">|; 

$key = q/d_0=(d_1+d_2)slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3716.png"
 ALT="$ d_0=(d_1+d_2)/2$">|; 

$key = q/K'=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2195.png"
 ALT="$ K'=1$">|; 

$key = q/a_i({bfx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6872.png"
 ALT="$ a_i({\bf x})$">|; 

$key = q/d^{,2}uslashdt^{,2}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img761.png"
 ALT="$ d^{ 2}u/dt^{ 2}&gt;0$">|; 

$key = q/displaystylea_i',b_i'={calR}_{ij},a_j,{calR}_{ik},b_k={calR}_{ij},{calR}_{ik},a_j,b_k=delta_{jk},a_j,b_k=a_j,b_j=a_i,b_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6855.png"
 ALT="$\displaystyle a_i' b_i' = {\cal R}_{ij} a_j {\cal R}_{ik} b_k ={\cal R}_{ij} {\cal R}_{ik} a_j b_k = \delta_{jk} a_j b_k = a_j b_j=a_i b_i,$">|; 

$key = q/u={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7218.png"
 ALT="$ u={\rm constant}$">|; 

$key = q/widehat{L}>0.30375;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img960.png"
 ALT="$ \widehat{L}&gt;0.30375$">|; 

$key = q/(partial^{,3}uslashpartialy^{,3})_{y=0}proptof'''(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3179.png"
 ALT="$ (\partial^{ 3} u/\partial y^{ 3})_{y=0}\propto f'''(0)=0$">|; 

$key = q/mleqn;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4264.png"
 ALT="$ m\leq n$">|; 

$key = q/displaystyle{mitPsi}(x_1,x_2,x_3)simeq-frac{G,M}{r}-frac{G,(I_1+I_2+I_3)}{2,r^{,frac{3,G,(I_1,x_1^{,2}+I_2,x_2^{,2}+I_3,x_3^{,2})}{2,r^{,5}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="577" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7288.png"
 ALT="$\displaystyle {\mit\Psi}(x_1,x_2,x_3) \simeq -\frac{G M}{r} - \frac{G (I_1+I_...
...}} + \frac{3 G (I_1 x_1^{ 2}+ I_2 x_2^{ 2}+I_3 x_3^{ 2})}{2 r^{ 5}},
$">|; 

$key = q/displaystylecdotd{bfS}=S_2,omega_2-S_1,omega_1=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1548.png"
 ALT="$\displaystyle \cdot d{\bf S} = S_2 \omega_2-S_1 \omega_1 = 0.$">|; 

$key = q/displaystyleC_L=frac{C_{L_0}}{sqrt{1-{rmMa}_infty^{,2}}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6036.png"
 ALT="$\displaystyle C_L = \frac{C_{L_0}}{\sqrt{1-{\rm Ma}_\infty^{ 2}}},$">|; 

$key = q/displaystylem,frac{dV_y}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1844.png"
 ALT="$\displaystyle m \frac{dV_y}{dt}$">|; 

$key = q/displaystyle=p_0-frac{1}{2},rholeft[V^{,2}+left(frac{{mitGamma}}{2pi,a}right)^2ro,(V_x^{,2}-V_y^{,2}),cos(2,theta)+2,rho,V_x,V_y,sin(2,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1837.png"
 ALT="$\displaystyle = p_0-\frac{1}{2} \rho\left[V^{ 2}+\left(\frac{{\mit\Gamma}}{2\...
...ho (V_x^{ 2}-V_y^{ 2}) \cos(2 \theta) + 2 \rho V_x V_y \sin(2 \theta)$">|; 

$key = q/displaystyleint_{z_1}^{z_2}{calL}(f,f_z),dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1248.png"
 ALT="$\displaystyle \int_{z_1}^{z_2} {\cal L}(f,f_z) dz,$">|; 

$key = q/C=(0,,-l);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2837.png"
 ALT="$ C= (0, - l)$">|; 

$key = q/rho,v,{calC},Stimesv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1417.png"
 ALT="$ \rho v {\cal C} S\times v$">|; 

$key = q/DEF;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2716.png"
 ALT="$ DEF$">|; 

$key = q/u_0=sinh^{-1}(1slashzeta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1273.png"
 ALT="$ u_0= \sinh^{-1}(1/\zeta)$">|; 

$key = q/displaystyleDproptoa^{,2}proptoleft[frac{sin(pi-{rmFr}^{,-2})}{pi-{rmFr}^{,-2}},frac{1}{1+pi,{rmFr}^{,2}}right]^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4002.png"
 ALT="$\displaystyle D\propto a^{ 2}\propto \left[\frac{\sin(\pi-{\rm Fr}^{ -2})}{\pi-{\rm Fr}^{ -2}} \frac{1}{1+\pi {\rm Fr}^{ 2}}\right]^{ 2},$">|; 

$key = q/{bfn}cdot{bfU}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3434.png"
 ALT="$ {\bf n}\cdot{\bf U}=0$">|; 

$key = q/displaystylev=frac{2,V_0}{n^{,2}}left[1-cos(pi,nslash2)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2490.png"
 ALT="$\displaystyle v = \frac{2 V_0}{n^{ 2}}\left[1-\cos(\pi n/2)\right].
$">|; 

$key = q/{bfa}cdot{bfb}=a_x,b_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6428.png"
 ALT="$ {\bf a}\cdot {\bf b} = a_x  b_x$">|; 

$key = q/displaystyleV=int_{x_1}^{x_2}dxint_{y_1(x)}^{y_2(x)}D(x,y),dy.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6490.png"
 ALT="$\displaystyle V = \int_{x_1}^{x_2} dx \int_{y_1(x)}^{y_2(x)} D(x,y) dy.$">|; 

$key = q/rho'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4108.png"
 ALT="$ \rho'=0$">|; 

$key = q/d{bfS}=(dS_x,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6637.png"
 ALT="$ d{\bf S} = (dS_x, 0, 0)$">|; 

$key = q/displaystylepsi=frac{Q}{4pi}left(frac{z-dslash2}{r_2}-frac{z+dslash2}{r_1}right)}right)-frac{d}{2}left(frac{1}{r_2}+frac{1}{r_1}right)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="565" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2631.png"
 ALT="$\displaystyle \psi = \frac{Q}{4\pi}\left(\frac{z-d/2}{r_2}-\frac{z+d/2}{r_1}\ri...
...frac{1}{r_1}\right)-\frac{d}{2}\left(\frac{1}{r_2}+\frac{1}{r_1}\right)\right].$">|; 

$key = q/infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$ \infty $">|; 

$key = q/displaystylesimeqfrac{2}{gamma-1}left(frac{gamma-1}{gamma+1}right)^{1slash2}left(frac{{rmMa}_1^{,2}-1}{{rmMa}_1^{,2}}right)(z-z_1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6157.png"
 ALT="$\displaystyle \simeq \frac{2}{\gamma-1}\left(\frac{\gamma-1}{\gamma+1}\right)^{1/2}\left(\frac{{\rm Ma}_1^{ 2}-1}{{\rm Ma}_1^{ 2}}\right)(z-z_1),$">|; 

$key = q/0.47399;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img921.png"
 ALT="$ 0.47399$">|; 

$key = q/zetall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5233.png"
 ALT="$ \zeta\ll 1$">|; 

$key = q/displaystyleepsilon,alphallsqrt{1-{rmMa}_infty^{,2}},~~frac{(1-{rmMa}_infty^{,2}}_infty^{,2},(gamma+1)},~~frac{1}{sqrt{1-{rmMa}_infty^{,2}}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5984.png"
 ALT="$\displaystyle \epsilon \alpha\ll \sqrt{1-{\rm Ma}_\infty^{ 2}},&nbsp;&nbsp;\frac{(1-{\r...
...{\rm Ma}_\infty^{ 2} (\gamma+1)},&nbsp;&nbsp; \frac{1}{\sqrt{1-{\rm Ma}_\infty^{ 2}}}.$">|; 

$key = q/displaystyle=-P,dy,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2384.png"
 ALT="$\displaystyle =-P dy,$">|; 

$key = q/displaystyle=frac{1}{z}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1946.png"
 ALT="$\displaystyle = \frac{1}{z}.$">|; 

$key = q/(dslash2),(cosalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2445.png"
 ALT="$ (d/2) (\cos\alpha$">|; 

$key = q/displaystylefrac{dP}{dx}=-G=-rho,g,sin,alpha,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3589.png"
 ALT="$\displaystyle \frac{dP}{dx} = -G= - \rho g \sin \alpha,$">|; 

$key = q/p_0slashp_{rmatm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5426.png"
 ALT="$ p_0/p_{\rm atm}$">|; 

$key = q/displaystyle|u_2|=frac{2,c_4}{gamma_4-1}left[1-left(frac{p_2}{p_4}right)^{(gamma_4-1)slashgamma_4}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5641.png"
 ALT="$\displaystyle \vert u_2\vert = \frac{2 c_4}{\gamma_4-1}\left[1-\left(\frac{p_2}{p_4}\right)^{(\gamma_4-1)/\gamma_4}\right],
$">|; 

$key = q/displaystyle=a_xcostheta+a_ysintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6286.png"
 ALT="$\displaystyle = a_x\cos\theta + a_y\sin \theta,$">|; 

$key = q/dzeta'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2080.png"
 ALT="$ d\zeta''$">|; 

$key = q/displaystyle-u,frac{partialv}{partialy}+v,frac{partialu}{partialy}=-u^{,2}frac{partial}{partialy}!left(frac{v}{u}right)simeq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3127.png"
 ALT="$\displaystyle -u \frac{\partial v}{\partial y} +v \frac{\partial u}{\partial y} =- u^{ 2}\frac{\partial}{\partial y}\!\left(\frac{v}{u}\right)\simeq 0.$">|; 

$key = q/displaystylefrac{U_0^{,2}}{L}left(V_x,frac{partialV_x}{partialX}+V_y,frac{partiax}{partialX^{,2}}+frac{partial^{,2}V_x}{partialY^{,2}}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="594" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2889.png"
 ALT="$\displaystyle \frac{U_0^{ 2}}{L}\left(V_x \frac{\partial V_x}{\partial X} + V...
... 2} V_x}{\partial X^{ 2}}+\frac{\partial^{ 2} V_x}{\partial Y^{ 2}}\right].$">|; 

$key = q/displaystyleJ_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2346.png"
 ALT="$\displaystyle J_i$">|; 

$key = q/h_xi=|nablaxi|^{,-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2691.png"
 ALT="$ h_\xi=\vert\nabla\xi\vert^{ -1}$">|; 

$key = q/displaystyle=left(frac{partialX}{partialW}right)_Zleftslashleft(frac{partialY}{partialW}right)_Zright.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5575.png"
 ALT="$\displaystyle = \left(\frac{\partial X}{\partial W}\right)_Z\left/\left(\frac{\partial Y}{\partial W}\right)_Z\right.,$">|; 

$key = q/displaystyle=c,{rme}^{,xi},sineta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2727.png"
 ALT="$\displaystyle = c {\rm e}^{ \xi} \sin\eta,$">|; 

$key = q/y>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2024.png"
 ALT="$ y&gt;0$">|; 

$key = q/{rmi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1934.png"
 ALT="$ {\rm i}$">|; 

$key = q/displaystylem_{rmadded}=frac{4}{3},pi,rho,a^{,3},G(e),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2781.png"
 ALT="$\displaystyle m_{\rm added} = \frac{4}{3} \pi \rho a^{ 3} G(e),$">|; 

$key = q/displaystyle{mitPsi}'=-frac{1}{2},omega^{,2},(x^{,2}+y^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img813.png"
 ALT="$\displaystyle {\mit\Psi}' = -\frac{1}{2} \omega^{ 2} (x^{ 2}+y^{ 2})$">|; 

$key = q/P=(x_0,,y_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6530.png"
 ALT="$ P=(x_0,  y_0)$">|; 

$key = q/displaystylex(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7378.png"
 ALT="$\displaystyle x(\theta)$">|; 

$key = q/rrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3846.png"
 ALT="$ r\rightarrow 0$">|; 

$key = q/displaystyledelta_1(x)=int_0^inftyleft(1-frac{u}{U}right)dy,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3085.png"
 ALT="$\displaystyle \delta_1(x) = \int_0^\infty \left(1-\frac{u}{U}\right)dy,$">|; 

$key = q/50,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1032.png"
 ALT="$ 50 {\rm m}$">|; 

$key = q/w(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3455.png"
 ALT="$ w(z)$">|; 

$key = q/displaystyleC=frac{2}{k^{,2}}-1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1167.png"
 ALT="$\displaystyle C = \frac{2}{k^{ 2}}-1,$">|; 

$key = q/displaystyle=a_{ssi}+a_{iss},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6929.png"
 ALT="$\displaystyle = a_{ssi}+a_{iss},$">|; 

$key = q/lambdalllambda_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4160.png"
 ALT="$ \lambda\ll \lambda_c$">|; 

$key = q/displaystylepsi(r,theta)=left{{array}{lll}({mitGamma}slash4pi),(r^{,2}slasha^{,2eqa[0.5ex]({mitGamma}slash2pi),ln(rslasha)&&r>a{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1715.png"
 ALT="$\displaystyle \psi(r,\theta)= \left\{\begin{array}{lll} ({\mit\Gamma}/4\pi) (r...
...{1cm}}&amp;r\leq a\ [0.5ex] ({\mit\Gamma}/2\pi) \ln(r/a)&amp;&amp;r&gt;a \end{array}\right..$">|; 

$key = q/displaystylen,(m,{bfa});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6246.png"
 ALT="$\displaystyle n (m {\bf a})$">|; 

$key = q/displaystylea_r=lambda_rint_{mitOmega}F,{mitPhi}_r,d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4825.png"
 ALT="$\displaystyle a_r = \lambda_r\int_{\mit\Omega} F {\mit\Phi}_r d{\mit\Omega}.$">|; 

$key = q/phi_0geqphigeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1325.png"
 ALT="$ \phi_0\geq \phi\geq 0$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashdipole.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img1700.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/dipole.eps}}
\end{figure}">|; 

$key = q/b_{ij}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6865.png"
 ALT="$ b_{ij}'$">|; 

$key = q/displaystylepsi(xi,eta)=-frac{1}{2},V,a^{,3},frac{sin^2eta}{c,{rme}^{,xi}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2740.png"
 ALT="$\displaystyle \psi(\xi,\eta) = -\frac{1}{2} V a^{ 3} \frac{\sin^2\eta}{c {\rm e}^{ \xi}}.$">|; 

$key = q/displaystyle=frac{1}{r^{,2}},frac{partial}{partialr},(r^{,2},A_r)+frac{1}{r,sint,A_theta)+frac{1}{r,sintheta},frac{partialA_phi}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="441" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7179.png"
 ALT="$\displaystyle =\frac{1}{r^{ 2}} \frac{\partial}{\partial r} (r^{ 2} A_r) +...
...eta A_\theta)+ \frac{1}{r \sin\theta} \frac{\partial A_\phi}{\partial \phi},$">|; 

$key = q/yslasha=-zeta-sqrt{zeta^{,2}-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2005.png"
 ALT="$ y/a=-\zeta-\sqrt{\zeta^{ 2}-1}$">|; 

$key = q/displaystyle=-ointP,sintheta,a,dtheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1784.png"
 ALT="$\displaystyle = -\oint P \sin\theta a d\theta.$">|; 

$key = q/alpha,delta_{ij},delta_{kl}+beta,delta_{ik},delta_{jl}+gamma,delta_{il},delta_{jk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6898.png"
 ALT="$ \alpha \delta_{ij} \delta_{kl} + \beta \delta_{ik} \delta_{jl} + \gamma \delta_{il} \delta_{jk}$">|; 

$key = q/displaystyle=frac{1}{2},X_0,sin^2beta,cosbeta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4048.png"
 ALT="$\displaystyle =\frac{1}{2} X_0 \sin^2\beta \cos\beta.$">|; 

$key = q/displaystylefrac{d{bfa}}{dt}=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6406.png"
 ALT="$\displaystyle \frac{d {\bf a}}{dt} =$">|; 

$key = q/v_theta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2608.png"
 ALT="$ v_\theta=0$">|; 

$key = q/displaystyle=cot^{-1}left[({rmMa}_1^{,2}-1)^{1slash2}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6144.png"
 ALT="$\displaystyle =\cot^{-1}\left[({\rm Ma}_1^{ 2}-1)^{1/2}\right],$">|; 

$key = q/displaystyle=m,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6014.png"
 ALT="$\displaystyle =m,$">|; 

$key = q/{mitOmega}_z=k,U_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3331.png"
 ALT="$ {\mit\Omega}_z = k U_0$">|; 

$key = q/0<kleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1302.png"
 ALT="$ 0&lt; k\leq 1$">|; 

$key = q/displaystyleF(z)=-Vleft(zeta,{rme}^{-{rmi},alpha}+frac{c^{,2}}{zeta},{rme}^{,{rmi},alpha}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2501.png"
 ALT="$\displaystyle F(z)= -V\left(\zeta {\rm e}^{-{\rm i} \alpha}+\frac{c^{ 2}}{\zeta} {\rm e}^{ {\rm i} \alpha}\right),
$">|; 

$key = q/sigma_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$ \sigma_{ij}$">|; 

$key = q/Lsim{calO}(alpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3524.png"
 ALT="$ L\sim{\cal O}(\alpha)$">|; 

$key = q/displaystyle=c,{rme}^{,xi},coseta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2725.png"
 ALT="$\displaystyle = c {\rm e}^{ \xi} \cos\eta,$">|; 

$key = q/{mitPsi}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4830.png"
 ALT="$ {\mit\Psi}_r$">|; 

$key = q/alpha_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6338.png"
 ALT="$ \alpha_y$">|; 

$key = q/displaystyleF(eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3167.png"
 ALT="$\displaystyle F(\eta)$">|; 

$key = q/omega=rho,{mitGamma}slash(m+m');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1850.png"
 ALT="$ \omega =\rho {\mit\Gamma}/(m+m')$">|; 

$key = q/x_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4326.png"
 ALT="$ x_i$">|; 

$key = q/alpha_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4437.png"
 ALT="$ \alpha_n$">|; 

$key = q/displaystylerho_1,u_1^{,2}+p_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5464.png"
 ALT="$\displaystyle \rho_1 u_1^{ 2}+p_1$">|; 

$key = q/CB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1392.png"
 ALT="$ CB$">|; 

$key = q/{bfv}(P_+)={bfU}(P);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3424.png"
 ALT="$ {\bf v}(P_+) = {\bf U}(P)$">|; 

$key = q/displaystyle=-frac{1}{sintheta}left[frac{partial}{partialtheta},(sintheta,costheta,eta)-frac{partial}{partialphi},(costheta,xi)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4770.png"
 ALT="$\displaystyle =-\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta} (\si...
...eta \cos\theta \eta)-\frac{\partial}{\partial\phi} (\cos\theta \xi)\right],$">|; 

$key = q/w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3984.png"
 ALT="$ w$">|; 

$key = q/displaystylep(a,theta,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1836.png"
 ALT="$\displaystyle p(a,\theta,t)$">|; 

$key = q/Aslasha^{,3}+Bslasha+(1slash2),V=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3845.png"
 ALT="$ A/a^{ 3}+B/a+(1/2) V=0$">|; 

$key = q/displaystylesigma_{rtheta}=mu,r,frac{d}{dr}!left(frac{v_theta}{r}right)=mu,r,frac{d{mitOmega}}{dr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3641.png"
 ALT="$\displaystyle \sigma_{r\theta} = \mu r \frac{d}{dr}\!\left(\frac{v_\theta}{r}\right) = \mu r \frac{d{\mit\Omega}}{dr}.$">|; 

$key = q/displaystylesinbeta=sin(mu+epsilon)simeqsinmu+epsilon,cosmu.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5733.png"
 ALT="$\displaystyle \sin\beta=\sin(\mu+\epsilon)\simeq \sin\mu + \epsilon \cos\mu.$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter15slashtip.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="366" BORDER="0"
 SRC="|."$dir".q|img6070.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter15/tip.eps}}
\end{figure}">|; 

$key = q/displaystylev_theta(r,theta,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1806.png"
 ALT="$\displaystyle v_\theta(r,\theta,t)$">|; 

$key = q/displaystylew(eta)=frac{{mitGamma}_0}{2,b}left(1-frac{eta}{pi}int_0^pifrac{dphi'{mitGamma}_0}{2,b}left(1-frac{|eta|}{sqrt{eta^{,2}-1}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="486" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3503.png"
 ALT="$\displaystyle w(\eta) = \frac{{\mit\Gamma}_0}{2 b}\left(1-\frac{\eta}{\pi}\int...
...{\mit\Gamma}_0}{2 b}\left(1-\frac{\vert\eta\vert}{\sqrt{\eta^{ 2}-1}}\right).$">|; 

$key = q/displaystylefrac{dF}{dz}=-V_0left(1-frac{a^{,2}}{z^{,2}}right)+{rmi},frac{mitGamma}{2pi,z}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1995.png"
 ALT="$\displaystyle \frac{dF}{dz}= -V_0\left(1-\frac{a^{ 2}}{z^{ 2}}\right) + {\rm i} \frac{\mit\Gamma}{2\pi z} = 0.$">|; 

$key = q/displaystylephi(r,theta,t)=-frac{mitGamma}{2pi,r}+V_x(t),frac{a^{,2}}{r},costheta+V_y(t),frac{a^{,2}}{r},sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="418" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1830.png"
 ALT="$\displaystyle \phi(r,\theta,t) = -\frac{\mit\Gamma}{2\pi r}+ V_x(t) \frac{a^{ 2}}{r} \cos\theta+ V_y(t) \frac{a^{ 2}}{r} \sin\theta.$">|; 

$key = q/displaystyle=left(k-frac{2}{k}right)tilde{F}(phi,k)+frac{2}{k},tilde{E}(phi,k),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1185.png"
 ALT="$\displaystyle = \left(k-\frac{2}{k}\right)\tilde{F}(\phi,k) + \frac{2}{k} \tilde{E}(\phi,k),$">|; 

$key = q/displaystyle=a_{sis}+a_{iss},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6931.png"
 ALT="$\displaystyle =a_{sis} + a_{iss},$">|; 

$key = q/displaystylefrac{rho^{,gamma}}{gamma-1},frac{D}{Dt}!left(frac{p}{rho^{,gamma}}right)=frac{kappa,M}R,nabla^{,2}!left(frac{p}{rho}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5113.png"
 ALT="$\displaystyle \frac{\rho^{ \gamma}}{\gamma-1} \frac{D}{D t}\!\left(\frac{p}{\...
...\gamma}}\right)= \frac{\kappa M}R  \nabla^{ 2}\!\left(\frac{p}{\rho}\right).$">|; 

$key = q/1<n<2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2483.png"
 ALT="$ 1&lt;n&lt;2$">|; 

$key = q/displaystyleF(z)=z^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1951.png"
 ALT="$\displaystyle F(z) = z^{ 2},$">|; 

$key = q/displaystyleepsilon_{mis},a_{sjk}+epsilon_{mjs},a_{isk}+epsilon_{mks},a_{ijs}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6908.png"
 ALT="$\displaystyle \epsilon_{mis} a_{sjk} + \epsilon_{mjs} a_{isk}+\epsilon_{mks} a_{ijs}=0.$">|; 

$key = q/displaystyle=0:;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3753.png"
 ALT="$\displaystyle =0:$">|; 

$key = q/-{bfF}=(rho,Q^{,2}slash4pi,a),{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2422.png"
 ALT="$ -{\bf F} = (\rho Q^{ 2}/4\pi a) {\bf e}_y$">|; 

$key = q/displaystylefrac{D{calE}}{Dt}-frac{p}{rho^{,2}},frac{Drho}{Dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img553.png"
 ALT="$\displaystyle \frac{D{\cal E}}{Dt} - \frac{p}{\rho^{ 2}} \frac {D\rho}{Dt}$">|; 

$key = q/{rmMa}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5394.png"
 ALT="$ {\rm Ma}=1$">|; 

$key = q/displaystyleJ=oint_CF(z),dz.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2340.png"
 ALT="$\displaystyle J = \oint_C F(z) dz.$">|; 

$key = q/p_2=p(z_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5162.png"
 ALT="$ p_2=p(z_2)$">|; 

$key = q/displaystylefrac{d{mitTheta}}{dt}+{mitPhi}_{mitTheta}=S_{mitTheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="$\displaystyle \frac{d{\mit\Theta}}{dt} +{\mit\Phi}_{\mit\Theta}= S_{\mit\Theta}.$">|; 

$key = q/displaystyle|zeta|(theta,phi)=left[zeta_c^{,2}(theta,phi)+zeta_s^{,2}(theta,phi)right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="49" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5028.png"
 ALT="$\displaystyle \vert\zeta\vert(\theta,\phi) = \left[\zeta_c^{ 2}(\theta,\phi)+\zeta_s^{ 2}(\theta,\phi)\right]^{1/2}.$">|; 

$key = q/z=x+{rmi},y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1947.png"
 ALT="$ z=x +{\rm i} y$">|; 

$key = q/{bfb}equiv(b_x,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6423.png"
 ALT="$ {\bf b} \equiv (b_x, 0, 0)$">|; 

$key = q/displaystyle=-a,{rme}^{,k,z},cos(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3938.png"
 ALT="$\displaystyle = -a {\rm e}^{ k z} \cos(\omega t-k x),$">|; 

$key = q/displaystyle_ycdotd{bfS},,-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$\displaystyle _y\cdot d{\bf S},  -$">|; 

$key = q/s,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img669.png"
 ALT="$ s c$">|; 

$key = q/displaystyleb_{ij}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6850.png"
 ALT="$\displaystyle b_{ij}'$">|; 

$key = q/nablatimes(r^{,n},{bfe}_r)={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7204.png"
 ALT="$ \nabla\times(r^{ n} {\bf e}_r)={\bf0}$">|; 

$key = q/displaystyle=left.frac{partial}{partialx}right|_eta+frac{m-1}{2},frac{eta}{x}left.frac{partial}{partialeta}right|_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2924.png"
 ALT="$\displaystyle = \left.\frac{\partial}{\partial x}\right\vert _\eta +\frac{m-1}{2} \frac{\eta}{x}\left.\frac{\partial}{\partial\eta}\right\vert _x,$">|; 

$key = q/displaystyleE(phi,k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1180.png"
 ALT="$\displaystyle E(\phi,k)$">|; 

$key = q/displaystyle=int_0^inftyleft[(U-u),frac{dU}{dx}+u,frac{partial(U-u)}{partialx}+(U-u),frac{partialu}{partialx}right]dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="428" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3083.png"
 ALT="$\displaystyle =\int_0^\infty\left[(U-u) \frac{dU}{dx} + u \frac{\partial(U-u)}{\partial x} +(U-u) \frac{\partial u}{\partial x}\right]dy$">|; 

$key = q/pi,a^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1596.png"
 ALT="$ \pi a^{ 2}$">|; 

$key = q/displaystyleskew{5}hat{P}_n^{,m}(x)=b_n^{,m},P_n^{,m}(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4920.png"
 ALT="$\displaystyle \skew{5}\hat{P}_n^{ m}(x) = b_n^{ m} P_n^{ m}(x),$">|; 

$key = q/-theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6342.png"
 ALT="$ -\theta$">|; 

$key = q/displaystyley'^{,2},(-rho,g,y+lambda),(1+y'^{,2})^{-1slash2}-(-rho,g,y+lambda),(1+y'^{,2})^{1slash2}=k,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7346.png"
 ALT="$\displaystyle y'^{ 2} (-\rho g y+\lambda) (1+y'^{ 2})^{-1/2} - (-\rho g y+\lambda) (1+y'^{ 2})^{1/2} = k,$">|; 

$key = q/dot{phantom{r}}=dslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img828.png"
 ALT="$ \dot{\phantom{r}}=d/dt$">|; 

$key = q/displaystyleV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img846.png"
 ALT="$\displaystyle V$">|; 

$key = q/displaystyle=(a_1^{,2}+u)^{1slash2},(a_2^{,2}+u)^{1slash2},(a_3^{,2}+u)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img862.png"
 ALT="$\displaystyle =(a_1^{ 2}+u)^{1/2} (a_2^{ 2}+u)^{1/2} (a_3^{ 2}+u)^{1/2}.$">|; 

$key = q/displaystyleP_2^{,0}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4265.png"
 ALT="$\displaystyle P_2^{ 0}(x)$">|; 

$key = q/intsqrt{1+y'^{,2}},dx=2pislash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7377.png"
 ALT="$ \int\sqrt{1+y'^{ 2}} dx=2\pi/3$">|; 

$key = q/overline{C}+overline{D},a^{,2}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3848.png"
 ALT="$ \overline{C}+ \overline{D} a^{ 2} = 0$">|; 

$key = q/AP'>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img744.png"
 ALT="$ AP'&gt;0$">|; 

$key = q/{calS}_0proptop_0slashrho_0^{,gamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5880.png"
 ALT="$ {\cal S}_0\propto p_0/\rho_0^{ \gamma}$">|; 

$key = q/theta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$ \theta _2$">|; 

$key = q/displaystyleint_Vfrac{partialrho}{partialt},dV+oint_Srho,{bfv}cdotd{bfS}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="$\displaystyle \int_V \frac{\partial \rho}{\partial t} dV + \oint_S\rho {\bf v}\cdot d{\bf S} = 0,$">|; 

$key = q/displaystylea_{ik}',b_{jk}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6860.png"
 ALT="$\displaystyle a_{ik}' b_{jk}'$">|; 

$key = q/displaystyleJ=8int_0^{pislash2}sintheta,dthetaint_0^inftyfrac{dt}{a+b,t^2}=4piint_0^{pislash2}frac{sintheta,dtheta}{(a,b)^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7260.png"
 ALT="$\displaystyle J = 8\int_0^{\pi/2}\sin\theta d\theta \int_0^\infty \frac{dt}{a+b t^2} = 4\pi\int_0^{\pi/2}\frac{\sin\theta d\theta}{(a b)^{1/2}},$">|; 

$key = q/(pi^{,2}slash4),V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2499.png"
 ALT="$ (\pi^{ 2}/4) V_0$">|; 

$key = q/displaystylec_1^{,2}=frac{g}{k}left(frac{1-s}{1+s}right)+frac{gamma,k}{rho,(1+s)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4190.png"
 ALT="$\displaystyle c_1^{ 2}= \frac{g}{k}\left(\frac{1-s}{1+s}\right) + \frac{\gamma k}{\rho (1+s)},$">|; 

$key = q/displaystyle=rho,{bfV}timesint_S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3438.png"
 ALT="$\displaystyle = \rho {\bf V}\times \int_S$">|; 

$key = q/displaystyle=left(frac{T}{T_0}right)^{1slash(gamma-1)}=left[1+frac{1}{2},(gamma-1),{rmMa}^{,2}right]^{-1slash(gamma-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5391.png"
 ALT="$\displaystyle =\left(\frac{T}{T_0}\right)^{1/(\gamma-1)}=\left[1+\frac{1}{2} (\gamma-1) {\rm Ma}^{ 2}\right]^{-1/(\gamma-1)}.$">|; 

$key = q/displaystyle=-frac{1}{rho},frac{partialp}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5876.png"
 ALT="$\displaystyle = -\frac{1}{\rho} \frac{\partial p}{\partial x},$">|; 

$key = q/hat{t}rightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3343.png"
 ALT="$ \hat{t}\rightarrow\infty$">|; 

$key = q/displaystylenu_{rmmax}=frac{pi}{2}left[left(frac{gamma+1}{gamma-1}right)^{1slash2}-1right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5776.png"
 ALT="$\displaystyle \nu_{\rm max} = \frac{\pi}{2}\left[\left(\frac{\gamma+1}{\gamma-1}\right)^{1/2}-1\right].$">|; 

$key = q/(partial^{,2}uslashpartialy^{,2})_{y=0}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3143.png"
 ALT="$ (\partial^{ 2} u/\partial y^{ 2})_{y=0}&gt;0$">|; 

$key = q/{muskipz@5mudividemuskipz@tw@mkernmuskipz@mathaccentV{hat}05E{mkern-muskipz@{zeta}mkernmuskipz@}mkern-muskipz@}{}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$ {\muskip \z@ 5mu\divide \muskip \z@ \tw@ \mkern \muskip \z@ \mathaccentV {hat}05E{\mkern -\muskip \z@ {\zeta }\mkern \muskip \z@ }\mkern -\muskip \z@ }{}_j$">|; 

$key = q/z=r,{rme}^{,{rmi},theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2043.png"
 ALT="$ z=r {\rm e}^{ {\rm i} \theta}$">|; 

$key = q/{bfV}=V,{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4065.png"
 ALT="$ {\bf V}=V {\bf e}_x$">|; 

$key = q/displaystyleesimeqsqrt{frac{5}{2}},frac{omega}{(G,Mslasha_0^{,3})^{1slash2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1071.png"
 ALT="$\displaystyle e\simeq \sqrt{\frac{5}{2}} \frac{\omega}{(G M/a_0^{ 3})^{1/2}}
$">|; 

$key = q/displaystyledy=-frac{2,k}{sin^4alpha},(1-cosalpha,cosomega+sinalpha,sinomega),sinomega,domega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2312.png"
 ALT="$\displaystyle dy = -\frac{2 k}{\sin^4\alpha} (1-\cos\alpha \cos\omega+\sin\alpha \sin\omega) \sin\omega d\omega,$">|; 

$key = q/displaystylefrac{d^{,2}{mitTheta}}{dtheta^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1734.png"
 ALT="$\displaystyle \frac{d^{ 2}{\mit\Theta}}{d\theta^{ 2}}$">|; 

$key = q/MslashR;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5186.png"
 ALT="$ M/R$">|; 

$key = q/displaystyle|u|llU_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3013.png"
 ALT="$\displaystyle \vert u\vert\ll U_0.$">|; 

$key = q/-{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6238.png"
 ALT="$ -{\bf a}$">|; 

$key = q/displaystylenablacdotnablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6685.png"
 ALT="$\displaystyle \nabla\cdot\nabla \phi$">|; 

$key = q/{multline}left(sum_{n=1,infty}p_n,frac{x_i}{r}-frac{mu}{r}left[r,frac{partialxi_3}bar{rho}}{rho},zeta_nright]left(frac{x_i}{r}right)_{r=a},{multline};LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="554" HEIGHT="161" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4406.png"
 ALT="\begin{multline}
\left(\sum_{n=1,\infty}p_n \frac{x_i}{r} - \frac{\mu}{r}\left[...
...ar{\rho}}{\rho} \zeta_n\right]\left(\frac{x_i}{r}\right)_{r=a} ,
\end{multline}">|; 

$key = q/displaystyle=(8,s,tantheta)^{1slash2},a,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img777.png"
 ALT="$\displaystyle = (8 s \tan\theta)^{1/2} a,$">|; 

$key = q/u_1=r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7105.png"
 ALT="$ u_1=r$">|; 

$key = q/f(r)=alpha,r^{,n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3773.png"
 ALT="$ f(r)=\alpha r^{ n}$">|; 

$key = q/displaystyleT_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5240.png"
 ALT="$\displaystyle T_c$">|; 

$key = q/p_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5419.png"
 ALT="$ p_e$">|; 

$key = q/displaystylephi(r,theta)=-frac{{mitGamma}}{2pi},theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1727.png"
 ALT="$\displaystyle \phi(r,\theta)= - \frac{{\mit\Gamma}}{2\pi} \theta.$">|; 

$key = q/z=(a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2116.png"
 ALT="$ z=(a$">|; 

$key = q/displaystyleu,frac{du}{dx}=-frac{1}{rho},frac{dp}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5452.png"
 ALT="$\displaystyle u \frac{du}{dx} =-\frac{1}{\rho} \frac{dp}{dx},$">|; 

$key = q/d^{,3}{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6505.png"
 ALT="$ d^{ 3}{\bf r}$">|; 

$key = q/delta{mitPhi}slash(g,a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4489.png"
 ALT="$ \delta{\mit\Phi}/(g a)$">|; 

$key = q/displaystyleM=48,rho,nu^{,2},alpha^{,3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3258.png"
 ALT="$\displaystyle M= 48 \rho \nu^{ 2} \alpha^{ 3}.
$">|; 

$key = q/{mitPhi}_r(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4804.png"
 ALT="$ {\mit\Phi}_r(\theta,\phi)$">|; 

$key = q/displaystylelambda=frac{b^{,2}}{12,s,c}-frac{c}{2},(1-s).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img674.png"
 ALT="$\displaystyle \lambda = \frac{b^{ 2}}{12 s c} - \frac{c}{2} (1-s).$">|; 

$key = q/displaystylec;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6090.png"
 ALT="$\displaystyle c$">|; 

$key = q/displaystylembox{if{j=l{and{k=mneqj{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6805.png"
 ALT="$\displaystyle \mbox{if $j=l$ and $k=m\neq j$}$">|; 

$key = q/12.658;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4630.png"
 ALT="$ 12.658$">|; 

$key = q/displaystylef=b,(1+f_z^{,2})^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1255.png"
 ALT="$\displaystyle f = b (1+f_z^{ 2})^{1/2}.$">|; 

$key = q/displaystyleh_n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4435.png"
 ALT="$\displaystyle h_n'$">|; 

$key = q/displaystyle{bfv}({bfr})=r^{,2},{mitOmega}(r),sin^2theta,,nablavarphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3883.png"
 ALT="$\displaystyle {\bf v}({\bf r}) = r^{ 2} {\mit\Omega}(r) \sin^2\theta  \nabla\varphi,
$">|; 

$key = q/displaystyle{bfA}cdot{bfB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7084.png"
 ALT="$\displaystyle {\bf A}\cdot{\bf B}$">|; 

$key = q/displaystyleK=langleint_{-d}^zetafrac{1}{2},rho,v^{,2},dzrangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3972.png"
 ALT="$\displaystyle K = \langle\int_{-d}^\zeta \frac{1}{2} \rho v^{ 2} dz\rangle,$">|; 

$key = q/delta{bfv}cdotd{bfS}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1612.png"
 ALT="$ \delta{\bf v}\cdot d {\bf S} = 0$">|; 

$key = q/displaystylesigma^{,2}=f^{,2}+c^{,2},k^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5091.png"
 ALT="$\displaystyle \sigma^{ 2} = f^{ 2}+c^{ 2} k^{ 2},
$">|; 

$key = q/displaystyleleft(frac{d^{,2}}{dr^{,2}}-frac{2}{r^{,2}}right)^2f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3768.png"
 ALT="$\displaystyle \left(\frac{d^{ 2}}{dr^{ 2}} - \frac{2}{r^{ 2}}\right)^2 f$">|; 

$key = q/displaystylefrac{dp}{dr}=rho,frac{v_theta^{,2}}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1721.png"
 ALT="$\displaystyle \frac{dp}{dr} = \rho \frac{v_\theta^{ 2}}{r},$">|; 

$key = q/skew{5}tilde{zeta}_{j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4674.png"
 ALT="$ \skew{5}\tilde{\zeta}_{j}$">|; 

$key = q/displaystylebeta=frac{2,m}{m+1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2941.png"
 ALT="$\displaystyle \beta = \frac{2 m}{m+1}.$">|; 

$key = q/deltaS_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6484.png"
 ALT="$ \delta S_i$">|; 

$key = q/u(theta,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4520.png"
 ALT="$ u(\theta,\phi,t)$">|; 

$key = q/displaystylez=r,{rme}^{,{rmi},theta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1935.png"
 ALT="$\displaystyle z = r {\rm e}^{ {\rm i} \theta},$">|; 

$key = q/exp(+k,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3917.png"
 ALT="$ \exp(+k z)$">|; 

$key = q/displaystyleu,frac{partialu}{partialx}+v,frac{partialu}{partialy}=U,frac{dU}{dx}+nu,frac{partial^{,2}u}{partialy^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3121.png"
 ALT="$\displaystyle u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = U \frac{dU}{dx} +\nu \frac{\partial^{ 2} u}{\partial y^{ 2}}.$">|; 

$key = q/M=29times10^{-3},{rmkg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5127.png"
 ALT="$ M = 29\times 10^{-3} {\rm kg}$">|; 

$key = q/displaystylesum_{i=1,3}left(frac{x_i+r,n_i}{a_i}right)^2=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7240.png"
 ALT="$\displaystyle \sum_{i=1,3}\left(\frac{x_i+r n_i}{a_i}\right)^2=1,$">|; 

$key = q/(nabla{bfn})cdot{bfn}equiv(1slash2),nabla(n^2)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1104.png"
 ALT="$ (\nabla{\bf n})\cdot {\bf n}\equiv (1/2) \nabla(n^2)=0$">|; 

$key = q/displaystyleV_x(X,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2902.png"
 ALT="$\displaystyle V_x(X,0)$">|; 

$key = q/displaystylefrac{1}{V},frac{dbar{F}}{dz'}simeq1+frac{{rmi},zeta}{sinalpha},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2320.png"
 ALT="$\displaystyle \frac{1}{V} \frac{d\bar{F}}{dz'}\simeq 1+\frac{{\rm i} \zeta}{\sin\alpha},$">|; 

$key = q/displaystyle=omega_z,{bfe}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1639.png"
 ALT="$\displaystyle = \omega_z {\bf e}_z,$">|; 

$key = q/displaystyle=mu_rint_{mitOmega}{mitPsi}',{mitPsi}_r,d{mitOmega}=-int_{mitOmega}{mitPsi}',D,{mitPsi}_r,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4904.png"
 ALT="$\displaystyle = \mu_r\int_{\mit\Omega}{\mit\Psi}' {\mit\Psi}_r d{\mit\Omega}= - \int_{\mit\Omega}{\mit\Psi}' D {\mit\Psi}_r d{\mit\Omega}$">|; 

$key = q/displaystylebeta=zeta^{,2}-frac{1}{4},zeta^{,4}+{calO}(zeta^{,6}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5234.png"
 ALT="$\displaystyle \beta = \zeta^{ 2} - \frac{1}{4} \zeta^{ 4} + {\cal O}(\zeta^{ 6}).$">|; 

$key = q/(v_x,,v_y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2181.png"
 ALT="$ (v_x, v_y)$">|; 

$key = q/displaystyleb^{,2}=frac{gamma-1}{gamma+1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6208.png"
 ALT="$\displaystyle b^{ 2}= \frac{\gamma-1}{\gamma+1},
$">|; 

$key = q/displaystylelambda_rint_{mitOmega}{mitPhi}',{mitPhi}_r,d{mitOmega}=-sum_{s=1,infty}left(beta_{r,,s},p_s+beta_{r,,-s},p_{-s}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4902.png"
 ALT="$\displaystyle \lambda_r\int_{\mit\Omega}{\mit\Phi}' {\mit\Phi}_r d{\mit\Omega} = -\sum_{s=1,\infty} \left(\beta_{r, s} p_s + \beta_{r, -s} p_{-s}\right).$">|; 

$key = q/dS_2=h_1,h_3,du_1,du_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7052.png"
 ALT="$ dS_2 = h_1 h_3 du_1 du_3$">|; 

$key = q/rslasharightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1763.png"
 ALT="$ r/a\rightarrow\infty$">|; 

$key = q/-[(m-m')slash(m+m')],(gslashomega);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1866.png"
 ALT="$ -[(m-m')/(m+m')] (g/\omega)$">|; 

$key = q/displaystylec^{,2}equivfrac{dp}{drho}=frac{gamma,p}{rho}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5349.png"
 ALT="$\displaystyle c^{ 2}\equiv \frac{dp}{d\rho} = \frac{\gamma p}{\rho}.$">|; 

$key = q/a_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6260.png"
 ALT="$ a_x$">|; 

$key = q/displaystyle(widetilde{nabla{bfA}})_{rtheta}=(widetilde{nabla{bfA}})_{thetar};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7153.png"
 ALT="$\displaystyle (\widetilde{\nabla {\bf A}})_{r\theta}=(\widetilde{\nabla {\bf A}})_{\theta r}$">|; 

$key = q/p_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5252.png"
 ALT="$ p_c$">|; 

$key = q/displaystyle=pmc_0,F(xmpc_0,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5361.png"
 ALT="$\displaystyle =\pm c_0 F(x\mp c_0 t),$">|; 

$key = q/skew{3}bar{rho},g,d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4401.png"
 ALT="$ \skew{3}\bar{\rho} g d$">|; 

$key = q/{mitDelta}p=p_2-p_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5729.png"
 ALT="$ {\mit\Delta}p=p_2-p_1$">|; 

$key = q/kappa_H=2.1times10^{-5},{rmm^2,s^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="$ \kappa_H = 2.1\times 10^{-5} {\rm m^2 s^{-1}}$">|; 

$key = q/displaystyle=-frac{partialz}{partialeta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2687.png"
 ALT="$\displaystyle =-\frac{\partial z}{\partial\eta}.$">|; 

$key = q/displaystylefrac{mu}{rho},nabla^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4386.png"
 ALT="$\displaystyle \frac{\mu}{\rho} \nabla^{ 2}$">|; 

$key = q/displaystyleK=frac{1}{2},rhoointphi,nablaphicdot{bfn},2pi,varpi,ds=-frac{1}{2},iC}phi,frac{1}{h_xi},frac{partialphi}{partialxi},2pi,varpi,ds.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="471" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2715.png"
 ALT="$\displaystyle K=\frac{1}{2} \rho\oint \phi \nabla\phi\cdot{\bf n} 2\pi \var...
...{ABC}\phi \frac{1}{h_\xi} \frac{\partial\phi}{\partial\xi} 2\pi \varpi ds.$">|; 

$key = q/c>l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2125.png"
 ALT="$ c&gt;l$">|; 

$key = q/H(krightarrow1)rightarrow1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3724.png"
 ALT="$ H(k\rightarrow 1)\rightarrow 1$">|; 

$key = q/d{bfr}times{bfe}_z=d{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1790.png"
 ALT="$ d{\bf r}\times {\bf e}_z =d{\bf S}$">|; 

$key = q/varphi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4254.png"
 ALT="$ \varphi'$">|; 

$key = q/h=z(x=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1221.png"
 ALT="$ h=z(x=0)$">|; 

$key = q/displaystylefrac{p}{p_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="$\displaystyle \frac{p}{p_0}$">|; 

$key = q/displaystyleD,P'=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4751.png"
 ALT="$\displaystyle D P'=0.$">|; 

$key = q/displaystyledelta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6791.png"
 ALT="$\displaystyle \delta_{ij}$">|; 

$key = q/displaystylerho(z)=rho_0,exp(-zslashH),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5131.png"
 ALT="$\displaystyle \rho(z)= \rho_0 \exp(-z/H),$">|; 

$key = q/displaystyle=theta^{,4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5208.png"
 ALT="$\displaystyle = \theta^{ 4}.$">|; 

$key = q/displaystyle=-int_0^piint_{-1}^{1}left[(1-mu^{,2}),frac{partial{mitPsi}_n^{,m}}{rac{partial{mitPhi}_{n'}^{,m'}}{partialphi}right]mu,dmu,dphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="535" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4951.png"
 ALT="$\displaystyle = -\int_0^\pi\int_{-1}^{1}\left[(1-\mu^{ 2})  \frac{\partial{\m...
...}  \frac{\partial{\mit\Phi}_{n'}^{ m'}}{\partial\phi}\right]\mu d\mu d\phi,$">|; 

$key = q/X(t)=(1slash2),a,t^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5671.png"
 ALT="$ X(t)=(1/2) a t^{ 2}$">|; 

$key = q/displaystylexi_z(x,0)=b,sin(k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4123.png"
 ALT="$\displaystyle \xi_z(x,0) = b \sin(k x),$">|; 

$key = q/displaystyle=left[frac{n',(n'+1)+m',(m'+1)}{m'+1}-2,m',frac{n,(n+1)-n',(n'+1)+m'}{m^{,2}+m'^{,2}}right]I_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4974.png"
 ALT="$\displaystyle = \left[\frac{n' (n'+1)+m' (m'+1)}{m'+1}-2 m' \frac{n (n+1)-n' (n'+1)+m'}{m^{ 2}+m'^{ 2}}\right]I_{n, n'}^{ m, m'}$">|; 

$key = q/displaystylea=sqrt{a_x^{,2}+a_y^{,2}+a_z^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6270.png"
 ALT="$\displaystyle a = \sqrt{a_x^{ 2} + a_y^{ 2} + a_z^{ 2}}.$">|; 

$key = q/k,dll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3955.png"
 ALT="$ k d\ll 1$">|; 

$key = q/displaystyleeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2164.png"
 ALT="$\displaystyle \eta$">|; 

$key = q/|{bfb}|=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6319.png"
 ALT="$ \vert{\bf b}\vert=0$">|; 

$key = q/displaystyle=frac{u_1}{u_2}=frac{gamma-1+(gamma+1),(p_2slashp_1)}{gamma+1+(gamma-1),(p_2slashp_1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5491.png"
 ALT="$\displaystyle = \frac{u_1}{u_2} =\frac{\gamma-1+(\gamma+1) (p_2/p_1)}{\gamma+1+(\gamma-1) (p_2/p_1)},$">|; 

$key = q/displaystyle{mitPhi}_0(rleqa);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4362.png"
 ALT="$\displaystyle {\mit\Phi}_0(r\leq a)$">|; 

$key = q/p_{-r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4926.png"
 ALT="$ p_{-r}$">|; 

$key = q/displaystyle=lim_{deltatrightarrow0}frac{deltambox{boldmaththeta}}{deltat},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6403.png"
 ALT="$\displaystyle = \lim_{\delta t\rightarrow 0} \frac{\delta \mbox{\boldmath$\theta$} }{\delta t},$">|; 

$key = q/{mitOmega}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3622.png"
 ALT="$ {\mit\Omega}_1$">|; 

$key = q/displaystylewsimleft(frac{d}{a}right)u,,left(frac{d}{a}right)vllu,v.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4478.png"
 ALT="$\displaystyle w \sim \left(\frac{d}{a}\right)u, \left(\frac{d}{a}\right)v \ll u, v.$">|; 

$key = q/displaystyle=v_g,t,cosbeta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4035.png"
 ALT="$\displaystyle =v_g t \cos\beta.$">|; 

$key = q/p_0neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1288.png"
 ALT="$ p_0\neq 0$">|; 

$key = q/nablatimes({bfa}times{bfb})=({bfb}cdotnabla),{bfa}-({bfa}cdotnabla),{bfb}+(nablacdot{bfb}),{bfa}-(nablacdot{bfa}),{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="478" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6989.png"
 ALT="$ \nabla\times({\bf a}\times {\bf b}) = ({\bf b}\cdot\nabla) {\bf a} - ({\bf a}...
...nabla) {\bf b} + (\nabla\cdot {\bf b}) {\bf a}
-(\nabla\cdot{\bf a}) {\bf b}$">|; 

$key = q/displaystyleleft(u^{,2}-c^{,2}right),frac{partialu}{partialx}+left(v^{,2}-c^{,2}ft(frac{partialu}{partialy}+frac{partialv}{partialx}right)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5889.png"
 ALT="$\displaystyle \left(u^{ 2}-c^{ 2}\right) \frac{\partial u}{\partial x} + \le...
... v\left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)=0.$">|; 

$key = q/displaystyle=nabla^{,2}A_r-frac{A_r}{r^{,2}}-frac{2}{r^{,2}},frac{partialA_theta}{partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7160.png"
 ALT="$\displaystyle =\nabla^{ 2} A_r - \frac{A_r}{r^{ 2}} - \frac{2}{r^{ 2}} \frac{\partial A_\theta}{\partial\theta},$">|; 

$key = q/displaystyle=-frac{g}{a},frac{partial}{partialtheta}left[zeta-(1+k_2-h_2),skew{5ar{zeta}_2-sum_{n=1,infty}(1+k_n'-h_n'),alpha_n,zeta_nright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4509.png"
 ALT="$\displaystyle = - \frac{g}{a} \frac{\partial}{\partial\theta}\left[\zeta-(1+k_...
...skew{5}\bar{\zeta}_2 -\sum_{n=1,\infty}(1+k_n'-h_n') \alpha_n \zeta_n\right],$">|; 

$key = q/{bfl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6388.png"
 ALT="$ {\bf l}$">|; 

$key = q/U(x)=U_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2980.png"
 ALT="$ U(x) = U_0$">|; 

$key = q/nablacdot(psi,nablavarphi)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2563.png"
 ALT="$ \nabla\cdot(\psi \nabla\varphi)=0$">|; 

$key = q/|theta_2|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$ \vert\theta _2\vert$">|; 

$key = q/T=273,{rmK};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5125.png"
 ALT="$ T=273 {\rm K}$">|; 

$key = q/{rmMa}_1=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5480.png"
 ALT="$ {\rm Ma}_1=1$">|; 

$key = q/displaystylef=frac{l^{,2},f_{xx}}{(1+f_x^{,2})^{3slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1160.png"
 ALT="$\displaystyle f = \frac{l^{ 2} f_{xx}}{(1+f_x^{ 2})^{3/2}}.$">|; 

$key = q/b>c_0ggdelta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3487.png"
 ALT="$ b&gt;c_0\gg\delta_0$">|; 

$key = q/displaystyle({bfa}times{bfb})times({bfc}times{bfd});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6679.png"
 ALT="$\displaystyle ({\bf a}\times {\bf b})\times ({\bf c}\times {\bf d})$">|; 

$key = q/displaystyle=mu,omega_varphi(a,theta)=frac{3,mu,V}{2,a},sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3804.png"
 ALT="$\displaystyle = \mu \omega_\varphi(a,\theta) = \frac{3 \mu V}{2 a} \sin\theta.$">|; 

$key = q/{mitPhi}_{rmtide}({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4329.png"
 ALT="$ {\mit\Phi}_{\rm tide}({\bf r})$">|; 

$key = q/{rmMa}_1=w_1slashc_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5683.png"
 ALT="$ {\rm Ma}_1=w_1/c_1$">|; 

$key = q/pislash2geqzetageq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2234.png"
 ALT="$ \pi/2\geq\zeta\geq 0$">|; 

$key = q/u=v_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4456.png"
 ALT="$ u=v_\theta$">|; 

$key = q/displaystylell1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3662.png"
 ALT="$\displaystyle \ll 1$">|; 

$key = q/displaystyleK_{rmcylinder}=frac{1}{2},m,V_x^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1825.png"
 ALT="$\displaystyle K_{\rm cylinder} = \frac{1}{2} m V_x^{ 2}.$">|; 

$key = q/displaystyler';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4295.png"
 ALT="$\displaystyle r'$">|; 

$key = q/displaystyleI_0=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3540.png"
 ALT="$\displaystyle I_0 = 0.
$">|; 

$key = q/4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$ 4$">|; 

$key = q/A=-(V'-c),zeta_0slashtanh(k,d');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4095.png"
 ALT="$ A=-(V'-c) \zeta_0/\tanh(k d')$">|; 

$key = q/widehat{mitGamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3344.png"
 ALT="$ \widehat{\mit\Gamma}$">|; 

$key = q/displaystyle{calC}_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5384.png"
 ALT="$\displaystyle {\cal C}_V$">|; 

$key = q/displaystyles^{,2}-s+frac{1}{6}>0:;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img764.png"
 ALT="$\displaystyle s^{ 2}-s+\frac{1}{6}&gt;0:$">|; 

$key = q/0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$ 0.5$">|; 

$key = q/U=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3197.png"
 ALT="$ U=0$">|; 

$key = q/displaystylefrac{D}{Dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="$\displaystyle \frac{D}{Dt}$">|; 

$key = q/bar{F}=skew{3}bar{phi}+{rmi},skew{3}bar{psi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2180.png"
 ALT="$ \bar{F}=\skew{3}\bar{\phi}+{\rm i} \skew{3}\bar{\psi}$">|; 

$key = q/R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="$ R$">|; 

$key = q/0.3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="$ 0.3$">|; 

$key = q/displaystylefrac{{calS}_4-{calS}_3}{{calR}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6189.png"
 ALT="$\displaystyle \frac{{\cal S}_4-{\cal S}_3}{{\cal R}}$">|; 

$key = q/displaystyled{calE}=k(T),dT,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5300.png"
 ALT="$\displaystyle d{\cal E} = k(T) dT,$">|; 

$key = q/displaystylesimeqfrac{(gamma+1)}{12,gamma^{,2}}left(frac{{mitDelta}p}{p_1}right)^{,3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5515.png"
 ALT="$\displaystyle \simeq \frac{(\gamma+1)}{12 \gamma^{ 2}}\left( \frac{{\mit\Delta}p}{p_1}\right)^{ 3},$">|; 

$key = q/displaystyle=-k^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5963.png"
 ALT="$\displaystyle =-k^{ 2},$">|; 

$key = q/{bfv}(P_-)={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3426.png"
 ALT="$ {\bf v}(P_-)={\bf0}$">|; 

$key = q/displaystylefrac{dp}{p}+frac{dv}{v}=frac{dT}{T}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5322.png"
 ALT="$\displaystyle \frac{dp}{p}+ \frac{dv}{v} = \frac{dT}{T}.$">|; 

$key = q/displaystyleW=|{bfF}|,(|{bfr}|,costheta)={bfF}cdot{bfr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6326.png"
 ALT="$\displaystyle W = \vert{\bf F}\vert  (\vert{\bf r}\vert \cos\theta) = {\bf F}\cdot {\bf r}.$">|; 

$key = q/displaystylefrac{d^{,2}F}{dz^{,2}}-k^{,2},F=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3916.png"
 ALT="$\displaystyle \frac{d^{ 2} F}{dz^{ 2}} - k^{ 2} F= 0,$">|; 

$key = q/|z|slasha<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1999.png"
 ALT="$ \vert z\vert/a&lt;1$">|; 

$key = q/displaystylev_x(y)=frac{G}{2,mu},y,(d-y)+U,left(frac{d-y}{d}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3586.png"
 ALT="$\displaystyle v_x(y) = \frac{G}{2 \mu} y (d-y) + U \left(\frac{d-y}{d}\right).$">|; 

$key = q/displaystyle{bfr}=x,{bfe}_z+y,{bfe}_y+z,{bfe}_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6258.png"
 ALT="$\displaystyle {\bf r} = x {\bf e}_z + y {\bf e}_y+z {\bf e}_z.$">|; 

$key = q/displaystyleA_{-r}=-int_{mitOmega}costhetaleft(xi,frac{partial{mitPsi}_r}{partiantheta},frac{partial{mitPsi}_r}{partialphi}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4908.png"
 ALT="$\displaystyle A_{-r}= -\int_{\mit\Omega}\cos\theta\left(\xi \frac{\partial{\mi...
...ta}{\sin\theta} \frac{\partial{\mit\Psi}_r}{\partial\phi}\right)d{\mit\Omega}.$">|; 

$key = q/({bfa}times{bfb})times{bfc}=({bfa}cdot{bfc}),{bfb}-({bfc}cdot{bfb}),{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6984.png"
 ALT="$ ({\bf a}\times {\bf b})\times {\bf c} = ({\bf a}\cdot{\bf c}) {\bf b} - ({\bf c}\cdot{\bf b}) {\bf a}$">|; 

$key = q/displaystyle=frac{2,omega^{,2},(z^{,4}-2,omega^{-1},z+3)}{z^{,4}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img789.png"
 ALT="$\displaystyle = \frac{2 \omega^{ 2} (z^{ 4}-2 \omega^{-1} z+3)}{z^{ 4}}.$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter02slashfig5.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img995.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter02/fig5.04.eps}}
\end{figure}">|; 

$key = q/displaystyle[{bfa}times({bfb}times{bfc})]_i=a_j,c_j,b_i-a_j,b_j,c_i=left[({bfa}cdot{bfc}),{bfb}-({bfa}cdot{bfb}),{bfc}right]_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="465" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6819.png"
 ALT="$\displaystyle [{\bf a}\times ({\bf b}\times {\bf c})]_i = a_j c_j b_i - a_j ...
...\left[({\bf a}\cdot{\bf c}) {\bf b} - ({\bf a}\cdot{\bf b}) {\bf c}\right]_i.$">|; 

$key = q/widehat{omega}=0.47399;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img919.png"
 ALT="$ \widehat{\omega} = 0.47399$">|; 

$key = q/displaystylefrac{tanh(k,d)}{k,d}=frac{V^{,2}}{g,d}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3983.png"
 ALT="$\displaystyle \frac{\tanh(k d)}{k d} = \frac{V^{ 2}}{g d}.$">|; 

$key = q/displaystyle|z|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1320.png"
 ALT="$\displaystyle \vert z\vert$">|; 

$key = q/theta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img939.png"
 ALT="$ \theta=0$">|; 

$key = q/displaystyle=beta,p,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5191.png"
 ALT="$\displaystyle = \beta p,$">|; 

$key = q/displaystyle{bfh}=alpha,{bfA}+beta,{bfB}+gamma,{bfC}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6740.png"
 ALT="$\displaystyle {\bf h} = \alpha {\bf A} + \beta {\bf B} + \gamma {\bf C}.
$">|; 

$key = q/{calT}={calE}+(1slash2),v^{,2}+{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5375.png"
 ALT="$ {\cal T} = {\cal E}+(1/2) v^{ 2}+{\mit\Psi}$">|; 

$key = q/theta=gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img937.png"
 ALT="$ \theta=\gamma$">|; 

$key = q/displaystyle=(a_x,costheta+a_y,sintheta),(b_x,costheta+b_y,sintheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6303.png"
 ALT="$\displaystyle = (a_x  \cos\theta+a_y \sin\theta) (b_x \cos\theta + b_y \sin\theta)$">|; 

$key = q/displaystylebeta_1'-beta_1simeq2left[left(frac{gamma+1}{4}right)left(frac{{rmMa}_1^{,2}}{{rmMa}_1^{,2}-1}right)-1right]theta_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6168.png"
 ALT="$\displaystyle \beta_1'-\beta_1\simeq 2\left[\left(\frac{\gamma+1}{4}\right)\left(\frac{{\rm Ma}_1^{ 2}}{{\rm Ma}_1^{ 2}-1}\right)-1\right]\theta_1,
$">|; 

$key = q/displaystyle{bfv}=v_theta(r),{bfe}_theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1703.png"
 ALT="$\displaystyle {\bf v} = v_\theta(r) {\bf e}_\theta.$">|; 

$key = q/displaystylepsi(xi,eta)=-frac{1}{2},V,b^{,2}left{frac{coshxi+sinh^2xi,lnleft[tane^{,2}),(1slash2)lnleft[(1-e)slash(1+e)right]}right}sin^2eta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="585" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2804.png"
 ALT="$\displaystyle \psi(\xi,\eta)= -\frac{1}{2} V b^{ 2}\left\{\frac{\cosh\xi + \...
...-1}+e^{ -2} (1-e^{ 2}) (1/2)\ln\left[(1-e)/(1+e)\right]}\right\}\sin^2\eta.$">|; 

$key = q/displaystylebeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1305.png"
 ALT="$\displaystyle \beta$">|; 

$key = q/displaystyle=frac{c_V}{M},T,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="$\displaystyle = \frac{c_V}{M} T,$">|; 

$key = q/i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="$ i$">|; 

$key = q/displaystyler-r_0=frac{R_0}{r_0},(p-p_0).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img721.png"
 ALT="$\displaystyle r -r_0 = \frac{R_0}{r_0} (p-p_0).$">|; 

$key = q/displaystylechi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="$\displaystyle \chi$">|; 

$key = q/displaystyleA_{ij}=-kappa,delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="$\displaystyle A_{ij} = -\kappa \delta_{ij},$">|; 

$key = q/13.5,{rmcm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1150.png"
 ALT="$ 13.5 {\rm cm}$">|; 

$key = q/displaystyle{rmdet}(A)=epsilon_{ijk},A_{i1},A_{j2},A_{k3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6997.png"
 ALT="$\displaystyle {\rm det}(A) = \epsilon_{ijk} A_{i1} A_{j2} A_{k3}.
$">|; 

$key = q/displaystylev';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5944.png"
 ALT="$\displaystyle v'$">|; 

$key = q/displaystyle{mitOmega}=lnleft(V,frac{dz}{dbar{F}}right)=lnleft(frac{V}{v}right)+{rmi},theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2192.png"
 ALT="$\displaystyle {\mit\Omega} =\ln\left(V \frac{dz}{d\bar{F}}\right)=\ln\left(\frac{V}{v}\right)+{\rm i} \theta.$">|; 

$key = q/displaystyle=-frac{partialphi}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1652.png"
 ALT="$\displaystyle = - \frac{\partial\phi}{\partial x},$">|; 

$key = q/displaystyleskew{3}ddot{r}=(M-M_0),omega^{,2},r_G>0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img834.png"
 ALT="$\displaystyle \skew{3}\ddot{r} = (M-M_0) \omega^{ 2} r_G&gt;0.$">|; 

$key = q/displaystylesimeqtheta_1'-theta_1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6205.png"
 ALT="$\displaystyle \simeq \theta_1'-\theta_1.$">|; 

$key = q/displaystylebeta(y)=frac{1}{2}int^yalpha(y'),dy'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3120.png"
 ALT="$\displaystyle \beta(y) = \frac{1}{2}\int^y \alpha(y') dy'.$">|; 

$key = q/displaystylef'''+f,f''+beta,(1-f'^{,2})=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2940.png"
 ALT="$\displaystyle f'''+f f'' + \beta (1-f'^{ 2}) = 0,$">|; 

$key = q/displaystylefrac{partialp}{partialz}+rho,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img819.png"
 ALT="$\displaystyle \frac{\partial p}{\partial z} +\rho g$">|; 

$key = q/displaystyle=h_i^{,-2},delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7060.png"
 ALT="$\displaystyle =h_i^{ -2} \delta_{ij},$">|; 

$key = q/displaystyle[({bfA}cdotnabla),{bfA}]_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7135.png"
 ALT="$\displaystyle [({\bf A}\cdot\nabla) {\bf A}]_r$">|; 

$key = q/displaystyle=left(frac{2}{gamma+1}right)^{gammaslash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5398.png"
 ALT="$\displaystyle = \left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)},$">|; 

$key = q/rho_1=rho(z_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5161.png"
 ALT="$ \rho_1=\rho(z_1)$">|; 

$key = q/{calC}_V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5324.png"
 ALT="$ {\cal C}_V$">|; 

$key = q/a<l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7338.png"
 ALT="$ a &lt; l$">|; 

$key = q/gslashomega^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1065.png"
 ALT="$ g/\omega^{ 2}$">|; 

$key = q/displaystyle=3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6794.png"
 ALT="$\displaystyle =3,$">|; 

$key = q/z_G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img648.png"
 ALT="$ z_G$">|; 

$key = q/u_2+du_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7043.png"
 ALT="$ u_2+d u_2$">|; 

$key = q/deltaz=deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1960.png"
 ALT="$ \delta z = \delta x$">|; 

$key = q/displaystyle=frac{a}{d},frac{partialxi}{partialt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4660.png"
 ALT="$\displaystyle = \frac{a}{d} \frac{\partial\xi}{\partial t},$">|; 

$key = q/s=rho'slashrho=1.225times10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4115.png"
 ALT="$ s=\rho'/\rho=1.225\times 10^{-3}$">|; 

$key = q/c_1=sqrt{gamma,p_1slashrho_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5790.png"
 ALT="$ c_1=\sqrt{\gamma p_1/\rho_1}$">|; 

$key = q/2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$ 2$">|; 

$key = q/displaystylemu_n^{,m}=n,(n+1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4943.png"
 ALT="$\displaystyle \mu_n^{ m} = n (n+1).$">|; 

$key = q/c,cotxi_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2465.png"
 ALT="$ c \cot \xi_0)$">|; 

$key = q/displaystyle{mitDelta}p=gamma,nablacdot{bfn},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1107.png"
 ALT="$\displaystyle {\mit\Delta} p = \gamma \nabla\cdot{\bf n},$">|; 

$key = q/displaystyle=a_3,b_1-a_1,b_3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6783.png"
 ALT="$\displaystyle = a_3 b_1-a_1 b_3,$">|; 

$key = q/dl=sqrt{dx^{,2}+dy^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6443.png"
 ALT="$ dl=\sqrt{dx^{ 2}+dy^{ 2}}$">|; 

$key = q/2,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img730.png"
 ALT="$ 2 a$">|; 

$key = q/displaystylerho,frac{partial{bfv}}{partialt}simeq-nablap_1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3902.png"
 ALT="$\displaystyle \rho \frac{\partial {\bf v}}{\partial t} \simeq - \nabla p_1.$">|; 

$key = q/M';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img966.png"
 ALT="$ M'$">|; 

$key = q/v_+>v_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1438.png"
 ALT="$ v_+&gt;v_c$">|; 

$key = q/displaystylephi=frac{Q}{4pi}left(frac{r_1-r_2}{r_1,r_2}right)rightarrowfrac{D}{4pi},frac{costheta}{r^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2630.png"
 ALT="$\displaystyle \phi = \frac{Q}{4\pi}\left(\frac{r_1-r_2}{r_1 r_2}\right)\rightarrow \frac{D}{4\pi} \frac{\cos\theta}{r^{ 2}}.$">|; 

$key = q/x=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$ x=0$">|; 

$key = q/v_x=sqrt{g,h},(1+u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3562.png"
 ALT="$ v_x=\sqrt{g h} (1+u)$">|; 

$key = q/T_0=273,{rmK};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5148.png"
 ALT="$ T_0=273 {\rm K}$">|; 

$key = q/B=(V'-c),zeta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4096.png"
 ALT="$ B=(V'-c) \zeta_0$">|; 

$key = q/displaystyledeltatheta_mleft(epsilon_{mis},a_{sj}+epsilon_{mjs},a_{is}right)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6900.png"
 ALT="$\displaystyle \delta \theta_m \left(\epsilon_{mis} a_{sj} + \epsilon_{mjs} a_{is}\right) = 0.$">|; 

$key = q/2pi,psi_{R'}-2pi,psi_R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2537.png"
 ALT="$ 2\pi \psi_{R'}-2\pi \psi_R$">|; 

$key = q/TslashT_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$ T/T_1$">|; 

$key = q/displaystyleF(xi)=-frac{1}{2},Bleft(coshxi+sinh^2xi,lnleft[tanh(xislash2)right]right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2801.png"
 ALT="$\displaystyle F(\xi)= -\frac{1}{2} B\left(\cosh\xi + \sinh^2\xi \ln\left[\tanh(\xi/2)\right]\right).$">|; 

$key = q/skew{5}tilde{phi}=-pislash4+thetaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1215.png"
 ALT="$ \skew{5}\tilde{\phi}=-\pi/4+\theta/2$">|; 

$key = q/9.63times10^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4632.png"
 ALT="$ 9.63\times 10^{-1}$">|; 

$key = q/pslashrho^{,gamma}={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5347.png"
 ALT="$ p/\rho^{ \gamma}={\rm constant}$">|; 

$key = q/P_l(mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2581.png"
 ALT="$ P_l(\mu$">|; 

$key = q/displaystyledyleft(int_0^{1-y}x,y^{,2},dxright)=y^{,2},dyleft[frac{x^{,2}}{2}right]^{1-y}_0=frac{y^{,2}}{2},(1-y)^2,dy.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6493.png"
 ALT="$\displaystyle dy\left(\int_0^{1-y} x  y^{ 2} dx\right) = y^{ 2} dy\left[ \frac{x^{ 2}}{2}\right]^{1-y}_0 = \frac{y^{ 2}}{2} (1-y)^2 dy.$">|; 

$key = q/displaystyle{bfF}=frac{rho}{8pi}int_Sint_{S'}frac{[mbox{boldmathOmega}_S({bfr}')a}_S({bfr})]times({bfr}-{bfr}')}{|{bfr}-{bfr}'|^{,3}},dS,dS'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3461.png"
 ALT="$\displaystyle {\bf F} = \frac{\rho}{8\pi}\int_S\int_{S'}\frac{[\mbox{\boldmath$...
...{\bf r})]\times ({\bf r}-{\bf r}')}{\vert{\bf r}-{\bf r}'\vert^{ 3}} dS dS'.$">|; 

$key = q/displaystylew(phi)=frac{{mitGamma}_0}{2,b}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3500.png"
 ALT="$\displaystyle w(\phi) = \frac{{\mit\Gamma}_0}{2 b}.$">|; 

$key = q/{bfr}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4226.png"
 ALT="$ {\bf r}'$">|; 

$key = q/displaystylefrac{partial^{,2}omega}{partialt^{,2}}+f^{,2},omega-g,d,nabla^{,2},omega=-f,g,nabla^{,2}skew{5}bar{zeta}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5075.png"
 ALT="$\displaystyle \frac{\partial^{ 2}\omega}{\partial t^{ 2}} + f^{ 2} \omega-g d \nabla^{ 2} \omega=-f g \nabla^{ 2}\skew{5}\bar{\zeta}',$">|; 

$key = q/displaystylez=-frac{2,a^{,2}}{9,V^{,2}},frac{dV}{dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2851.png"
 ALT="$\displaystyle z = - \frac{2 a^{ 2}}{9 V^{ 2}} \frac{dV}{dt},
$">|; 

$key = q/rleqa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1905.png"
 ALT="$ r\leq a$">|; 

$key = q/displaystylelambda=frac{kappa^{,2},A_0}{V_0}-z_{GH},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img657.png"
 ALT="$\displaystyle \lambda = \frac{\kappa^{ 2} A_0}{V_0} -z_{GH},$">|; 

$key = q/displaystyleX-{rmi},Y=frac{1}{2},{rmi^{,2}},rhoointleft[V^{,2},R,{rme}^{,{rmi},(1})right]dtheta=-{rmi},{rme}^{,{rmi},alpha},rho,V,{mitGamma},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="622" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3286.png"
 ALT="$\displaystyle X -{\rm i} Y = \frac{1}{2} {\rm i^{ 2}} \rho\oint\left[V^{ 2...
...\right] d\theta = -{\rm i} {\rm e}^{ {\rm i} \alpha} \rho V {\mit\Gamma},$">|; 

$key = q/rho,S_1,v_1,{bfv}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1391.png"
 ALT="$ \rho S_1 v_1 {\bf v}_1$">|; 

$key = q/displaystylee_{13}equiv(1-a_3^{,2}slasha_1^{,2})^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img873.png"
 ALT="$\displaystyle e_{13} \equiv (1-a_3^{ 2}/a_1^{ 2})^{1/2}.$">|; 

$key = q/displaystyle{rmRe};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="$\displaystyle {\rm Re}$">|; 

$key = q/displaystylez>frac{3}{2},omega.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img790.png"
 ALT="$\displaystyle z &gt; \frac{3}{2} \omega.$">|; 

$key = q/1slash9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$ 1/9$">|; 

$key = q/displaystyletheta=-frac{m}{1+m},frac{pi}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3226.png"
 ALT="$\displaystyle \theta = - \frac{m}{1+m} \frac{\pi}{2}.$">|; 

$key = q/displaystylephi(xi,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2789.png"
 ALT="$\displaystyle \phi(\xi,\eta)$">|; 

$key = q/displaystyleX_i=-frac{x_j,sigma_{ij}}{r}=p,frac{x_i}{r}-frac{mu}{r}left[r,frac{pxi_i}{partialr}-xi_i+frac{partial(r,xi_r)}{partialx_i}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4403.png"
 ALT="$\displaystyle X_i= -\frac{x_j \sigma_{ij}}{r} = p \frac{x_i}{r} - \frac{\mu}{...
...ial \xi_i}{\partial r}-\xi_i + \frac{\partial (r \xi_r)}{\partial x_i}\right],$">|; 

$key = q/displaystyle=(1+beta)left[frac{2,gamma+(gamma-1),beta}{2,gamma+(gamma+1),beta}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5534.png"
 ALT="$\displaystyle =(1+\beta)\left[\frac{2 \gamma+(\gamma-1) \beta}{2 \gamma+(\gamma+1) \beta}\right],$">|; 

$key = q/d{bfr}equiv(dx,,dy,,dz);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6474.png"
 ALT="$ d{\bf r} \equiv (dx, dy, dz)$">|; 

$key = q/displaystylesimeq-frac{2,k}{zeta,sinalpha}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2329.png"
 ALT="$\displaystyle \simeq -\frac{2 k}{\zeta \sin\alpha}.$">|; 

$key = q/displaystylelambda={rmi},expleft(-frac{pi}{2},frac{bar{F}}{skew{3}bar{psi}_1}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2206.png"
 ALT="$\displaystyle \lambda = {\rm i} \exp\left(-\frac{\pi}{2} \frac{\bar{F}}{\skew{3}\bar{\psi}_1}\right).$">|; 

$key = q/xslasha=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2004.png"
 ALT="$ x/a=0$">|; 

$key = q/r_G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img823.png"
 ALT="$ r_G$">|; 

$key = q/displaystyle-costheta,xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4778.png"
 ALT="$\displaystyle -\cos\theta \xi$">|; 

$key = q/displaystylefrac{1}{{rmMa}_1^{,2},sin^2beta}=left(frac{gamma+1}{2}right)frac{tan(beta-theta)}{tanbeta}-left(frac{gamma-1}{2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5713.png"
 ALT="$\displaystyle \frac{1}{{\rm Ma}_1^{ 2} \sin^2\beta}=\left(\frac{\gamma+1}{2}\right)\frac{\tan(\beta-\theta)}{\tan\beta} - \left(\frac{\gamma-1}{2}\right),$">|; 

$key = q/displaystyle=-frac{partial(u,d)}{partialx}-frac{partial(v,d)}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5062.png"
 ALT="$\displaystyle =-\frac{\partial (u d)}{\partial x} -\frac{\partial (v d)}{\partial y},$">|; 

$key = q/rho<rho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5374.png"
 ALT="$ \rho&lt;\rho_0$">|; 

$key = q/zetasimeqz-l^{,2}slashz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3314.png"
 ALT="$ \zeta\simeq z-l^{ 2}/z$">|; 

$key = q/{mitPsi}'=-(1slash2),omega^{,2},r^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img814.png"
 ALT="$ {\mit\Psi}' =-(1/2) \omega^{ 2} r^{ 2}$">|; 

$key = q/displaystylef_z=pmleft(frac{f^{,2}}{b^{,2}}-1right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1256.png"
 ALT="$\displaystyle f_z=\pm \left(\frac{f^{ 2}}{b^{ 2}}-1\right)^{1/2},$">|; 

$key = q/displaystyleF(z)=-frac{Q}{2pi},ln(z-b)-frac{Q}{2pi},ln(z-c)+frac{Q}{2pi},lnz=-frac{Q}{2pi}lnleft[frac{(z-b),(z-c)}{z}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="622" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2042.png"
 ALT="$\displaystyle F(z) = -\frac{Q}{2\pi} \ln(z-b)-\frac{Q}{2\pi} \ln(z-c)+ \frac{Q}{2\pi} \ln z = - \frac{Q}{2\pi}\ln\left[\frac{(z-b) (z-c)}{z}\right].$">|; 

$key = q/sin^2delta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4310.png"
 ALT="$ \sin^2\delta$">|; 

$key = q/displaystyleM_ast=4pi,rho_c,a^{,3},y_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5226.png"
 ALT="$\displaystyle M_\ast = 4\pi \rho_c a^{ 3} y_1,$">|; 

$key = q/displaystylex';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2326.png"
 ALT="$\displaystyle x'$">|; 

$key = q/displaystyle{bfF}({bfr},t),dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$\displaystyle {\bf F}({\bf r},t) d V.$">|; 

$key = q/|zeta|slash|skew{5}tilde{zeta}_6|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5046.png"
 ALT="$ \vert\zeta\vert/\vert\skew{5}\tilde{\zeta}_6\vert$">|; 

$key = q/displaystyle=muleft(frac{partialv_r}{partialz}+frac{partialv_z}{partialr}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="$\displaystyle =\mu\left(\frac{\partial v_r}{\partial z} + \frac{\partial v_z}{\partial r}\right),$">|; 

$key = q/(dTslashdr)_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6540.png"
 ALT="$ (dT/dr)_{\rm max}$">|; 

$key = q/displaystylephantom{=}+frac{1}{12},P_2^{,2}(costheta),P_2^{,2}(costheta'),cosleft[2,(varphi-varphi')right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4262.png"
 ALT="$\displaystyle \phantom{=}+ \frac{1}{12} P_2^{ 2}(\cos\theta) P_2^{ 2}(\cos\theta') \cos\left[2 (\varphi-\varphi')\right].$">|; 

$key = q/e_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img957.png"
 ALT="$ e_{12}$">|; 

$key = q/displaystyle=left(frac{2}{gamma+1}right),frac{c_0^{,2}}{a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5678.png"
 ALT="$\displaystyle = \left(\frac{2}{\gamma+1}\right) \frac{c_0^{ 2}}{a}.$">|; 

$key = q/displaystyle{bfF}=rho,{mitGamma},{bfV}times{bfe}_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2430.png"
 ALT="$\displaystyle {\bf F} =\rho {\mit\Gamma} {\bf V}\times{\bf e}_z.$">|; 

$key = q/phi(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1949.png"
 ALT="$ \phi(x,y)$">|; 

$key = q/p_0<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1336.png"
 ALT="$ p_0&lt;0$">|; 

$key = q/displaystylefrac{d}{dxi}left({rme}^{-xi},frac{dF}{dxi}right)-2,{rme}^{-xi},F=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2736.png"
 ALT="$\displaystyle \frac{d}{d\xi}\left({\rm e}^{-\xi} \frac{dF}{d\xi}\right) -2 {\rm e}^{-\xi} F = 0,$">|; 

$key = q/z_M=kappa_x^{,2},A_0slashV_0-h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img647.png"
 ALT="$ z_M=\kappa_x^{ 2} A_0/V_0- h$">|; 

$key = q/displaystyleB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2738.png"
 ALT="$\displaystyle B$">|; 

$key = q/displaystyletanbeta=frac{u_1}{v},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5702.png"
 ALT="$\displaystyle \tan\beta = \frac{u_1}{v},$">|; 

$key = q/h(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1520.png"
 ALT="$ h(x)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter12slashampk1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5047.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter12/ampk1.eps}}
\end{figure}">|; 

$key = q/u=v_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5408.png"
 ALT="$ u=v_x$">|; 

$key = q/displaystyledelta_{ij}'={calR}_{ik},{calR}_{jl},delta_{kl}={calR}_{ik},{calR}_{jk}=delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6867.png"
 ALT="$\displaystyle \delta_{ij}' = {\cal R}_{ik} {\cal R}_{jl} \delta_{kl}= {\cal R}_{ik} {\cal R}_{jk} = \delta_{ij},$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter02slashfig3.04.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img771.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter02/fig3.04.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{dX}{dbeta}=frac{X}{tanbeta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4044.png"
 ALT="$\displaystyle \frac{dX}{d\beta} = \frac{X}{\tan\beta}.$">|; 

$key = q/displaystylesimfrac{1}{delta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3017.png"
 ALT="$\displaystyle \sim \frac{1}{\delta},$">|; 

$key = q/v(theta,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4521.png"
 ALT="$ v(\theta,\phi,t)$">|; 

$key = q/displaystyletheta=frac{chi}{T}+frac{{bfq}cdot{bfq}}{kappa,T^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img528.png"
 ALT="$\displaystyle \theta = \frac{\chi}{T} + \frac{{\bf q}\cdot{\bf q}}{\kappa T^2}.
$">|; 

$key = q/displaystyleskew{5}bar{zeta}_j(theta,phi,t)=skew{5}tilde{zeta}_{j},P_2^{,m_j}(costheta),{rme}^{,{rmi},(m_j,phi+sigma_j,t)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4539.png"
 ALT="$\displaystyle \skew{5}\bar{\zeta}_j(\theta,\phi,t) =\skew{5}\tilde{\zeta}_{j} P_2^{ m_j}(\cos\theta) {\rm e}^{ {\rm i} (m_j \phi+\sigma_j t)}.$">|; 

$key = q/displaystyle{bfA}=-int_Sp,{bfn},dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3430.png"
 ALT="$\displaystyle {\bf A} =- \int_S p {\bf n} dS,$">|; 

$key = q/displaystyleT,sin(alpha+delta)+L,cosdelta=W+D,sindelta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3520.png"
 ALT="$\displaystyle T \sin(\alpha+\delta) + L \cos\delta = W + D \sin\delta,$">|; 

$key = q/{calL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3760.png"
 ALT="$ {\cal L}$">|; 

$key = q/j=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4637.png"
 ALT="$ j=0$">|; 

$key = q/displaystyle2,a_{ssi}+a_{iss}+a_{sis};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6930.png"
 ALT="$\displaystyle 2 a_{ssi} +a_{iss} +a_{sis}$">|; 

$key = q/{mitGamma}_{n,r}^{,m}rightarrow{mitGamma}_{n,n'}^{,m,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4980.png"
 ALT="$ {\mit\Gamma}_{n,r}^{ m}\rightarrow {\mit\Gamma}_{n,n'}^{ m,m'}$">|; 

$key = q/displaystyle=intdA=pi,a_1,a_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img839.png"
 ALT="$\displaystyle = \int dA = \pi a_1 a_2,$">|; 

$key = q/displaystylesimeqskew{5}tilde{phi}+frac{k^{,2}}{4},(skew{5}tilde{phi}+sinskew{5}tilde{phi},cosskew{5}tilde{phi}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1210.png"
 ALT="$\displaystyle \simeq \skew{5}\tilde{\phi} + \frac{k^{ 2}}{4} (\skew{5}\tilde{\phi}+\sin\skew{5}\tilde{\phi} \cos\skew{5}\tilde{\phi}),$">|; 

$key = q/displaystyle=rho,V^{,2},b-rhoint_0^{k,V},frac{partialskew{3}bar{phi}}{partialy},dskew{3}bar{phi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2277.png"
 ALT="$\displaystyle =\rho V^{ 2} b-\rho\int_0^{k V} \frac{\partial\skew{3}\bar{\phi}}{\partial y} d\skew{3}\bar{\phi}.$">|; 

$key = q/-sinepsilon,zeta_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4557.png"
 ALT="$ -\sin\epsilon \zeta_E$">|; 

$key = q/displaystyler=a,xi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5210.png"
 ALT="$\displaystyle r = a \xi,$">|; 

$key = q/kappa_T=1slashp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5608.png"
 ALT="$ \kappa_T = 1/p$">|; 

$key = q/c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$ c$">|; 

$key = q/displaystyle1-{rme}^{-zslashH}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5256.png"
 ALT="$\displaystyle 1 - {\rm e}^{-z/H}.
$">|; 

$key = q/displaystyleomega=(g,d)^{1slash2},k,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3951.png"
 ALT="$\displaystyle \omega= (g d)^{1/2} k,$">|; 

$key = q/displaystyle{bf0}=nablap+rho,nabla{mitPsi}+rho,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img807.png"
 ALT="$\displaystyle {\bf0} = \nabla p + \rho \nabla{\mit\Psi} + \rho $">|; 

$key = q/V_p=u_1-u_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5530.png"
 ALT="$ V_p=u_1-u_2$">|; 

$key = q/u_1=w_1,sinbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5685.png"
 ALT="$ u_1=w_1 \sin\beta$">|; 

$key = q/{rmMa}_2>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$ {\rm Ma}_2&gt;1$">|; 

$key = q/r^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1741.png"
 ALT="$ r^{ m}$">|; 

$key = q/rho_c=15.6times10^4,{rmkg,m^{-3}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5248.png"
 ALT="$ \rho_c = 15.6\times 10^4 {\rm kg m^{-3}}$">|; 

$key = q/displaystylefrac{{calS}_2-{calS}_1}{calR};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5690.png"
 ALT="$\displaystyle \frac{{\cal S}_2-{\cal S}_1}{\cal R}$">|; 

$key = q/u_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6978.png"
 ALT="$ u_{ij}$">|; 

$key = q/psi_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2017.png"
 ALT="$ \psi_3$">|; 

$key = q/4slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5196.png"
 ALT="$ 4/3$">|; 

$key = q/displaystyleoint_C{bfA}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2356.png"
 ALT="$\displaystyle \oint_C {\bf A}\cdot d{\bf r}$">|; 

$key = q/v_theta(r)=partialpsislashpartialr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1716.png"
 ALT="$ v_\theta(r)=\partial\psi/\partial r$">|; 

$key = q/displaystyle=nabla^{,2}phi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6686.png"
 ALT="$\displaystyle =\nabla^{ 2}\phi,$">|; 

$key = q/displaystylephantom{=}left.left.+frac{8pi^2,V,B,{rme}^{,{rmi},alpha}-{mitGamma}^{,2}}{4pi^2}+{calO}(R^{,-1})right]dthetaright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3291.png"
 ALT="$\displaystyle \phantom{=}\left.\left.+ \frac{8\pi^2 V B {\rm e}^{ {\rm i} \alpha}-{\mit\Gamma}^{ 2}}{4\pi^2}+{\cal O}(R^{ -1})\right] d\theta\right) ,$">|; 

$key = q/K,arightarrowK';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2172.png"
 ALT="$ K a\rightarrow K'$">|; 

$key = q/displaystyleF_r=lambda_rint_{mitOmega}{calF},{mitPhi}_r,d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4915.png"
 ALT="$\displaystyle F_r = \lambda_r\int_{\mit\Omega} {\cal F} {\mit\Phi}_r d{\mit\Omega}.$">|; 

$key = q/displaystyle=frac{1}{2},rho,V^{,2}int_{+1}^{,-1}left(1-frac{v_y^{,2}}{V^{,2}}rigrtialskew{3}bar{phi}},frac{dskew{3}bar{phi}}{dlambda},dlambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2332.png"
 ALT="$\displaystyle = \frac{1}{2} \rho V^{ 2}\int_{+1}^{ -1}\left(1-\frac{v_y^{ ...
... y}{\partial\skew{3}\bar{\phi}} \frac{d\skew{3}\bar{\phi}}{d\lambda} d\lambda$">|; 

$key = q/B';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2255.png"
 ALT="$ B'$">|; 

$key = q/({bfv}cdotnabla),{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3646.png"
 ALT="$ ({\bf v}\cdot\nabla) {\bf v}$">|; 

$key = q/displaystylefrac{partialV_x}{partialX}+frac{partialV_y}{partialY};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2898.png"
 ALT="$\displaystyle \frac{\partial V_x}{\partial X} + \frac{\partial V_y}{\partial Y}$">|; 

$key = q/displaystylelambda={rmi},expleft(-frac{pi}{2},frac{bar{F}}{skew{3}bar{psi}_1}rig}left(V,frac{dz}{dbar{F}}-frac{1}{V},frac{dbar{F}}{dz}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="456" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2207.png"
 ALT="$\displaystyle \lambda = {\rm i} \exp\left(-\frac{\pi}{2} \frac{\bar{F}}{\skew...
...\rm i}}{2}\left(V \frac{dz}{d\bar{F}}-\frac{1}{V} \frac{d\bar{F}}{dz}\right).$">|; 

$key = q/displaystylefrac{dS}{dt}+{mitPhi}_S={mitTheta}_S,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img525.png"
 ALT="$\displaystyle \frac{dS}{dt} + {\mit\Phi}_S = {\mit\Theta}_S,
$">|; 

$key = q/displaystylealphaleft[n,(n-1)-2right],[(n-2),(n-3)-2]=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3774.png"
 ALT="$\displaystyle \alpha\left[n (n-1)-2\right] [(n-2) (n-3)-2] = 0,$">|; 

$key = q/{bfb}=(1,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6353.png"
 ALT="$ {\bf b} = (1, 0, 0)$">|; 

$key = q/G(u_0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1276.png"
 ALT="$ G(u_0)=0$">|; 

$key = q/theta_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1698.png"
 ALT="$ \theta_0=0$">|; 

$key = q/|alpha|leqfleq|beta|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1296.png"
 ALT="$ \vert\alpha\vert\leq f\leq \vert\beta\vert$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{AppendixAslashfigA.19.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="365" HEIGHT="251" BORDER="0"
 SRC="|."$dir".q|img6526.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{AppendixA/figA.19.eps}}
\end{figure}">|; 

$key = q/displaystyle=r,costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1940.png"
 ALT="$\displaystyle =r \cos\theta,$">|; 

$key = q/displaystylepsi(r,theta)rightarrow-V_0,r,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1764.png"
 ALT="$\displaystyle \psi(r,\theta)\rightarrow -V_0 r \sin\theta$">|; 

$key = q/f_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="$ f_i$">|; 

$key = q/f(eta)>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3174.png"
 ALT="$ f(\eta)&gt;1$">|; 

$key = q/mu_2=sin^{-1}(1slash{rmMa}_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5763.png"
 ALT="$ \mu_2=\sin^{-1}(1/{\rm Ma}_2)$">|; 

$key = q/overline{rho}=rhoslashrho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="$ \overline{\rho}=\rho/\rho_0$">|; 

$key = q/displaystylenablatimes{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7073.png"
 ALT="$\displaystyle \nabla\times {\bf A}$">|; 

$key = q/9.97times10^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4635.png"
 ALT="$ 9.97\times 10^{-1}$">|; 

$key = q/displaystylep_r=frac{1}{3},alpha,T^{,4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5181.png"
 ALT="$\displaystyle p_r = \frac{1}{3} \alpha T^{ 4}$">|; 

$key = q/j=8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$ j=8$">|; 

$key = q/displaystyleM={rmRe}left(-frac{1}{2},rhooint_Cleft|frac{dF}{dz}right|^{,2}z,dskew{2}bar{z}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2411.png"
 ALT="$\displaystyle M = {\rm Re}\left(-\frac{1}{2} \rho \oint_C \left\vert\frac{dF}{dz}\right\vert^{ 2} z d\skew{2}\bar{z}\right),$">|; 

$key = q/lambda=z_M-z_G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img651.png"
 ALT="$ \lambda= z_M-z_G$">|; 

$key = q/displaystyleI_{n+1}+I_{n-1}=2,cosphi,I_n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3542.png"
 ALT="$\displaystyle I_{n+1} + I_{n-1} = 2 \cos\phi I_n,
$">|; 

$key = q/u(x,y)rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3245.png"
 ALT="$ u(x,y)\rightarrow 0$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashcyl.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img3217.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/cyl.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter05slashmove.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="431" HEIGHT="277" BORDER="0"
 SRC="|."$dir".q|img1798.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter05/move.eps}}
\end{figure}">|; 

$key = q/displaystyle=frac{1}{r}frac{partialA_theta}{partialtheta}+frac{A_r}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7150.png"
 ALT="$\displaystyle =\frac{1}{r} \frac{\partial A_\theta}{\partial \theta}+ \frac{A_r}{r},$">|; 

$key = q/displaystylefrac{{mitDelta}p}{p_1}simeq-left(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmM(gamma+1),{rmMa}_1^{,4}-4,({rmMa}_1^{,2}-1)right]|theta|^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="620" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6166.png"
 ALT="$\displaystyle \frac{{\mit\Delta}p}{p_1} \simeq -\left(\frac{\gamma {\rm Ma}_1^...
...amma+1) {\rm Ma}_1^{ 4}-4 ({\rm Ma}_1^{ 2}-1)\right]\vert\theta\vert^{ 2}
$">|; 

$key = q/S,(p-p_0)=S,rho,g,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1416.png"
 ALT="$ S (p-p_0)= S \rho g h$">|; 

$key = q/z_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5159.png"
 ALT="$ z_2$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter06slashimage3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="529" HEIGHT="252" BORDER="0"
 SRC="|."$dir".q|img2048.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter06/image3.eps}}
\end{figure}">|; 

$key = q/displaystyleH=G({rmFr}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1445.png"
 ALT="$\displaystyle H = G({\rm Fr}),$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashcross.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="285" BORDER="0"
 SRC="|."$dir".q|img6191.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/cross.eps}}
\end{figure}">|; 

$key = q/p=p_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6219.png"
 ALT="$ p=p_\infty$">|; 

$key = q/displaystyle={calR}_{ij},a_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6849.png"
 ALT="$\displaystyle = {\cal R}_{ij} a_j,$">|; 

$key = q/r=sqrt{x^{,2}+y^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1936.png"
 ALT="$ r=\sqrt{x^{ 2}+y^{ 2}}$">|; 

$key = q/25^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1135.png"
 ALT="$ 25^\circ$">|; 

$key = q/nablaf;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6574.png"
 ALT="$ \nabla f$">|; 

$key = q/displaystylezeta=frac{w}{2}left[coshleft(pi,frac{z}{w}right)-1right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2161.png"
 ALT="$\displaystyle \zeta = \frac{w}{2}\left[\cosh\left(\pi \frac{z}{w}\right)-1\right],$">|; 

$key = q/displaystylepsi(x,y)=frac{{mitGamma}}{2pi},lnleft[frac{x^{,2}+(y-a)^2}{x^{,2}+(y+a)^2}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2034.png"
 ALT="$\displaystyle \psi(x,y) =\frac{{\mit\Gamma}}{2\pi} \ln\left[\frac{x^{ 2}+(y-a)^2}{x^{ 2}+(y+a)^2}\right].$">|; 

$key = q/h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img646.png"
 ALT="$ h$">|; 

$key = q/R=3.84times10^8,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4573.png"
 ALT="$ R= 3.84\times 10^8 {\rm m}$">|; 

$key = q/displaystyle=-omega^{,2},Y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1847.png"
 ALT="$\displaystyle = -\omega^{ 2} Y,$">|; 

$key = q/d{bfr}=(dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2350.png"
 ALT="$ d{\bf r}=(dx$">|; 

$key = q/{bfv}cdotnablapsi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img542.png"
 ALT="$ {\bf v}\cdot\nabla\psi = 0$">|; 

$key = q/displaystylefrac{M}{M'}llfrac{rho'}{rho},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1018.png"
 ALT="$\displaystyle \frac{M}{M'} \ll \frac{\rho'}{\rho},$">|; 

$key = q/displaystylev_p=(g,d)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4017.png"
 ALT="$\displaystyle v_p = (g d)^{1/2},$">|; 

$key = q/t_0=pi,deltaslash(k,V);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3339.png"
 ALT="$ t_0= \pi \delta/(k V)$">|; 

$key = q/displaystyle=-frac{1}{a},frac{partial}{partialtheta}left[g,(zeta_b+zeta)+delta{mitPhi}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4491.png"
 ALT="$\displaystyle = - \frac{1}{a} \frac{\partial}{\partial\theta}\left[g (\zeta_b+\zeta)+ \delta{\mit\Phi}\right],$">|; 

$key = q/displaystylefrac{partialu}{partialt}-f,v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5063.png"
 ALT="$\displaystyle \frac{\partial u}{\partial t} -f v$">|; 

$key = q/m'=pi,a^{,2},rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1818.png"
 ALT="$ m'=\pi a^{ 2} \rho$">|; 

$key = q/displaystyle{bfS}=sum_i{bfS}_i.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6346.png"
 ALT="$\displaystyle {\bf S} = \sum_i {\bf S}_i.$">|; 

$key = q/v_y=sqrt{g,h},w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3563.png"
 ALT="$ v_y=\sqrt{g h} w$">|; 

$key = q/gsimeq9.8,{rmm,s}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img569.png"
 ALT="$ g\simeq 9.8 {\rm m s}^{-2}$">|; 

$key = q/displaystylesimeq1-frac{theta}{sqrt{{rmMa}_1^{,2}-1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5743.png"
 ALT="$\displaystyle \simeq 1-\frac{\theta}{\sqrt{{\rm Ma}_1^{ 2}-1}},$">|; 

$key = q/stackrel{displaystylerightarrow}{CD};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="30" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6704.png"
 ALT="$ \stackrel{\displaystyle \rightarrow}{CD}$">|; 

$key = q/displaystylelambda=frac{0.47}{U^{,6}},frac{dU}{dx}int_0^xU^{,5}(x'),dx'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3208.png"
 ALT="$\displaystyle \lambda= \frac{0.47}{U^{ 6}} \frac{dU}{dx}\int_0^x U^{ 5}(x') dx'.$">|; 

$key = q/e_E=0.0167;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4602.png"
 ALT="$ e_E= 0.0167$">|; 

$key = q/beta=6.58times10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5237.png"
 ALT="$ \beta=6.58\times 10^{-4}$">|; 

$key = q/displaystylefrac{dP}{dx}=-G,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3576.png"
 ALT="$\displaystyle \frac{dP}{dx} =- G,$">|; 

$key = q/displaystyle{bff}={bfB}+{bfW},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img578.png"
 ALT="$\displaystyle {\bf f}= {\bf B} + {\bf W},$">|; 

$key = q/displaystylesimeq{rmMa}_infty^{,2},(gamma+1),frac{u'}{U},frac{partialu'}{partialx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5915.png"
 ALT="$\displaystyle \simeq {\rm Ma}_\infty^{ 2} (\gamma+1) \frac{u'}{U} \frac{\partial u'}{\partial x}.$">|; 

$key = q/displaystyle=lambda_1+lambda_2+lambda_3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7011.png"
 ALT="$\displaystyle =\lambda_1+\lambda_2+\lambda_3,$">|; 

$key = q/displaystylez+{rmi},varpi=c,sinh(xi+{rmi},eta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2852.png"
 ALT="$\displaystyle z+{\rm i} \varpi = c \sinh(\xi+{\rm i} \eta).
$">|; 

$key = q/rho_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1362.png"
 ALT="$ \rho_1$">|; 

$key = q/displaystylealpha(x)=alpha_0+alpha_C(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5830.png"
 ALT="$\displaystyle \alpha(x)= \alpha_0+\alpha_C(x)$">|; 

$key = q/displaystyleoint{bfA}cdotd{bfr}=left(frac{partialA_z}{partialy}-frac{partialA_y}{partialz}right)dS_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6636.png"
 ALT="$\displaystyle \oint {\bf A} \cdot d{\bf r} = \left(\frac{\partial A_z}{\partial y}- \frac{\partial A_y}{\partial z}\right) dS_x$">|; 

$key = q/displaystylebar{F}=frac{k,V}{(lambda-cosalpha)^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2289.png"
 ALT="$\displaystyle \bar{F} = \frac{k V}{(\lambda-\cos\alpha)^2}.$">|; 

$key = q/delta'(x,y)simeqdelta(x,y)-theta,y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img636.png"
 ALT="$ \delta'(x,y)\simeq \delta(x,y) - \theta y$">|; 

$key = q/displaystylefrac{dw}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3567.png"
 ALT="$\displaystyle \frac{dw}{dt}$">|; 

$key = q/displaystyle{bf0}=nablap+rho,nabla({mitPsi}+{mitPsi}'+{mitPsi}''),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img978.png"
 ALT="$\displaystyle {\bf0 } = \nabla p + \rho \nabla({\mit\Psi}+{\mit\Psi}'+{\mit\Psi}'') ,$">|; 

$key = q/w(z)=-v_y(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3479.png"
 ALT="$ w(z)=-v_y(z)$">|; 

$key = q/k_n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4447.png"
 ALT="$ k_n'$">|; 

$key = q/omega_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4606.png"
 ALT="$ \omega_E$">|; 

$key = q/{mitDelta}rho=rho_2-rho_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5516.png"
 ALT="$ {\mit\Delta}\rho =\rho_2-\rho_1$">|; 

$key = q/displaystyle=frac{mitGamma}{2pi},lnr.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1986.png"
 ALT="$\displaystyle =\frac{\mit\Gamma}{2\pi} \ln r.$">|; 

$key = q/displaystyleT_c=frac{p_c}{rho_c},frac{m_p}{k},(1-beta),mu.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5202.png"
 ALT="$\displaystyle T_c = \frac{p_c}{\rho_c} \frac{m_p}{k} (1-\beta) \mu.$">|; 

$key = q/displaystyle=frac{1}{2},frac{x}{sqrt{{rmMa}_infty^{,2}-1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6073.png"
 ALT="$\displaystyle = \frac{1}{2} \frac{x}{\sqrt{{\rm Ma}_\infty^{ 2}-1}},$">|; 

$key = q/(0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2019.png"
 ALT="$ (0,$">|; 

$key = q/v_2=V-u_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1514.png"
 ALT="$ v_2=V-u_2$">|; 

$key = q/Lsim1,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="$ L\sim 1 {\rm m}$">|; 

$key = q/r_H=r_G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img833.png"
 ALT="$ r_H=r_G$">|; 

$key = q/displaystylefrac{delta(L)}{L}=frac{1.72}{{rmRe}^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2990.png"
 ALT="$\displaystyle \frac{\delta(L)}{L} = \frac{1.72}{{\rm Re}^{1/2}},$">|; 

$key = q/displaystyleh,rho,v^{,2},cosalpha,d-h_1,rho,v^{,2},frac{h_1}{2}+h_2,rho,v^{,2},frac{h_2}{2}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1897.png"
 ALT="$\displaystyle h \rho v^{ 2} \cos\alpha d- h_1 \rho v^{ 2} \frac{h_1}{2} + h_2 \rho v^{ 2} \frac{h_2}{2} = 0,$">|; 

$key = q/{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="$ {\mit\Psi}$">|; 

$key = q/{mitGamma}(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3451.png"
 ALT="$ {\mit\Gamma}(z)$">|; 

$key = q/oint{bfA}cdotd{bfr}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6625.png"
 ALT="$ \oint {\bf A}\cdot d{\bf r}=0$">|; 

$key = q/displaystyle=frac{2}{pi}int_{sin^{-1}(lambda)}^{pislash2}cosleft(frac{2,alpha}{pfrac{2,alpha}{pi},betaright)frac{dbeta}{tanbeta}right]^{,-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="552" HEIGHT="81" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2516.png"
 ALT="$\displaystyle =\frac{2}{\pi}\int_{\sin^{-1}(\lambda)}^{\pi/2}\cos\left(\frac{2\...
...left(\frac{2 \alpha}{\pi} \beta\right)\frac{d\beta}{\tan\beta}\right]^{ -1},$">|; 

$key = q/{mitGamma}_{n,r}^m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4931.png"
 ALT="$ {\mit\Gamma}_{n,r}^m$">|; 

$key = q/displaystyle{bfv}=v_theta(r),{bfe}_theta=r,{mitOmega}(r),{bfe}_theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3626.png"
 ALT="$\displaystyle {\bf v} = v_\theta(r) {\bf e}_\theta = r {\mit\Omega}(r) {\bf e}_\theta,$">|; 

$key = q/displaystyleskew{3}hat{beta}_{-n,,-n'}^{,-m,,-m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5004.png"
 ALT="$\displaystyle \skew{3}\hat{\beta}_{-n, -n'}^{ -m, -m'}$">|; 

$key = q/displaystyle(1-{rmMa}_infty^{,2}),frac{partial^{,2}phi}{partialx^{,2}}+frac{partial^{,2}phi}{partialy^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5945.png"
 ALT="$\displaystyle (1-{\rm Ma}_\infty^{ 2}) \frac{\partial^{ 2} \phi}{\partial x^{ 2}} + \frac{\partial^{ 2}\phi}{\partial y^{ 2}}$">|; 

$key = q/displaystyletimes{bfa}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6407.png"
 ALT="$\displaystyle \times {\bf a}.$">|; 

$key = q/displaystylefrac{p_2-p_1}{p_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6192.png"
 ALT="$\displaystyle \frac{p_2-p_1}{p_1}$">|; 

$key = q/displaystylenablaP=-G,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3608.png"
 ALT="$\displaystyle \nabla P =- G {\bf e}_z$">|; 

$key = q/0leqphileqpislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1301.png"
 ALT="$ 0\leq\phi\leq \pi/2$">|; 

$key = q/90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$ 90^\circ $">|; 

$key = q/ds=sqrt{dx^{,2}+dy^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7343.png"
 ALT="$ ds = \sqrt{dx^{ 2}+dy^{ 2}}$">|; 

$key = q/displaystylef_x=int_0^lmuleft(frac{partialv_x}{partialy}right)_{y=0},dx=-frac{2,rac{d_1}{d_2}right)-3left(frac{d_1-d_2}{d_1+d_2}right)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="525" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3711.png"
 ALT="$\displaystyle f_x = \int_0^l\mu\left(\frac{\partial v_x}{\partial y}\right)_{y=...
...[2\ln\left(\frac{d_1}{d_2}\right)-3\left(\frac{d_1-d_2}{d_1+d_2}\right)\right].$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashfree2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img2238.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/free2.eps}}
\end{figure}">|; 

$key = q/displaystyle=m,(m+2,n+1),r^{,m-2},{calR}_n.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4336.png"
 ALT="$\displaystyle = m (m+2 n+1) r^{ m-2} {\cal R}_n.$">|; 

$key = q/sigma_{ii}slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="$ \sigma_{ii}/3$">|; 

$key = q/displaystyleu_e=sqrt{{calR},T_0left(frac{2,gamma}{gamma-1}right)left[1-left(frac{p_e}{p_0}right)^{(gamma-1)slashgamma}right]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="91" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5432.png"
 ALT="$\displaystyle u_e=\sqrt{{\cal R} T_0\left(\frac{2 \gamma}{\gamma-1}\right)\left[1-\left(\frac{p_e}{p_0}\right) ^{(\gamma-1)/\gamma}\right]}.$">|; 

$key = q/p_2slashp_1=rho_2slashrho_1=u_2slashu_1=T_2slashT_1=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5502.png"
 ALT="$ p_2/p_1=\rho_2/\rho_1=u_2/u_1=T_2/T_1=1$">|; 

$key = q/displaystylem,frac{d{bfV}}{dt}={bfF}+{bfF}_{rmext},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1820.png"
 ALT="$\displaystyle m \frac{d{\bf V}}{dt} = {\bf F} + {\bf F}_{\rm ext},$">|; 

$key = q/displaystyle=frac{6}{beta},(1+k_2-h_2),skew{5}tilde{zeta}_j,delta_{n,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4697.png"
 ALT="$\displaystyle =\frac{6}{\beta} (1+k_2-h_2) \skew{5}\tilde{\zeta}_j \delta_{n 2}.$">|; 

$key = q/f(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6441.png"
 ALT="$ f(x,y)$">|; 

$key = q/displaystyle=frac{1}{r^{,2}},frac{partial(r^{,2},v_r)}{partialr}+frac{1}{r,sinthta}+frac{1}{r,sintheta},frac{partialv_varphi}{partialvarphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img511.png"
 ALT="$\displaystyle =\frac{1}{r^{ 2}} \frac{\partial (r^{ 2} v_r)}{\partial r} +\...
...\theta} + \frac{1}{r \sin\theta} \frac{\partial v_\varphi}{\partial \varphi},$">|; 

$key = q/displaystylexi_i=sum_{n=1,infty}xi_{i,n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4410.png"
 ALT="$\displaystyle \xi_i = \sum_{n=1,\infty} \xi_{i n},$">|; 

$key = q/displaystyle=-frac{g}{a},frac{partial}{partialtheta}left[zeta-(1+k_2-h_2),skew{5}bar{zeta}_2-ointG_F,zeta,d{mitOmega}'right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4516.png"
 ALT="$\displaystyle =- \frac{g}{a} \frac{\partial}{\partial\theta}\left[\zeta-(1+k_2-h_2) \skew{5}\bar{\zeta}_2 -\oint G_F \zeta d{\mit\Omega}'\right],$">|; 

$key = q/displaystyleV_c=0.56,(g,l)^{1slash2}=1.75,[l({rmm})]^{1slash2},{rmmslashs}=3.4,[l({rmm})]^{1slash2},{rmkts}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="456" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4008.png"
 ALT="$\displaystyle V_c = 0.56 (g l)^{1/2} = 1.75 [l({\rm m})]^{1/2} {\rm m/s} = 3.4 [l({\rm m})]^{1/2} {\rm kts}.$">|; 

$key = q/displaystyleg'(x)=-frac{U}{lambda}left(frac{dy}{dx}right)_L.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6054.png"
 ALT="$\displaystyle g'(x) = -\frac{U}{\lambda}\left(\frac{dy}{dx}\right)_L.$">|; 

$key = q/displaystylea=frac{gamma}{p_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1237.png"
 ALT="$\displaystyle a = \frac{\gamma}{p_0}.$">|; 

$key = q/displaystylephantom{=}+frac{h}{h_i,h_j^{,2}}left[frac{A_j}{h_i^{,2}},frac{partia,frac{partial}{partialu_j}!left(frac{h_j^{,2}}{h}right)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="429" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7100.png"
 ALT="$\displaystyle \phantom{=}+\frac{h}{h_i h_j^{ 2}}\left[\frac{A_j}{h_i^{ 2}} ...
... u_j} \frac{\partial}{\partial u_j}\!\left( \frac{h_j^{ 2}}{h}\right) \right]$">|; 

$key = q/displaystyle{mitLambda}=frac{delta^{,2}}{nu},frac{dU}{dx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3158.png"
 ALT="$\displaystyle {\mit\Lambda} = \frac{\delta^{ 2}}{\nu} \frac{dU}{dx}.$">|; 

$key = q/displaystylefrac{w_2^{,2}}{w_1^{,2}}=frac{u_2^{,2}+v^{,2}}{u_1^{,2}+v^{,2}}=fracta-theta)+1}{tan^2beta+1}=frac{cos^2beta}{cos^2(beta-theta)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="525" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5739.png"
 ALT="$\displaystyle \frac{w_2^{ 2}}{w_1^{ 2}}=\frac{u_2^{ 2}+v^{ 2}}{u_1^{ 2}+v^...
...tan^2(\beta-\theta)+1}{\tan^2\beta+1}=\frac{\cos^2\beta}{\cos^2(\beta-\theta)},$">|; 

$key = q/displaystyle{mitGamma}_0simeqpi,V,c_0,alpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3491.png"
 ALT="$\displaystyle {\mit\Gamma}_0\simeq \pi V c_0 \alpha.$">|; 

$key = q/displaystyle{bfA}times{bfB};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7086.png"
 ALT="$\displaystyle {\bf A}\times {\bf B}$">|; 

$key = q/displaystylefrac{rho^{,(gamma-1)slash2}-rho_2^{,(gamma-1)slash2}}{rho_1^{,(gamma-1)slash2}-rho_2^{,(gamma-1)slash2}}=frac{x-x_2}{x_1-x_2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5571.png"
 ALT="$\displaystyle \frac{\rho^{ (\gamma-1)/2}-\rho_2^{ (\gamma-1)/2}}{\rho_1^{ (\gamma-1)/2}-\rho_2^{ (\gamma-1)/2}}=\frac{x-x_2}{x_1-x_2}$">|; 

$key = q/{rmMa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="$ {\rm Ma}$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashrect.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="499" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3041.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/rect.eps}}
\end{figure}">|; 

$key = q/displaystylea;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2127.png"
 ALT="$\displaystyle a$">|; 

$key = q/displaystylerho,u,frac{d{calE}}{dx}+frac{d}{dx},(p,u)=u,frac{dp}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5456.png"
 ALT="$\displaystyle \rho u \frac{d{\cal E}}{dx}+\frac{d}{dx} (p u) = u \frac{dp}{dx},$">|; 

$key = q/displaystyle=oint_{C}F(z),dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2363.png"
 ALT="$\displaystyle =\oint_{C}F(z) dz,$">|; 

$key = q/displaystyle{bfA}={bfgrad},V.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6557.png"
 ALT="$\displaystyle {\bf A} = {\bf grad}  V.$">|; 

$key = q/yrightarrow-y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2236.png"
 ALT="$ y\rightarrow -y$">|; 

$key = q/displaystyleint_P^Qx,y^{,2},dl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6449.png"
 ALT="$\displaystyle \int_P^Q x y^{ 2} dl$">|; 

$key = q/displaystylephi(r,theta)=R(r),{mitTheta}(theta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2573.png"
 ALT="$\displaystyle \phi(r,\theta)= R(r) {\mit\Theta}(\theta).$">|; 

$key = q/displaystylepsi=frac{Q}{4pi}left(costheta_2-costheta_1right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2617.png"
 ALT="$\displaystyle \psi = \frac{Q}{4\pi}\left(\cos\theta_2-\cos\theta_1\right),$">|; 

$key = q/xigeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2493.png"
 ALT="$ \xi\geq 0$">|; 

$key = q/displaystyle=frac{2}{A},L_0,alpha^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3514.png"
 ALT="$\displaystyle = \frac{2}{A} L_0 \alpha^2,$">|; 

$key = q/displaystyle{rmdet}(A_{ij}-lambda,delta_{ij})=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7005.png"
 ALT="$\displaystyle {\rm det}(A_{ij}-\lambda \delta_{ij}) = 0.
$">|; 

$key = q/displaystyle(nabla^{,2}{bfA})_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7159.png"
 ALT="$\displaystyle (\nabla^{ 2}{\bf A})_r$">|; 

$key = q/cslash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3353.png"
 ALT="$ c/4$">|; 

$key = q/oint{bfv}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1704.png"
 ALT="$ \oint {\bf v}\cdot d{\bf r}$">|; 

$key = q/lambda_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7009.png"
 ALT="$ \lambda_3$">|; 

$key = q/{mitPhi}_S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img526.png"
 ALT="$ {\mit\Phi}_S$">|; 

$key = q/displaystylev_theta(rrightarrowinfty,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3757.png"
 ALT="$\displaystyle v_\theta(r\rightarrow\infty,\theta)$">|; 

$key = q/displaystylefrac{D}{Dt}left(frac{p}{rho^{,gamma}}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5337.png"
 ALT="$\displaystyle \frac{D}{Dt}\left(\frac{p}{\rho^{ \gamma}}\right)$">|; 

$key = q/{bfa}cdot{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6301.png"
 ALT="$ {\bf a} \cdot {\bf b}$">|; 

$key = q/20^circ,{rmC};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="$ 20^\circ {\rm C}$">|; 

$key = q/displaystyleu'=-frac{partialphi}{partialx}=-frac{1}{m},frac{partialskew{3}bar{phi}}{partialxi}=frac{bar{u}}{sqrt{1-{rmMa}_infty^{,2}}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6040.png"
 ALT="$\displaystyle u' =-\frac{\partial\phi}{\partial x} =-\frac{1}{m} \frac{\partia...
...ew{3}\bar{\phi}}{\partial\xi} = \frac{\bar{u}}{\sqrt{1-{\rm Ma}_\infty^{ 2}}},$">|; 

$key = q/displaystyleV=int_{y_1}^{y_2}dyleft[int_{x_1(y)}^{x_2(y)}D(x,y),dxright]equivint!int_SD(x,y),dx,dy.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6489.png"
 ALT="$\displaystyle V = \int_{y_1}^{y_2} dy\left[\int_{x_1(y)}^{x_2(y)} D(x,y) dx\right] \equiv \int\!\int_S D(x,y) dx dy.$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashfree1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img2217.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/free1.eps}}
\end{figure}">|; 

$key = q/r>a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1906.png"
 ALT="$ r&gt;a$">|; 

$key = q/displaystyleF_x=F_y=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2669.png"
 ALT="$\displaystyle F_x=F_y=0,$">|; 

$key = q/r=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4314.png"
 ALT="$ r=$">|; 

$key = q/deltaJ=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7333.png"
 ALT="$ \delta J = 0$">|; 

$key = q/displaystyle=frac{pi}{4},rho,V,b,{mitGamma}_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3509.png"
 ALT="$\displaystyle =\frac{\pi}{4} \rho V b {\mit\Gamma}_0,$">|; 

$key = q/omega,r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1914.png"
 ALT="$ \omega r$">|; 

$key = q/2,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1915.png"
 ALT="$ 2 \omega$">|; 

$key = q/D{bfv}slashDt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3610.png"
 ALT="$ D{\bf v}/Dt=0$">|; 

$key = q/slashDt={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1571.png"
 ALT="$ /Dt={\bf0}$">|; 

$key = q/displaystyle=q_1int_{x_L}^{x_R}left[C_{pL}left(-frac{dy_L}{dx}right)+C_{pU}left(frac{dy_U}{dy}right)right]dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5846.png"
 ALT="$\displaystyle = q_1\int_{x_L}^{x_R} \left[C_{pL}\left(-\frac{dy_L}{dx}\right)+C_{pU}\left(\frac{dy_U}{dy}\right)\right]dx.$">|; 

$key = q/displaystyleF(z)=-V_0left(z+frac{a^{,2}}{z}right)+{rmi},frac{mitGamma}{2pi},lnleft(frac{z}{a}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1988.png"
 ALT="$\displaystyle F(z)= -V_0\left(z+\frac{a^{ 2}}{z}\right) + {\rm i} \frac{\mit\Gamma}{2\pi} \ln\left(\frac{z}{a}\right),$">|; 

$key = q/2,a^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2115.png"
 ALT="$ 2 a^{ 2})$">|; 

$key = q/theta=2,sin^{-1}(1slashsqrt{3});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2824.png"
 ALT="$ \theta=2 \sin^{-1}(1/\sqrt{3})$">|; 

$key = q/rho_2<rho(z_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5165.png"
 ALT="$ \rho_2&lt;\rho(z_2)$">|; 

$key = q/q=2slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3252.png"
 ALT="$ q=2/3$">|; 

$key = q/displaystyled(x)=d_1-alpha,x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3693.png"
 ALT="$\displaystyle d(x) = d_1 -\alpha x,$">|; 

$key = q/ggg{mitOmega}^{,2},a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4399.png"
 ALT="$ g\gg {\mit\Omega}^{ 2} a$">|; 

$key = q/1slash4<s<9slash32;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img801.png"
 ALT="$ 1/4&lt;s&lt;9/32$">|; 

$key = q/psi(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3732.png"
 ALT="$ \psi(r,\theta)$">|; 

$key = q/displaystylefrac{1}{U_0^{,2},x^{,2,m-1}},frac{dU^{,2}}{dx}=2,m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2938.png"
 ALT="$\displaystyle \frac{1}{U_0^{ 2} x^{ 2 m-1}} \frac{dU^{ 2}}{dx}=2 m$">|; 

$key = q/displaystyle[p]_{z=0_-}^{z=0_+}=gamma,frac{partial^{,2}zeta}{partialx^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4141.png"
 ALT="$\displaystyle [p]_{z=0_-}^{z=0_+} = \gamma \frac{\partial^{ 2}\zeta}{\partial x^{ 2}}.$">|; 

$key = q/displaystyle=int_{mitOmega}left(frac{partial{mitPhi}_r''}{partialtheta},frac{parialphi},frac{partial{mitPhi}_s}{partialphi}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4872.png"
 ALT="$\displaystyle =\int_{\mit\Omega} \left(\frac{\partial {\mit\Phi}_r''}{\partial\...
...}{\partial\phi} \frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega},$">|; 

$key = q/displaystyleint_0^1left(frac{y^{,2}}{2}-y^3+frac{y^{,4}}{2}right)dy=frac{1}{60}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6494.png"
 ALT="$\displaystyle \int_0^1 \left(\frac{y^{ 2}}{2}-y^3+ \frac{y^{ 4}}{2}\right)dy = \frac{1}{60}.$">|; 

$key = q/A=(1slash4),V,a^{,3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3779.png"
 ALT="$ A=(1/4) V a^{ 3}$">|; 

$key = q/lambda_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4808.png"
 ALT="$ \lambda_r$">|; 

$key = q/gamma=1.4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$ \gamma =1.4$">|; 

$key = q/displaystylev_x(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3870.png"
 ALT="$\displaystyle v_x(y)$">|; 

$key = q/displaystyle{mitGamma}(phi)=sum_{nu=1,3,5,cdots}{mitGamma}_nu,sin(nu,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3546.png"
 ALT="$\displaystyle {\mit\Gamma}(\phi) = \sum_{\nu=1,3,5,\cdots} {\mit\Gamma}_\nu \sin(\nu \phi),
$">|; 

$key = q/90^{circ};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2431.png"
 ALT="$ 90^{\circ}$">|; 

$key = q/m_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4524.png"
 ALT="$ m_j$">|; 

$key = q/D_1,{rme}^{,{rmi},alpha_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2454.png"
 ALT="$ D_1 {\rm e}^{ {\rm i} \alpha_1}$">|; 

$key = q/displaystyle{bfV}cdot{bfe}_r=left.{bfv}cdot{bfe}_rright|_{r=a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1801.png"
 ALT="$\displaystyle {\bf V}\cdot{\bf e}_r = \left. {\bf v}\cdot{\bf e}_r\right\vert _{r=a},$">|; 

$key = q/(2,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2049.png"
 ALT="$ (2 a$">|; 

$key = q/displaystyle=int_{mitOmega}costhetaleft(-eta,frac{partial{mitPhi}_r}{partialthetintheta},frac{partial{mitPhi}_r}{partialphi}right)d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4899.png"
 ALT="$\displaystyle = \int_{\mit\Omega}\cos\theta\left(-\eta \frac{\partial{\mit\Phi...
...\xi}{\sin\theta} \frac{\partial{\mit\Phi}_r}{\partial\phi}\right)d{\mit\Omega}$">|; 

$key = q/r^{,3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1917.png"
 ALT="$ r^{ 3}$">|; 

$key = q/1slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$ 1/3$">|; 

$key = q/displaystyleskew{3}hat{delta}_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4996.png"
 ALT="$\displaystyle \skew{3}\hat{\delta}_{n, n'}^{ m, m'}$">|; 

$key = q/displaystyleC_psimeqfrac{2}{U},frac{partialphi}{partialx}=frac{2}{U,m},frac{partialskew{3}bar{phi}}{partialxi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6033.png"
 ALT="$\displaystyle C_p\simeq \frac{2}{U} \frac{\partial\phi}{\partial x} = \frac{2}{U m} \frac{\partial\skew{3}\bar{\phi}}{\partial\xi}.$">|; 

$key = q/displaystyleskew{3}bar{phi}=frac{k,V}{(lambda-cosalpha)^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2302.png"
 ALT="$\displaystyle \skew{3}\bar{\phi} = \frac{k V}{(\lambda-\cos\alpha)^2}.$">|; 

$key = q/displaystylekappa_T-kappa_{{calS}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5604.png"
 ALT="$\displaystyle \kappa_T - \kappa_{{\cal S}}$">|; 

$key = q/f_j=sigma_jslash2,{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4609.png"
 ALT="$ f_j=\sigma_j/2 {\mit\Omega}$">|; 

$key = q/P'={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4788.png"
 ALT="$ P'={\rm constant}$">|; 

$key = q/displaystylea_{ij}'={calR}_{ik},{calR}_{jl},a_{kl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6841.png"
 ALT="$\displaystyle a_{ij}' = {\cal R}_{ik} {\cal R}_{jl} a_{kl},$">|; 

$key = q/displaystyleleft.pright|_{z=d+zeta_b+zeta}=p_{rmatm}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4485.png"
 ALT="$\displaystyle \left.p\right\vert _{z=d+\zeta_b+\zeta} = p_{\rm atm}.$">|; 

$key = q/m=5.97times10^{24},{rmkg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4570.png"
 ALT="$ m=5.97\times 10^{24} {\rm kg}$">|; 

$key = q/displaystylee_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img931.png"
 ALT="$\displaystyle e_{12}$">|; 

$key = q/displaystyle=frac{1}{3}left[tan^{-1}left(frac{sqrt{-D}}{R}right)+{mitDelta}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6110.png"
 ALT="$\displaystyle = \frac{1}{3}\left[\tan^{-1}\left(\frac{\sqrt{-D}}{R}\right)+{\mit\Delta}\right].$">|; 

$key = q/(rho,g)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4023.png"
 ALT="$ (\rho g)^{1/2}$">|; 

$key = q/displaystylewidetilde{nabla{bfA}}=frac{1}{2}left[nabla{bfA}+(nabla{bfA})^Tright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7141.png"
 ALT="$\displaystyle \widetilde{\nabla {\bf A}} = \frac{1}{2}\left[\nabla {\bf A} + (\nabla{\bf A})^T\right].$">|; 

$key = q/displaystylepsi(xi,eta)=-frac{2,V_0,c}{n},frac{sin(2,xislashn)}{cosh(2,etaslashn)-cos(2,xislashn)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2482.png"
 ALT="$\displaystyle \psi(\xi,\eta) = -\frac{2 V_0 c}{n} \frac{\sin(2 \xi/n)}{\cosh(2 \eta/n)-\cos(2 \xi/n)}.
$">|; 

$key = q/c_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5554.png"
 ALT="$ c_2$">|; 

$key = q/displaystylezeta_b(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4638.png"
 ALT="$\displaystyle \zeta_b (\theta)$">|; 

$key = q/B'=rho_0,V_0',g=W+rho_0,A_0,g,delta_1=(1+delta_1slashdelta_0),W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img614.png"
 ALT="$ B'=\rho_0 V_0' g=W+ \rho_0 A_0 g \delta_1=(1+\delta_1/\delta_0) W$">|; 

$key = q/displaystylefrac{partialp}{partialx}=c^{,2},frac{partialrho}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5348.png"
 ALT="$\displaystyle \frac{\partial p}{\partial x} =c^{ 2} \frac{\partial \rho}{\partial x},$">|; 

$key = q/kll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1208.png"
 ALT="$ k\ll 1$">|; 

$key = q/displaystyleskew{3}hat{beta}_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4998.png"
 ALT="$\displaystyle \skew{3}\hat{\beta}_{n, n'}^{ m, m'}$">|; 

$key = q/p_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$ p_3$">|; 

$key = q/displaystyle-lambda,f'(x)=-U,epsilon,alpha,cos(alpha,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5992.png"
 ALT="$\displaystyle -\lambda f'(x)=-U \epsilon \alpha \cos(\alpha x),$">|; 

$key = q/n>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6242.png"
 ALT="$ n&gt;0$">|; 

$key = q/psi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1922.png"
 ALT="$ \psi=0$">|; 

$key = q/dl=z,sec^2phi,dphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3474.png"
 ALT="$ dl= z \sec^2\phi d\phi$">|; 

$key = q/displaystyle-frac{sinalpha,tanbeta,cosgamma,(1+sin^2beta)}{cos^2alpha};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img941.png"
 ALT="$\displaystyle -\frac{\sin\alpha \tan\beta \cos\gamma (1+\sin^2\beta)}{\cos^2\alpha}$">|; 

$key = q/v_p=omegaslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3930.png"
 ALT="$ v_p=\omega/k$">|; 

$key = q/displaystylefrac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi}-frac{partialphi}{partialt}=C(t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1591.png"
 ALT="$\displaystyle \frac{p}{\rho} + \frac{1}{2} v^{ 2} + {\mit\Psi} - \frac{\partial\phi}{\partial t} = C(t),$">|; 

$key = q/displaystyle=frac{(3+n)}{2,(2,n+3),(n+1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4427.png"
 ALT="$\displaystyle = \frac{(3+n)}{2 (2 n+3) (n+1)},$">|; 

$key = q/displaystyle=(a_y,b_z-a_z,b_y),costheta+(a_z,b_x-a_x,b_z),sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6361.png"
 ALT="$\displaystyle = (a_y  b_z - a_z  b_y)  \cos\theta + (a_z  b_x-a_x  b_z) \sin\theta$">|; 

$key = q/displaystyle=sum_{n=1,infty}left[(1+k_2),skew{5}bar{zeta}_2,delta_{n,2}+(1+k_n'),alpha_n,zeta_nright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4506.png"
 ALT="$\displaystyle = \sum_{n=1,\infty}\left[(1+k_2) \skew{5}\bar{\zeta}_2 \delta_{n 2} + (1+k_n') \alpha_n \zeta_n\right].$">|; 

$key = q/N_A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5294.png"
 ALT="$ N_A$">|; 

$key = q/displaystyle=p_0-left[rho',g+frac{rho',k,(V'-c)^2}{tanh(k,d')}right]zeta_0,sin(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4105.png"
 ALT="$\displaystyle =p_0-\left[\rho' g+\frac{\rho' k (V'-c)^2}{\tanh(k d')}\right]\zeta_0 \sin(\omega t-k x).$">|; 

$key = q/z=H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5134.png"
 ALT="$ z=H$">|; 

$key = q/y=sinhxi,sineta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7222.png"
 ALT="$ y=\sinh\xi \sin\eta$">|; 

$key = q/alpha=alpha(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3263.png"
 ALT="$ \alpha=\alpha(x)$">|; 

$key = q/displaystylefrac{Dmbox{boldmathomega}}{Dt}=(mbox{boldmathomega}cdotnabla),{bfv}+nu,nabla^{,2}mbox{boldmathomega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img532.png"
 ALT="$\displaystyle \frac{D\mbox{\boldmath $\omega$}}{Dt} = (\mbox{\boldmath $\omega$}\cdot\nabla) {\bf v} + \nu \nabla^{ 2}\mbox{\boldmath $\omega$}.
$">|; 

$key = q/displaystylec_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5613.png"
 ALT="$\displaystyle c_p$">|; 

$key = q/v_yrightarrow-V,cosalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2294.png"
 ALT="$ v_y\rightarrow -V \cos\alpha$">|; 

$key = q/y=d(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3664.png"
 ALT="$ y=d(x)$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{AppendixAslashfigA.21.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="226" BORDER="0"
 SRC="|."$dir".q|img6577.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{AppendixA/figA.21.eps}}
\end{figure}">|; 

$key = q/displaystyle=(delta_{ik}-delta{theta}_l,epsilon_{lik}),(delta_{jk}-deltatheta_m,elta_{jk},epsilon_{lik}-deltatheta_m,delta_{ik},epsilon_{mjk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="552" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6830.png"
 ALT="$\displaystyle = (\delta_{ik}-\delta{\theta}_l \epsilon_{lik}) (\delta_{jk}-\d...
...eta_l \delta_{jk} \epsilon_{lik} -\delta\theta_m \delta_{ik} \epsilon_{mjk}$">|; 

$key = q/displaystyleint_{S_1}{bfA}cdotd{bfS}=int_{S_2}{bfA}cdotd{bfS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6591.png"
 ALT="$\displaystyle \int_{S_1} {\bf A}\cdot d{\bf S} = \int_{S_2} {\bf A}\cdot d{\bf S},$">|; 

$key = q/displaystyle=frac{D}{2pi},frac{cos(theta-theta_0)}{r^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1695.png"
 ALT="$\displaystyle = \frac{D}{2\pi} \frac{\cos(\theta-\theta_0)}{r^{ 2}},$">|; 

$key = q/v_x(0)=U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3585.png"
 ALT="$ v_x(0)=U$">|; 

$key = q/thetarightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img622.png"
 ALT="$ \theta\rightarrow 0$">|; 

$key = q/deltaI;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7311.png"
 ALT="$ \delta I$">|; 

$key = q/dF=dphi+{rmi},dpsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2403.png"
 ALT="$ dF = d\phi+{\rm i} d\psi$">|; 

$key = q/displaystyle=frac{M}{5},(a_1^{,2}+a_2^{,2}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7284.png"
 ALT="$\displaystyle = \frac{M}{5}  (a_1^{ 2}+a_2^{ 2}).$">|; 

$key = q/skew{5}tilde{zeta}_8=8.95times10^{-2},{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5041.png"
 ALT="$ \skew{5}\tilde{\zeta}_8=8.95\times 10^{-2} {\rm m}$">|; 

$key = q/a_0';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1016.png"
 ALT="$ a_0'$">|; 

$key = q/displaystylen_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4138.png"
 ALT="$\displaystyle n_z$">|; 

$key = q/displaystylec^{,2}-left(frac{2,V',s}{1+s}right)c+frac{V'^{,2},s}{1+s}=c_1^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4189.png"
 ALT="$\displaystyle c^{ 2} - \left(\frac{2 V' s}{1+s}\right)c + \frac{V'^{ 2} s}{1+s} =c_1^{ 2},$">|; 

$key = q/Y=eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4756.png"
 ALT="$ Y=\eta$">|; 

$key = q/displaystyle={bfv}_1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4075.png"
 ALT="$\displaystyle = {\bf v}_1.$">|; 

$key = q/displaystyle=oint_S{bff}cdot{bfe}_z,dS;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3806.png"
 ALT="$\displaystyle = \oint_S {\bf f}\cdot{\bf e}_z dS$">|; 

$key = q/displaystyleB=V,(l^{,2},{rme}^{,{rmi},alpha}-a^{,2},{rme}^{-{rmi},alpha});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3317.png"
 ALT="$\displaystyle B = V (l^{ 2} {\rm e}^{ {\rm i} \alpha} -a^{ 2} {\rm e}^{-{\rm i} \alpha})$">|; 

$key = q/zeta_n=zeta_{t,n}-zeta_{b,n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4358.png"
 ALT="$ \zeta_n=\zeta_{t n}-\zeta_{b n}$">|; 

$key = q/displaystyle=-frac{1}{a,sintheta}left[frac{partial}{partialtheta},(sintheta,u,d)+frac{partial(v,d)}{partialphi}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5053.png"
 ALT="$\displaystyle = - \frac{1}{a \sin\theta}\left[\frac{\partial}{\partial\theta} (\sin\theta u d)+\frac{\partial (v d)}{\partial\phi}\right],$">|; 

$key = q/displaystylefrac{d{bfa}}{dt}=lim_{deltatrightarrow0}left[frac{{bfa}(t+deltat)-{bfa}(t)}{deltat}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6433.png"
 ALT="$\displaystyle \frac{d {\bf a}}{dt} = \lim_{\delta t\rightarrow 0} \left[\frac{{\bf a}(t+\delta t) - {\bf a}(t)} {\delta t}\right].$">|; 

$key = q/b_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6863.png"
 ALT="$ b_{ij}$">|; 

$key = q/i=i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6923.png"
 ALT="$ i=i$">|; 

$key = q/displaystyle=-frac{partial{mitPhi}'}{partialtheta}-frac{1}{sintheta},frac{partial{mitPsi}'}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4777.png"
 ALT="$\displaystyle = -\frac{\partial{\mit\Phi}'}{\partial\theta} -\frac{1}{\sin\theta} \frac{\partial{\mit\Psi}'}{\partial\phi},$">|; 

$key = q/displaystyle={rmsgn}(beta),|alpha|,(1-k^{,2})^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1306.png"
 ALT="$\displaystyle ={\rm sgn}(\beta) \vert\alpha\vert (1-k^{ 2})^{1/2},$">|; 

$key = q/displaystylephantom{=}+int_{mitOmega}frac{1}{sintheta}left(frac{partial{mitPhi}_ltheta},frac{partial{mitPsi}_s}{partialphi}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4879.png"
 ALT="$\displaystyle \phantom{=}+\int_{\mit\Omega}\frac{1}{\sin\theta}\left(\frac{\par...
...\partial\theta} \frac{\partial{\mit\Psi}_s}{\partial\phi}\right)d{\mit\Omega}.$">|; 

$key = q/displaystylesqrt{{rmMa}^{,2}-1},frac{dw}{w}=left(frac{b^{,2}-1}{b^{,2}+tan^2mu}right)dmu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6207.png"
 ALT="$\displaystyle \sqrt{{\rm Ma}^{ 2}-1} \frac{dw}{w}=\left(\frac{b^{ 2}-1}{b^{ 2}+\tan^2\mu}\right)d\mu,
$">|; 

$key = q/A_i={bfe}_icdot{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7041.png"
 ALT="$ A_i={\bf e}_i\cdot {\bf A}$">|; 

$key = q/displaystylepsi(r,theta)=frac{1}{4},V,a^2,sin^2thetaleft[left(frac{overline{mu}}overline{mu}}right)frac{r}{a}+2left(frac{r}{a}right)^2right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="532" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3856.png"
 ALT="$\displaystyle \psi(r,\theta) =\frac{1}{4} V a^2 \sin^2\theta\left[\left(\fra...
...u}}{\mu+\overline{\mu}}\right)\frac{r}{a} + 2\left(\frac{r}{a}\right)^2\right],$">|; 

$key = q/displaystylev_r(a,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3751.png"
 ALT="$\displaystyle v_r(a,\theta)$">|; 

$key = q/S_{mitTheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="$ S_{\mit\Theta}$">|; 

$key = q/{bfF}=F_r(r),{bfe}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7205.png"
 ALT="$ {\bf F} = F_r(r) {\bf e}_r$">|; 

$key = q/y=pm1.5,X_1^{,1slash3},x^{,2slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4225.png"
 ALT="$ y=\pm 1.5 X_1^{ 1/3} x^{ 2/3}$">|; 

$key = q/v=v_phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4457.png"
 ALT="$ v=v_\phi$">|; 

$key = q/{bfv}({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4451.png"
 ALT="$ {\bf v}({\bf r})$">|; 

$key = q/displaystylex_2=left[c_1-frac{1}{2},(gamma+1),V_pright]t.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5566.png"
 ALT="$\displaystyle x_2=\left[c_1-\frac{1}{2} (\gamma+1) V_p\right]t.$">|; 

$key = q/displaystyle=x',costheta-y',sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6567.png"
 ALT="$\displaystyle = x'  \cos\theta - y' \sin\theta,$">|; 

$key = q/displaystyle=-frac{partialphi}{partialy}=U,epsilon,alpha,cos[alpha,(x-lambda,y)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5996.png"
 ALT="$\displaystyle =- \frac{\partial\phi}{\partial y}=U \epsilon \alpha \cos[\alpha (x-\lambda y)].$">|; 

$key = q/{bfV}=nablavarphitimesnablapsi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2709.png"
 ALT="$ {\bf V} = \nabla\varphi\times\nabla\psi_0$">|; 

$key = q/displaystyle=frac{alpha^{,2}-beta^{,2}}{alpha^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1300.png"
 ALT="$\displaystyle = \frac{\alpha^{ 2}-\beta^{ 2}}{\alpha^{ 2}},$">|; 

$key = q/displaystyle=|alpha|left[E(phi,k)-(1-k^{,2})^{1slash2},F(phi,k)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1335.png"
 ALT="$\displaystyle = \vert\alpha\vert\left[E(\phi,k) -(1-k^{ 2})^{1/2} F(\phi,k)\right].$">|; 

$key = q/displaystyletheta_s=frac{pi}{2},frac{(-lambda_s)}{0.47+4,(-lambda_s)}simeq13^circ.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3230.png"
 ALT="$\displaystyle \theta_s = \frac{\pi}{2} \frac{(-\lambda_s)}{0.47+4 (-\lambda_s)} \simeq 13^\circ.$">|; 

$key = q/x+dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2377.png"
 ALT="$ x+dx$">|; 

$key = q/R_p=(1-e),R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4288.png"
 ALT="$ R_p=(1-e) R$">|; 

$key = q/beta_0,lnr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1750.png"
 ALT="$ \beta_0 \ln r$">|; 

$key = q/displaystylev'(x,y)=v'(x,0)+left(frac{partialv'}{partialy}right)_{y=0}y+cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5940.png"
 ALT="$\displaystyle v'(x,y) = v'(x,0) + \left(\frac{\partial v'}{\partial y}\right)_{y=0} y+\cdots$">|; 

$key = q/dhslashdr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6521.png"
 ALT="$ dh/dr$">|; 

$key = q/displaystylehsimeqfrac{gamma,costheta}{rho,g,d}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1218.png"
 ALT="$\displaystyle h \simeq \frac{\gamma \cos\theta}{\rho g d}.$">|; 

$key = q/displaystylephantom{frac{beta_{n,,-n'}^{,m,,-m'}}{c_n^{,m},c_{n'}^{,m'}}=}left.+n',(n'+2),(n'-m'+1),I_{n,,n'+1}^{,m,,m'}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4978.png"
 ALT="$\displaystyle \phantom{\frac{\beta_{n, -n'}^{ m, -m'}}{c_n^{ m} c_{n'}^{ m'}}=} \left.+n' (n'+2) (n'-m'+1) I_{n, n'+1}^{ m, m'} \right],$">|; 

$key = q/displaystylez={rmi},c,cot(zetaslash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2460.png"
 ALT="$\displaystyle z ={\rm i} c \cot(\zeta/2),
$">|; 

$key = q/0.2113<s<0.25;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img803.png"
 ALT="$ 0.2113&lt;s&lt;0.25$">|; 

$key = q/displaystyle{mitPhi}_n^{,m}(theta,phi)=a_n^{,m},skew{5}hat{P}_n^{,m}(costheta),cos(m,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4937.png"
 ALT="$\displaystyle {\mit\Phi}_n^{ m}(\theta,\phi) = a_n^{ m} \skew{5}\hat{P}_n^{ m}(\cos\theta) \cos(m \phi),$">|; 

$key = q/displaystylenablatimes{bfv}={bf0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img543.png"
 ALT="$\displaystyle \nabla\times {\bf v} = {\bf0},
$">|; 

$key = q/displaystylez=frac{omega^{,2}}{2,g},r^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img822.png"
 ALT="$\displaystyle z = \frac{\omega^{ 2}}{2 g} r^{ 2},$">|; 

$key = q/a<b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2443.png"
 ALT="$ a&lt;b$">|; 

$key = q/displaystylev_1,S_1=v_2,S_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1357.png"
 ALT="$\displaystyle v_1 S_1=v_2 S_2.$">|; 

$key = q/displaystyleC_p=frac{2}{gamma,{rmMa}_1^{,2}},frac{p-p_1}{p_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5791.png"
 ALT="$\displaystyle C_p =\frac{2}{\gamma {\rm Ma}_1^{ 2}} \frac{p-p_1}{p_1},$">|; 

$key = q/0leqetaleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3166.png"
 ALT="$ 0\leq \eta\leq 1$">|; 

$key = q/epsilon=(r_+-r_-)slashr_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1349.png"
 ALT="$ \epsilon = (r_+-r_-)/r_+$">|; 

$key = q/displaystyle=k,g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4210.png"
 ALT="$\displaystyle =k g,$">|; 

$key = q/displaystylep(a,theta,t)=p_0-frac{1}{2},rho,V_x^{,2}+rho,V_x^{,2},cos(2,theta)+rho,a,frac{dV_x}{dt},costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1811.png"
 ALT="$\displaystyle p(a,\theta,t) = p_0-\frac{1}{2} \rho V_x^{ 2} + \rho V_x^{ 2} \cos(2 \theta) + \rho a \frac{dV_x}{dt} \cos\theta.$">|; 

$key = q/v'slashVll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5979.png"
 ALT="$ v'/V\ll 1$">|; 

$key = q/displaystyle{rmMa}_p=frac{u_1-u_2}{c_1}=frac{u_1}{c_1}left(1-frac{u_2}{u_1}right)={rmMa}_sleft(1-frac{rho_1}{rho_2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5538.png"
 ALT="$\displaystyle {\rm Ma}_p =\frac{u_1-u_2}{c_1}= \frac{u_1}{c_1}\left(1-\frac{u_2}{u_1}\right) = {\rm Ma}_s\left(1-\frac{\rho_1}{\rho_2}\right),$">|; 

$key = q/lambda=-0.1567;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3203.png"
 ALT="$ \lambda=-0.1567$">|; 

$key = q/zrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2819.png"
 ALT="$ z\rightarrow\infty$">|; 

$key = q/sigma_{rtheta}(a_-,theta)=sigma_{rtheta}(a_+,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3850.png"
 ALT="$ \sigma_{r\theta}(a_-,\theta)=\sigma_{r\theta}(a_+,\theta)$">|; 

$key = q/displaystylep_0,a^{,3}+4,gamma,a^{,2}=p_0,(a_1^{,3}+a_2^{,3})+4,gamma,(a_1^{,2}+a_2^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1351.png"
 ALT="$\displaystyle p_0 a^{ 3}+ 4 \gamma a^{ 2}= p_0 (a_1^{ 3}+a_2^{ 3})+4 \gamma (a_1^{ 2}+a_2^{ 2}),
$">|; 

$key = q/displaystyle({bfa}times{bfb})_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6780.png"
 ALT="$\displaystyle ({\bf a}\times {\bf b})_1$">|; 

$key = q/displaystyled{calE}=T,d{calS}-p,dv.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5577.png"
 ALT="$\displaystyle d{\cal E} = T d{\cal S} -p dv.
$">|; 

$key = q/displaystyleleft(h-a-frac{2}{3},frac{a^{,3}}{R^{,2}}right)frac{W}{s}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1055.png"
 ALT="$\displaystyle \left(h-a-\frac{2}{3} \frac{a^{ 3}}{R^{ 2}}\right)\frac{W}{s}.
$">|; 

$key = q/nablacdot{bfv}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1567.png"
 ALT="$ \nabla\cdot{\bf v}=0$">|; 

$key = q/displaystyleQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3681.png"
 ALT="$\displaystyle Q$">|; 

$key = q/APD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4031.png"
 ALT="$ APD$">|; 

$key = q/displaystyleu(x,infty);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3076.png"
 ALT="$\displaystyle u(x,\infty)$">|; 

$key = q/k=C=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1220.png"
 ALT="$ k=C =1$">|; 

$key = q/displaystylesimeqfrac{gamma-1}{2}left(frac{{rmMa}_1^{,2}}{{rmMa}_1^{,2}-1}right)|theta|,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6159.png"
 ALT="$\displaystyle \simeq \frac{\gamma-1}{2}\left(\frac{{\rm Ma}_1^{ 2}}{{\rm Ma}_1^{ 2}-1}\right)\vert\theta\vert,$">|; 

$key = q/phi=3pislash4-thetaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1200.png"
 ALT="$ \phi=3\pi/4-\theta/2$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter15slashoshock.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img5710.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter15/oshock.eps}}
\end{figure}">|; 

$key = q/skew{3}bar{phi}=-phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2183.png"
 ALT="$ \skew{3}\bar{\phi}=-\phi$">|; 

$key = q/m=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$ m=4$">|; 

$key = q/displaystyleH=E+p,V.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5316.png"
 ALT="$\displaystyle H = E + p V.$">|; 

$key = q/{multline}(beta^{,-1},lambda_r-f_j^{,2}),p_r+sum_{s=1,infty}left[({{rm{i},f_j,beskew{5}tilde{zeta}_j,{mitGamma}_{2,,r}^{,m_j}}{b_2^{,m_j}},{multline};LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="537" HEIGHT="121" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4927.png"
 ALT="\begin{multline}
(\beta^{ -1} \lambda_r-f_j^{ 2}) p_r + \sum_{s=1,\infty}\le...
...ew{5}\tilde{\zeta}_j {\mit\Gamma}_{2, r}^{ m_j}}{b_2^{ m_j}},
\end{multline}">|; 

$key = q/int_P^Q{bfA}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6551.png"
 ALT="$ \int_P^Q {\bf A}\cdot d{\bf r}$">|; 

$key = q/displaystyle{bfq}cdotnablaf=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5935.png"
 ALT="$\displaystyle {\bf q}\cdot\nabla f = 0,$">|; 

$key = q/displaystylezeta=frac{1}{2},(b+c)+frac{1}{2},(b-c),cosh[K',(z-z_0)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2154.png"
 ALT="$\displaystyle \zeta = \frac{1}{2} (b+c)+\frac{1}{2} (b-c) \cosh[K' (z-z_0)].$">|; 

$key = q/y=sinu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1227.png"
 ALT="$ y=\sin u$">|; 

$key = q/displaystyle{rmMa}_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5542.png"
 ALT="$\displaystyle {\rm Ma}_s$">|; 

$key = q/p=p_0+rho,g,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1412.png"
 ALT="$ p=p_0+\rho g h$">|; 

$key = q/displaystyle=frac{1}{2}left(5,mu^{,3}-3,muright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2591.png"
 ALT="$\displaystyle = \frac{1}{2}\left(5 \mu^{ 3}-3 \mu\right),$">|; 

$key = q/displaystyle=frac{M}{5},(a_2^{,2}+a_3^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7280.png"
 ALT="$\displaystyle =\frac{M}{5} (a_2^{ 2}+a_3^{ 2}),$">|; 

$key = q/a_3=a_0,(cosgamma)^{2slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img890.png"
 ALT="$ a_3=a_0 (\cos\gamma)^{2/3}$">|; 

$key = q/displaystyle{bfL}=rho,Vint_{-bslash2}^{bslash2}{mitGamma}(z),dz,,{bfe}_perp.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3452.png"
 ALT="$\displaystyle {\bf L} = \rho V\int_{-b/2}^{b/2}{\mit\Gamma}(z) dz  {\bf e}_\perp.$">|; 

$key = q/displaystyleq=dE+p,dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5297.png"
 ALT="$\displaystyle q = dE+p dV.$">|; 

$key = q/xirightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2704.png"
 ALT="$ \xi\rightarrow\infty$">|; 

$key = q/displaystylerho,frac{D}{Dt}!left({calE}+frac{1}{2},v_i,v_iright)=v_i,F_i+frac{pa}!left(v_i,sigma_{ij}+kappa,frac{partialT}{partialx_j}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="417" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="$\displaystyle \rho \frac{D}{D t}\!\left({\cal E}+ \frac{1}{2} v_i v_i\right)...
... x_j}\!\left(v_i \sigma_{ij} + \kappa \frac{\partial T}{\partial x_j}\right),$">|; 

$key = q/zeta=zeta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2059.png"
 ALT="$ \zeta=\zeta_0$">|; 

$key = q/displaystylerho,frac{partialmbox{boldmathomega}}{partialt}simeq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3903.png"
 ALT="$\displaystyle \rho \frac{\partial \mbox{\boldmath$\omega$}}{\partial t} \simeq 0,$">|; 

$key = q/displaystylefrac{gamma}{2}intleft(frac{partialzeta}{partialx}right)^2,dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1342.png"
 ALT="$\displaystyle \frac{\gamma}{2}\int \left(\frac{\partial \zeta}{\partial x}\right)^2 dx
$">|; 

$key = q/displaystyle(n+1),psi,sqrt{1+k^{,2}}slashr,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1921.png"
 ALT="$\displaystyle (n+1) \psi \sqrt{1+k^{ 2}}/r,
$">|; 

$key = q/displaystylefrac{1}{f_x}=mpfrac{C-f^{,2}slash2,l^{,2}}{[1-(C-f^{,2}slash2,l^{,2})^2]^{1slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1165.png"
 ALT="$\displaystyle \frac{1}{f_x} = \mp \frac{C-f^{ 2}/2 l^{ 2}}{[1-(C-f^{ 2}/2 l^{ 2})^2]^{1/2}}.$">|; 

$key = q/OAP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1572.png"
 ALT="$ OAP$">|; 

$key = q/displaystylerho_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5242.png"
 ALT="$\displaystyle \rho_c$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashimage1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="337" BORDER="0"
 SRC="|."$dir".q|img2018.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/image1.eps}}
\end{figure}">|; 

$key = q/z=-bslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3400.png"
 ALT="$ z=-b/2$">|; 

$key = q/displaystylesigma_{ij}=-p,delta_{ij}+rho,nuleft(frac{partialv_i}{partialx_j}+frac{partialv_j}{partialx_i}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3056.png"
 ALT="$\displaystyle \sigma_{ij} = -p \delta_{ij} + \rho \nu\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right).$">|; 

$key = q/p,slashrho^{,gamma}={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5388.png"
 ALT="$ p /\rho^{ \gamma}={\rm constant}$">|; 

$key = q/displaystylefrac{h}{2,a}>sqrt{3},sin(s,pi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1046.png"
 ALT="$\displaystyle \frac{h}{2 a}&gt; \sqrt{3} \sin(s \pi),
$">|; 

$key = q/(x_1slasha_1)^2+(x_2slasha_2)^2+(x_3slasha_3)^2=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7278.png"
 ALT="$ (x_1/a_1)^2+(x_2/a_2)^2+ (x_3/a_3)^2=1$">|; 

$key = q/displaystyle=tan^{-1}left[({rmMa}_1^{,2}-1)^{1slash2}right]-tan^{-1}left[({rmMa}_1}right]+left(frac{gamma+1}{gamma-1}right)^{1slash2}(z-z_1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="632" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6132.png"
 ALT="$\displaystyle =\tan^{-1}\left[({\rm Ma}_1^{ 2}-1)^{1/2}\right] -\tan^{-1}\left...
...{\tan z}{\tan z_1}\right] +\left(\frac{\gamma+1}{\gamma-1}\right)^{1/2}(z-z_1),$">|; 

$key = q/displaystylenablacdot{bfn}=-frac{2}{R}=-frac{2,costheta}{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1140.png"
 ALT="$\displaystyle \nabla\cdot{\bf n} = -\frac{2}{R} =- \frac{2 \cos\theta}{a},$">|; 

$key = q/{bfa}cdot{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6312.png"
 ALT="$ {\bf a} \cdot {\bf a}$">|; 

$key = q/({bfA}cdotnabla),{bfB}equiv({bfA}cdotnablaB_x,,{bfA}cdotnablaB_y,,{bfA}cdotnablaB_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6671.png"
 ALT="$ ({\bf A}\cdot\nabla) {\bf B}\equiv
({\bf A}\cdot\nabla B_x,  {\bf A}\cdot\nabla B_y, {\bf A}\cdot\nabla B_z)$">|; 

$key = q/B'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4022.png"
 ALT="$ B''$">|; 

$key = q/v_z=v_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6765.png"
 ALT="$ v_z= v_3$">|; 

$key = q/z=+dslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2615.png"
 ALT="$ z=+d/2$">|; 

$key = q/{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6257.png"
 ALT="$ {\bf e}_y$">|; 

$key = q/displaystyleint_0^inftyomega,dy=-U(x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2945.png"
 ALT="$\displaystyle \int_0^\infty \omega dy = -U(x).$">|; 

$key = q/displaystylet_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5675.png"
 ALT="$\displaystyle t_\ast$">|; 

$key = q/(x,,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6025.png"
 ALT="$ (x, y)$">|; 

$key = q/displaystylefrac{r}{a}=frac{1}{|sin(thetaslash2)|}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2818.png"
 ALT="$\displaystyle \frac{r}{a}=\frac{1}{\vert\sin(\theta/2)\vert}.
$">|; 

$key = q/displaystyle=c,(a^{,2}+x^{,2}-y^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1903.png"
 ALT="$\displaystyle = c (a^{ 2}+x^{ 2}-y^{ 2})$">|; 

$key = q/epsilon_{mik};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6902.png"
 ALT="$ \epsilon_{mik}$">|; 

$key = q/displaystylefrac{c}{d};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6080.png"
 ALT="$\displaystyle \frac{c}{d}$">|; 

$key = q/alpha_C(x_L)=alpha_C(x_R)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5829.png"
 ALT="$ \alpha_C(x_L)=\alpha_C(x_R)=0$">|; 

$key = q/displaystylefrac{dF}{dz}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1994.png"
 ALT="$\displaystyle \frac{dF}{dz} = 0$">|; 

$key = q/{rmRe}=U_0,Lslashnu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3002.png"
 ALT="$ {\rm Re} = U_0 L/\nu$">|; 

$key = q/-{mitGamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2032.png"
 ALT="$ -{\mit\Gamma}$">|; 

$key = q/z=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img573.png"
 ALT="$ z=0$">|; 

$key = q/displaystylenabla^{,2}{calR}_n=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4321.png"
 ALT="$\displaystyle \nabla^{ 2} {\cal R}_n = 0$">|; 

$key = q/c<a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2040.png"
 ALT="$ c&lt;a$">|; 

$key = q/z_1leqzleqz_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1240.png"
 ALT="$ z_1\leq z\leq z_2$">|; 

$key = q/displaystyle=frac{1}{sintheta}left[frac{partial}{partialtheta}left(costheta,fract(costheta,frac{partial{mitPsi}_r}{partialtheta}right)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4856.png"
 ALT="$\displaystyle = \frac{1}{\sin\theta}\left[ \frac{\partial}{\partial\theta}\left...
...hi}\left(\cos\theta \frac{\partial{\mit\Psi}_r}{\partial\theta}\right)\right],$">|; 

$key = q/costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5057.png"
 ALT="$ \cos\theta$">|; 

$key = q/displaystyle=frac{2}{sqrt{{rmMa}_infty^{,2}-1}}left(frac{dy_U}{dx}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6058.png"
 ALT="$\displaystyle = \frac{2}{\sqrt{{\rm Ma}_\infty^{ 2}-1}}\left(\frac{dy_U}{dx}\right),$">|; 

$key = q/displaystyle=m,y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6007.png"
 ALT="$\displaystyle =m y.$">|; 

$key = q/displaystyleepsilon_{ijk},{calR}_{il},{calR}_{jm},{calR}_{kn}=epsilon_{lmn}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6840.png"
 ALT="$\displaystyle \epsilon_{ijk} {\cal R}_{il} {\cal R}_{jm} {\cal R}_{kn} = \epsilon_{lmn}.$">|; 

$key = q/{mitSigma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3405.png"
 ALT="$ {\mit\Sigma}$">|; 

$key = q/displaystyle{bfv}=v_z(x,y),{bfe}_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3879.png"
 ALT="$\displaystyle {\bf v} = v_z(x,y) {\bf e}_z.
$">|; 

$key = q/displaystyle=tan^{-1}left(frac{y}{x},frac{p}{[p^{,2}-1]^{1slash2}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2139.png"
 ALT="$\displaystyle = \tan^{-1}\left(\frac{y}{x} \frac{p}{[p^{ 2}-1]^{1/2}}\right),$">|; 

$key = q/displaystylefrac{1}{2},{bfU}(P)={bfV}+{bfv}_{mitSigma}(P)+{bfv}_S(P).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3429.png"
 ALT="$\displaystyle \frac{1}{2} {\bf U}(P) = {\bf V} + {\bf v}_{\mit\Sigma}(P)+ {\bf v}_S(P).$">|; 

$key = q/OB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2279.png"
 ALT="$ OB$">|; 

$key = q/displaystyle{rmFr}_2^{,2}=frac{8,{rmFr}_1^{,2}}{[(1+8,{rmFr}_1^{,2})^{1slash2}-1]^{,3}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1608.png"
 ALT="$\displaystyle {\rm Fr}_2^{ 2} =\frac{8 {\rm Fr}_1^{ 2}}{[(1+8 {\rm Fr}_1^{ 2})^{1/2}-1]^{ 3}}.
$">|; 

$key = q/gamma_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5628.png"
 ALT="$ \gamma_1$">|; 

$key = q/displaystyleint_0^piP_n^{,m}(costheta),P_{n'}^{,m}(costheta),sintheta,dtheta=left(frac{2}{2,n+1}right)frac{(n+m)!}{(n-m)!},delta_{n,n'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="498" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4513.png"
 ALT="$\displaystyle \int_0^\pi P_n^{ m}(\cos\theta) P_{n'}^{ m}(\cos\theta) \sin\...
... d\theta = \left(\frac{2}{2 n+1}\right)\frac{(n+m)!}{(n-m)!} \delta_{n n'}.$">|; 

$key = q/displaystyle=rho_2,u_2^{,2}+p_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5465.png"
 ALT="$\displaystyle = \rho_2 u_2^{ 2}+p_2,$">|; 

$key = q/displaystyleF_z=m',g-frac{1}{2},m',frac{dV_z}{dr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2670.png"
 ALT="$\displaystyle F_z = m' g - \frac{1}{2} m' \frac{dV_z}{dr},$">|; 

$key = q/u(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5666.png"
 ALT="$ u(x,t)$">|; 

$key = q/partialpsislashpartialr=partial^{,2}psislashpartialtheta^{,2}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1675.png"
 ALT="$ \partial\psi/\partial r=\partial^{ 2}\psi/\partial\theta^{ 2}=0$">|; 

$key = q/displaystyleomega_z=frac{partialv_y}{partialx}-frac{partialv_x}{partialy}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1640.png"
 ALT="$\displaystyle \omega_z = \frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}.$">|; 

$key = q/displaystylef(r)=frac{V,(r-a)^2,(2,r+a)}{4,r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3781.png"
 ALT="$\displaystyle f(r) = \frac{V (r-a)^2 (2 r+a)}{4 r},$">|; 

$key = q/displaystyled_{ji}=d_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$\displaystyle d_{ji} = d_{ij}.$">|; 

$key = q/displaystylebar{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img746.png"
 ALT="$\displaystyle \bar{x}$">|; 

$key = q/{rmFr}_1>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1509.png"
 ALT="$ {\rm Fr}_1&gt;1$">|; 

$key = q/displaystyle=frac{1}{r,sintheta},frac{partialA_phi}{partialphi}+frac{A_r}{r}+frac{cottheta,A_theta}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7190.png"
 ALT="$\displaystyle =\frac{1}{r \sin\theta} \frac{\partial A_\phi}{\partial \phi}+ \frac{A_r}{r} + \frac{\cot\theta A_\theta}{r},$">|; 

$key = q/displaystylenabla^{,2}phi=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1588.png"
 ALT="$\displaystyle \nabla^{ 2}\phi = 0.$">|; 

$key = q/displaystylec(x)=c_1+frac{1}{2},(gamma-1),u(x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5556.png"
 ALT="$\displaystyle c(x)= c_1+\frac{1}{2} (\gamma-1) u(x).$">|; 

$key = q/tanh(x)rightarrowx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3952.png"
 ALT="$ \tanh(x)\rightarrow x$">|; 

$key = q/h<H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1059.png"
 ALT="$ h&lt;H$">|; 

$key = q/displaystyleu_0(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3136.png"
 ALT="$\displaystyle u_0(0)$">|; 

$key = q/displaystyle=left(frac{37}{315}-frac{{mitLambda}}{945}-frac{{mitLambda}^{,2}}{9072}right)^2{mitLambda},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3190.png"
 ALT="$\displaystyle = \left(\frac{37}{315}-\frac{{\mit\Lambda}}{945}-\frac{{\mit\Lambda}^{ 2}}{9072}\right)^2{\mit\Lambda},$">|; 

$key = q/{bfb}equiv(b_x,,b_y,,b_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6271.png"
 ALT="$ {\bf b}\equiv (b_x, b_y, b_z)$">|; 

$key = q/displaystylefrac{dh}{dr}=left(frac{dh}{dr}right)_{rmmax},costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6523.png"
 ALT="$\displaystyle \frac{dh}{dr}= \left(\frac{dh}{dr}\right)_{\rm max}  \cos\theta,$">|; 

$key = q/displaystylez=zeta(x,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4134.png"
 ALT="$\displaystyle z = \zeta(x,t),$">|; 

$key = q/displaystylesimeq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3893.png"
 ALT="$\displaystyle \simeq 0,$">|; 

$key = q/displaystyle{bff}=int_S(p_1-p_2),{bfn},dS+gammaoint_C{bft}times{bfn},dr,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1089.png"
 ALT="$\displaystyle {\bf f} =\int_S (p_1-p_2) {\bf n} dS + \gamma \oint_C {\bf t}\times {\bf n} dr,$">|; 

$key = q/displaystyle=frac{1}{2}left[nabla({bfA}cdot{bfB})-nablatimes({bfA}times{bfB})-(nablacdot{bfA}),{bfB}+(nablacdot{bfB}),{bfA}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="460" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7082.png"
 ALT="$\displaystyle = \frac{1}{2}\left[\nabla({\bf A}\cdot{\bf B}) - \nabla\times ({\...
...\bf B}) - (\nabla\cdot{\bf A}) {\bf B} + (\nabla\cdot{\bf B}) {\bf A} \right.$">|; 

$key = q/displaystyle=V,sintheta-frac{3}{4},V,sintheta,frac{a}{r}+{calO}left(frac{a}{r}right)^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3828.png"
 ALT="$\displaystyle = V \sin\theta - \frac{3}{4} V \sin\theta \frac{a}{r}+{\cal O}\left(\frac{a}{r}\right)^2.$">|; 

$key = q/displaystylefrac{partial{bfv}}{partialt}+({bfv}cdotnabla),{bfv}=-nablaleft(frac{p}{rho}+{mitPsi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1562.png"
 ALT="$\displaystyle \frac{\partial{\bf v}}{\partial t} + ({\bf v}\cdot \nabla) {\bf v} = - \nabla\left(\frac{p}{\rho}+{\mit\Psi}\right),$">|; 

$key = q/displaystyle=-B,cosh^2xiint_{xi}^inftyfrac{dxi'}{cosh^3xi'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2765.png"
 ALT="$\displaystyle =-B \cosh^2\xi \int_{\xi}^\infty \frac{d\xi'}{\cosh^3\xi'}$">|; 

$key = q/displaystylec_v=left(frac{partial{calE}}{partialT}right)_v=Tleft(frac{partial{calS}}{partialT}right)_v.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5594.png"
 ALT="$\displaystyle c_v =\left(\frac{\partial {\cal E}}{\partial T}\right)_v = T\left(\frac{\partial{\cal S}}{\partial T}\right)_v.
$">|; 

$key = q/vsimeq-partialpsislashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5108.png"
 ALT="$ v\simeq -\partial\psi/\partial x$">|; 

$key = q/displaystyledelta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6204.png"
 ALT="$\displaystyle \delta$">|; 

$key = q/displaystyle=1-frac{2}{pi}int_{sin^{-1}(lambda)}^{pislash2}sinleft(frac{2,alpha}frac{2,alpha}{pi},betaright)frac{dbeta}{tanbeta}right]^{,-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="580" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2517.png"
 ALT="$\displaystyle =1-\frac{2}{\pi}\int_{\sin^{-1}(\lambda)}^{\pi/2}\sin\left(\frac{...
...left(\frac{2 \alpha}{\pi} \beta\right)\frac{d\beta}{\tan\beta}\right]^{ -1},$">|; 

$key = q/displaystylephantom{=}left.+frac{2,sinalpha,sinbeta,cosbeta,cosgamma}{cos^2alpha}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1012.png"
 ALT="$\displaystyle \phantom{=}\left. +\frac{2 \sin\alpha \sin\beta \cos\beta \cos\gamma}{\cos^2\alpha}\right],$">|; 

$key = q/u=partialpsislashpartialy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3236.png"
 ALT="$ u=\partial\psi/\partial y$">|; 

$key = q/widehat{P}=widehat{P}(X);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2895.png"
 ALT="$ \widehat{P}=\widehat{P}(X)$">|; 

$key = q/displaystyleA_n,mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4426.png"
 ALT="$\displaystyle A_n \mu$">|; 

$key = q/displaystylesum_{i=1,3}frac{x_i^{,2}}{a_i^{,2}+lambda}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7297.png"
 ALT="$\displaystyle \sum_{i=1,3} \frac{x_i^{ 2}}{a_i^{ 2}+\lambda} = 1.
$">|; 

$key = q/displaystyleW=int_{infty}^aleft(-frac{1}{x^{,2}}right),dx=left[frac{1}{x}right]_{infty}^a=frac{1}{a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6481.png"
 ALT="$\displaystyle W = \int_{\infty}^a \left(-\frac{1}{x^{ 2}}\right) dx = \left[\frac{1}{x}\right]_{\infty}^a =\frac{1}{a}.$">|; 

$key = q/displaystylepsi(r,theta)=frac{Q}{2pi},theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1679.png"
 ALT="$\displaystyle \psi(r,\theta) = \frac{Q}{2\pi} \theta.$">|; 

$key = q/displaystyleg,H+frac{1}{2},V^{,2}=g,(H+h)+frac{1}{2},v^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1455.png"
 ALT="$\displaystyle g H + \frac{1}{2} V^{ 2} = g (H+h)+\frac{1}{2} v^{ 2}.$">|; 

$key = q/2.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1038.png"
 ALT="$ 2.5$">|; 

$key = q/displaystylepsi=frac{Q}{4pi},costheta=frac{Q}{4pi},frac{z}{(varpi^{,2}+z^{,2})^{1slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2611.png"
 ALT="$\displaystyle \psi = \frac{Q}{4\pi} \cos\theta = \frac{Q}{4\pi} \frac{z}{(\varpi^{ 2}+z^{ 2})^{1/2}}.$">|; 

$key = q/displaystyle=frac{2,{rmMa}_1^{,2}+1}{{rmMa}_1^{,4}}+left[left(frac{gamma+1}{2}right)^2+frac{gamma-1}{{rmMa}_1^{,2}}right]sin^2theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6091.png"
 ALT="$\displaystyle = \frac{2 {\rm Ma}_1^{ 2}+1}{{\rm Ma}_1^{ 4}}+\left[\left(\frac{\gamma+1}{2}\right)^2+\frac{\gamma-1}{{\rm Ma}_1^{ 2}}\right]\sin^2\theta,$">|; 

$key = q/displaystyle=(a_x^{,2}+a_y^{,2}+a_z^{,2}),(b_x^{,2}+b_y^{,2}+b_z^{,2})-(a_x,b_x+a_y,b_y+a_z,b_z)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="464" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6374.png"
 ALT="$\displaystyle = (a_x^{ 2}+a_y^{ 2}+a_z^{ 2}) (b_x^{ 2}+b_y^{ 2}+b_z^{ 2}) - (a_x  b_x + a_y  b_y + a_z  b_z)^2$">|; 

$key = q/cos(2,beta)equivcos^2beta-sin^2beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5707.png"
 ALT="$ \cos(2 \beta)\equiv \cos^2\beta-\sin^2\beta$">|; 

$key = q/g=9.82,{rmm,s}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4575.png"
 ALT="$ g=9.82 {\rm m s}^{-2}$">|; 

$key = q/-sinepsilon,zeta_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4554.png"
 ALT="$ -\sin\epsilon \zeta_M$">|; 

$key = q/c=1.5,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2142.png"
 ALT="$ c=1.5 l$">|; 

$key = q/overline{p}=pslash(rho,V_0^{,2}+rho,g,L+rho,nu,V_0slashL);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="$ \overline{p} = p/(\rho V_0^{ 2} +\rho g L+ \rho \nu V_0/L)$">|; 

$key = q/U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3326.png"
 ALT="$ U$">|; 

$key = q/displaystylephi(r,theta)=alpha_0-beta_0,theta+sum_{m>0}(alpha_m,r^{,m}-beta_m,r^{-m}),cos[m,(theta-theta_0)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="485" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1759.png"
 ALT="$\displaystyle \phi(r,\theta) = \alpha_0-\beta_0 \theta+\sum_{m&gt;0}(\alpha_m r^{ m}-\beta_m r^{-m}) \cos[m (\theta-\theta_0)].$">|; 

$key = q/displaystylefrac{1}{r^{,2}},frac{partial}{partialr}left(r^{,2},frac{partialphi}{altheta}left(sintheta,frac{partialphi}{partialtheta}right)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2572.png"
 ALT="$\displaystyle \frac{1}{r^{ 2}} \frac{\partial}{\partial r}\left(r^{ 2} \fra...
...{\partial\theta}\left(\sin\theta \frac{\partial\phi}{\partial\theta}\right)=0.$">|; 

$key = q/displaystylepsi(r,theta)=V_0,aleft[-gamma,lnleft(frac{r}{a}right)-left(frac{r}{a}-frac{a}{r}right)sinthetaright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="366" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1768.png"
 ALT="$\displaystyle \psi(r,\theta) = V_0 a\left[-\gamma \ln\left(\frac{r}{a}\right)-\left(\frac{r}{a}-\frac{a}{r}\right)\sin\theta\right],$">|; 

$key = q/P_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3418.png"
 ALT="$ P_-$">|; 

$key = q/v_g=v_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4168.png"
 ALT="$ v_g=v_p$">|; 

$key = q/displaystyle=(p_2'-p_2),c,sinalpha_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5784.png"
 ALT="$\displaystyle =(p_2'-p_2) c \sin\alpha_0,$">|; 

$key = q/displaystyleU=frac{1}{4},rho,g,a^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3980.png"
 ALT="$\displaystyle U = \frac{1}{4} \rho g a^{ 2}.$">|; 

$key = q/displaystylen_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4136.png"
 ALT="$\displaystyle n_x$">|; 

$key = q/rho_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5251.png"
 ALT="$ \rho_c$">|; 

$key = q/3.5,{rmcm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5038.png"
 ALT="$ 3.5 {\rm cm}$">|; 

$key = q/AslashA_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$ A/A_1$">|; 

$key = q/displaystyle{mitPsi}'(theta,phi_pm,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4775.png"
 ALT="$\displaystyle {\mit\Psi}'(\theta,\phi_\pm,t)$">|; 

$key = q/displaystyle{bfv}({bfr})=frac{1}{4pi}intfrac{mbox{boldmathGamma}({bfr}')times({bfr}-{bfr}')}{|{bfr}-{bfr}'|^{,3}},dl',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3391.png"
 ALT="$\displaystyle {\bf v}({\bf r}) = \frac{1}{4\pi}\int\frac{\mbox{\boldmath$\Gamma$}({\bf r}')\times ({\bf r}-{\bf r}')}{\vert{\bf r}-{\bf r}'\vert^{ 3}} dl',$">|; 

$key = q/nabla;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6573.png"
 ALT="$ \nabla$">|; 

$key = q/displaystylefrac{partialv}{partialt}+2,{mitOmega},costheta,u+2,{mitOmega},sintheta,w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4463.png"
 ALT="$\displaystyle \frac{\partial v}{\partial t} + 2 {\mit\Omega} \cos\theta u+2 {\mit\Omega} \sin\theta w$">|; 

$key = q/4.97times10^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4623.png"
 ALT="$ 4.97\times 10^{-1}$">|; 

$key = q/displaystyle{mitPhi}_{rmext}(r,theta,phi)={mitPhi}_{rmtide}(r,theta,phi)+chi_2(r,theta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4389.png"
 ALT="$\displaystyle {\mit\Phi}_{\rm ext}(r,\theta,\phi) = {\mit\Phi}_{\rm tide}(r,\theta,\phi) + \chi_2(r,\theta),$">|; 

$key = q/displaystyley'^{,2}=left(frac{4,k}{sin^2alpha}right)x'=left(frac{8,b,sin^2alpha}{pi,sinalpha+4}right)x'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2330.png"
 ALT="$\displaystyle y'^{ 2} =\left(\frac{4 k}{\sin^2\alpha}\right)x' = \left(\frac{8 b \sin^2\alpha}{\pi \sin\alpha+4}\right)x'.$">|; 

$key = q/displaystyle=-frac{1}{sintheta},frac{partial{mitPhi}_r'}{partialphi}+frac{partial{mitPsi}_r''}{partialtheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4849.png"
 ALT="$\displaystyle = -\frac{1}{\sin\theta} \frac{\partial{\mit\Phi}_r'}{\partial\phi} + \frac{\partial{\mit\Psi}_r''}{\partial\theta}.$">|; 

$key = q/displaystyle2,(a_{ijkl}+a_{iklj}+a_{iljk})+3,(a_{ijlk}+a_{ikjl}+a_{ilkj})=(lambd{ij},delta_{kl}+delta_{ik},delta_{jl}+delta_{il},delta_{jk}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="687" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6970.png"
 ALT="$\displaystyle 2 (a_{ijkl}+a_{iklj}+a_{iljk}) + 3 (a_{ijlk}+a_{ikjl} +a_{ilkj}...
...,(\delta_{ij} \delta_{kl} +\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}).$">|; 

$key = q/displaystyle2,a_{iss}+a_{sis}+a_{ssi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6926.png"
 ALT="$\displaystyle 2 a_{iss} + a_{sis} + a_{ssi}$">|; 

$key = q/displaystyleq_i=A_{ij},frac{partialT}{partialx_j},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="$\displaystyle q_i = A_{ij} \frac{\partial T}{\partial x_j},$">|; 

$key = q/0leqetaleqa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2492.png"
 ALT="$ 0\leq \eta\leq a$">|; 

$key = q/aleftrightarrowb;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img687.png"
 ALT="$ a\leftrightarrow b$">|; 

$key = q/P_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3417.png"
 ALT="$ P_+$">|; 

$key = q/nabla^{,2}psi_1=nabla^{,2}psi_2=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2012.png"
 ALT="$ \nabla^{ 2}\psi_1=
\nabla^{ 2}\psi_2=0$">|; 

$key = q/{figure}{epsfysize{=1.25in{centerline{epsffile{AppendixAslashfigA.09.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="134" BORDER="0"
 SRC="|."$dir".q|img6382.png"
 ALT="\begin{figure}
\epsfysize =1.25in
\centerline{\epsffile{AppendixA/figA.09.eps}}
\end{figure}">|; 

$key = q/displaystyle{bfv}({bfr})=frac{1}{4pi}intfrac{mbox{boldmathomega}({bfr}')times({bfr}-{bfr}')}{|{bfr}-{bfr}'|^{,3}},d^{,3}{bfr}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3389.png"
 ALT="$\displaystyle {\bf v}({\bf r}) = \frac{1}{4\pi}\int\frac{\mbox{\boldmath$\omega...
...)\times ({\bf r}-{\bf r}')}{\vert{\bf r}-{\bf r}'\vert^{ 3}} d^{ 3}{\bf r}'.$">|; 

$key = q/X(theta,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4731.png"
 ALT="$ X(\theta,\phi,t)$">|; 

$key = q/displaystyleoverline{left(frac{dy}{dx}right)^2}=frac{1}{l}int_0^lleft(frac{dy}{dx}right)^2dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6002.png"
 ALT="$\displaystyle \overline{\left(\frac{dy}{dx}\right)^2}=\frac{1}{l}\int_0^l\left(\frac{dy}{dx}\right)^2 dx.$">|; 

$key = q/displaystylelambda=sqrt{{rmMa}_infty^{,2}-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6045.png"
 ALT="$\displaystyle \lambda = \sqrt{{\rm Ma}_\infty^{ 2}-1}.$">|; 

$key = q/{bfE}({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7210.png"
 ALT="$ {\bf E}({\bf r})$">|; 

$key = q/kappa_x^{,2},A_0slashV_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img654.png"
 ALT="$ \kappa_x^{ 2} A_0/V_0&gt;0$">|; 

$key = q/displaystylef=frac{2,rho,k,V^{,2}}{sin^3alpha}int_0^pisin^2omega,domega=frac{pi,rho,k,V^{,2}}{sin^3alpha}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2335.png"
 ALT="$\displaystyle f = \frac{2 \rho k V^{ 2}}{\sin^3\alpha}\int_0^\pi\sin^2\omega d\omega = \frac{\pi \rho k V^{ 2}}{\sin^3\alpha}.$">|; 

$key = q/displaystylem=S_1,v_1,deltat,rho_1=S_2,v_2,deltat,rho_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1369.png"
 ALT="$\displaystyle m= S_1 v_1 \delta t \rho_1=S_2 v_2 \delta t \rho_2.$">|; 

$key = q/displaystyleH-d(x)=F(v),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1525.png"
 ALT="$\displaystyle H- d(x) = F(v),$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter15slasharb.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="302" BORDER="0"
 SRC="|."$dir".q|img5815.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter15/arb.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter04slashbernoulli.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="279" BORDER="0"
 SRC="|."$dir".q|img1360.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter04/bernoulli.eps}}
\end{figure}">|; 

$key = q/f'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3160.png"
 ALT="$ f''$">|; 

$key = q/nuslashU_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3024.png"
 ALT="$ \nu/U_0$">|; 

$key = q/0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$ 0)$">|; 

$key = q/widetilde{nabla{bfA}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7146.png"
 ALT="$ \widetilde{\nabla {\bf A}}$">|; 

$key = q/P(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2870.png"
 ALT="$ P(x)$">|; 

$key = q/displaystylefrac{T}{T_c}=theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5203.png"
 ALT="$\displaystyle \frac{T}{T_c}= \theta,$">|; 

$key = q/displaystyle=left[1-frac{1}{2},(gamma-1)left(frac{v}{c_0}right)^2right]^{,gammaslash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5618.png"
 ALT="$\displaystyle = \left[1-\frac{1}{2} (\gamma-1)\left(\frac{v}{c_0}\right)^2\right]^{ \gamma/(\gamma-1)},$">|; 

$key = q/displaystyledelta_{ij},a_i,b_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6799.png"
 ALT="$\displaystyle \delta_{ij} a_i b_j$">|; 

$key = q/y+dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2378.png"
 ALT="$ y+dy$">|; 

$key = q/displaystylelambda={rmi},sinh{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2254.png"
 ALT="$\displaystyle \lambda = {\rm i} \sinh {\mit\Omega},$">|; 

$key = q/displaystyleint_Vleft[frac{partialrho}{partialt}+nablacdot(rho,{bfv})right]dV=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$\displaystyle \int_V\left[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho {\bf v})\right]dV = 0.$">|; 

$key = q/{bfa}times({bfb}times{bfc});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6419.png"
 ALT="$ {\bf a}\times({\bf b} \times {\bf c})$">|; 

$key = q/displaystyleint_Vleft[frac{partial(rho,v_i)}{partialt}+frac{partial(rho,v_i,v_j)dV=int_Vleft(F_i+frac{partialsigma_{ij}}{partialx_j}right)dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="$\displaystyle \int_V\left[\frac{\partial (\rho v_i)}{\partial t} + \frac{\part...
...right]dV = \int_V\left(F_i + \frac{\partial\sigma_{ij}}{\partial x_j}\right)dV.$">|; 

$key = q/theta=theta_ssimeq108^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3222.png"
 ALT="$ \theta=\theta_s\simeq 108^\circ$">|; 

$key = q/displaystylekappa_T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5598.png"
 ALT="$\displaystyle \kappa_T$">|; 

$key = q/displaystyle=-frac{1}{3},b+2sqrt{-Q},cosphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6106.png"
 ALT="$\displaystyle = -\frac{1}{3} b+2\sqrt{-Q} \cos\phi,$">|; 

$key = q/|{bfr}-{bfr}'|=z,secphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3475.png"
 ALT="$ \vert{\bf r}-{\bf r}'\vert= z \sec\phi$">|; 

$key = q/eta=eta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2474.png"
 ALT="$ \eta=\eta_0$">|; 

$key = q/|{bfr}|=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6721.png"
 ALT="$ \vert{\bf r}\vert = a$">|; 

$key = q/-U_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3043.png"
 ALT="$ -U_0$">|; 

$key = q/alphall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3690.png"
 ALT="$ \alpha\ll 1$">|; 

$key = q/zeta_0+dzeta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2066.png"
 ALT="$ \zeta_0+d\zeta'$">|; 

$key = q/displaystylenablacdot({bfa}times{bfb});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6889.png"
 ALT="$\displaystyle \nabla\cdot ({\bf a}\times {\bf b})$">|; 

$key = q/displaystyle={rmi}^{,n+n'},theta_{n,,n'}^{,m,,m'},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5009.png"
 ALT="$\displaystyle = {\rm i}^{ n+n'}  \theta_{n, n'}^{ m, m'},$">|; 

$key = q/p_2slashp_1-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5712.png"
 ALT="$ p_2/p_1-1$">|; 

$key = q/displaystyle=-frac{1}{rho},frac{partialp}{partialy}+nuleft(frac{partial^{,2}v_y}{partialx^{,2}}+frac{partial^{,2}v_y}{partialy^{,2}}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2864.png"
 ALT="$\displaystyle =-\frac{1}{\rho} \frac{\partial p}{\partial y} + \nu\left(\frac{...
... 2} v_y}{\partial x^{ 2}}+\frac{\partial^{ 2} v_y}{\partial y^{ 2}}\right),$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="193" BORDER="0"
 SRC="|."$dir".q|img6284.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.05.eps}}
\end{figure}">|; 

$key = q/l<c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2503.png"
 ALT="$ l&lt;c$">|; 

$key = q/displaystyleC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7244.png"
 ALT="$\displaystyle C$">|; 

$key = q/u_1+du_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7042.png"
 ALT="$ u_1+d u_1$">|; 

$key = q/{mitTheta}(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1739.png"
 ALT="$ {\mit\Theta}(\theta)$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{Chapter04slashtube.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="616" HEIGHT="199" BORDER="0"
 SRC="|."$dir".q|img1398.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{Chapter04/tube.eps}}
\end{figure}">|; 

$key = q/e=e_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4599.png"
 ALT="$ e=e_E$">|; 

$key = q/displaystyle=-frac{1}{r},frac{partialpsi}{partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1645.png"
 ALT="$\displaystyle = - \frac{1}{r} \frac{\partial \psi}{\partial\theta},$">|; 

$key = q/delta(-x,y)=delta(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img629.png"
 ALT="$ \delta(-x,y)=\delta(x,y)$">|; 

$key = q/{bfq}=u,{bfe}_x+v,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5873.png"
 ALT="$ {\bf q} = u {\bf e}_x+v {\bf e}_y$">|; 

$key = q/tan(2,phi)=1slashtantheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1199.png"
 ALT="$ \tan(2 \phi)=1/\tan\theta$">|; 

$key = q/displaystylephi(r,theta)=-frac{Q}{2pi},lnr.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1677.png"
 ALT="$\displaystyle \phi(r,\theta) = -\frac{Q}{2\pi} \ln r.$">|; 

$key = q/displaystyleleft(a^{,3}-r^{,3}right)frac{sin^2theta}{r}={rmconstant},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2849.png"
 ALT="$\displaystyle \left(a^{ 3}-r^{ 3}\right)\frac{\sin^2\theta}{r} = {\rm constant},
$">|; 

$key = q/displaystyle=F_{rmc}=frac{3}{2}left(frac{Q^{,2}}{g}right)^{1slash3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1529.png"
 ALT="$\displaystyle =F_{\rm c}=\frac{3}{2}\left(\frac{Q^{ 2}}{g}\right)^{1/3},$">|; 

$key = q/displaystylefrac{partialphi}{partialeta}=-frac{1}{varpi},frac{partialpsi}{partialxi}=-frac{1}{2},V,a^{,3},frac{sineta}{(c,{rme}^{,xi})^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2741.png"
 ALT="$\displaystyle \frac{\partial\phi}{\partial\eta} = -\frac{1}{\varpi} \frac{\par...
...artial\xi} = -\frac{1}{2} V a^{ 3} \frac{\sin\eta}{(c {\rm e}^{ \xi})^2},$">|; 

$key = q/displaystylez={rmi},c,cot(zetaslash2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2478.png"
 ALT="$\displaystyle z ={\rm i} c \cot(\zeta/2).
$">|; 

$key = q/({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3933.png"
 ALT="$ ({\bf r},t)$">|; 

$key = q/{mitLambda};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3171.png"
 ALT="$ {\mit\Lambda}$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter03slashfig6.07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="371" BORDER="0"
 SRC="|."$dir".q|img1261.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter03/fig6.07.eps}}
\end{figure}">|; 

$key = q/182.62;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4612.png"
 ALT="$ 182.62$">|; 

$key = q/displaystylefrac{1}{rho_0},frac{partialrho_1}{partialt}+frac{partialu_1}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5355.png"
 ALT="$\displaystyle \frac{1}{\rho_0} \frac{\partial \rho_1}{\partial t} +\frac{\partial u_1}{\partial x}$">|; 

$key = q/displaystyle=frac{tanlambda}{cosdelta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4293.png"
 ALT="$\displaystyle =\frac{\tan\lambda}{\cos\delta},$">|; 

$key = q/displaystylefrac{d}{dr}left(frac{p_0}{rho}+{mitPhi}_0+{mitchi}_0right)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4391.png"
 ALT="$\displaystyle \frac{d}{dr}\left(\frac{p_0}{\rho}+{\mit\Phi}_0+{\mit\chi}_0\right) = 0,$">|; 

$key = q/displaystylefrac{partialw}{partialt}-2,{mitOmega},sintheta,v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4465.png"
 ALT="$\displaystyle \frac{\partial w}{\partial t} - 2 {\mit\Omega} \sin\theta v$">|; 

$key = q/2,theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5718.png"
 ALT="$ 2 \theta$">|; 

$key = q/z=pm0.65,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1260.png"
 ALT="$ z=\pm 0.65 c$">|; 

$key = q/displaystyle=-ointp(a,theta,t),costheta,a,dtheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1813.png"
 ALT="$\displaystyle =-\oint p(a,\theta,t) \cos\theta a d\theta,$">|; 

$key = q/BC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2218.png"
 ALT="$ BC$">|; 

$key = q/displaystylephantom{=}-frac{rho,{mitGamma}}{pi,a}left(V_x,sintheta-V_y,costhetaright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1839.png"
 ALT="$\displaystyle \phantom{=} -\frac{\rho {\mit\Gamma}}{\pi a}\left(V_x \sin\theta-V_y \cos\theta\right).$">|; 

$key = q/displaystyle=left[1+left(frac{gamma-1}{gamma+1}right)left({rmMa}^{,2}-1right)right]^{-gammaslash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5404.png"
 ALT="$\displaystyle =\left[1+\left(\frac{\gamma-1}{\gamma+1}\right)\left({\rm Ma}^{ 2}-1\right)\right]^{-\gamma/(\gamma-1)},$">|; 

$key = q/p_1=p(z_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5160.png"
 ALT="$ p_1=p(z_1)$">|; 

$key = q/displaystylev_y(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3011.png"
 ALT="$\displaystyle v_y(x,y)$">|; 

$key = q/{calR}_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6879.png"
 ALT="$ {\cal R}_{ij}$">|; 

$key = q/displaystyle=-frac{1}{2},V,c^2,{rme}^{,2,xi_0},sin^2eta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2734.png"
 ALT="$\displaystyle = -\frac{1}{2} V c^2 {\rm e}^{ 2 \xi_0} \sin^2\eta,$">|; 

$key = q/zeta=zeta_0,sin(omega,t-k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4170.png"
 ALT="$ \zeta=\zeta_0 \sin(\omega t-k x)$">|; 

$key = q/displaystyleV,frac{dz}{dbar{F}}=-{rmi},lambda+(1-lambda^{,2})^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2208.png"
 ALT="$\displaystyle V \frac{dz}{d\bar{F}} = -{\rm i} \lambda+(1-\lambda^{ 2})^{1/2},$">|; 

$key = q/displaystylelim_{deltatrightarrow0}frac{rho({bfr}+{bfv},deltat,t+deltat)-rho({bf=frac{partialrho}{partialt}+{bfv}cdotnablarho=frac{Drho}{Dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="$\displaystyle \lim_{\delta t\rightarrow 0} \frac{\rho({\bf r}+{\bf v} \delta t...
...\frac{\partial \rho}{\partial t} + {\bf v}\cdot\nabla \rho = \frac{D \rho}{Dt},$">|; 

$key = q/displaystyle=T_1left(frac{cosz}{cosz_1}right)^{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6140.png"
 ALT="$\displaystyle =T_1\left(\frac{\cos z}{\cos z_1}\right)^{2},$">|; 

$key = q/k,l<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4184.png"
 ALT="$ k l&lt;1$">|; 

$key = q/h_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7047.png"
 ALT="$ h_i$">|; 

$key = q/displaystyle=2sum_{i=1,3}frac{x_i,n_i}{a_i^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7243.png"
 ALT="$\displaystyle = 2\sum_{i=1,3}\frac{x_i n_i}{a_i^{ 2}},$">|; 

$key = q/displaystyleg(theta)=frac{cos(alpha+theta)}{f(theta)}-frac{(c^{,2}-delta^{,2}),costheta,sintheta}{h^{,2}(theta)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3361.png"
 ALT="$\displaystyle g(\theta) = \frac{\cos(\alpha+\theta)}{f(\theta)}-\frac{(c^{ 2}-\delta^{ 2}) \cos\theta \sin\theta}{h^{ 2}(\theta)}.$">|; 

$key = q/{mitOmega}=7.29times10^{-5},{rmrad.,s}^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4574.png"
 ALT="$ {\mit\Omega}=7.29\times 10^{-5} {\rm rad. s}^{-1}$">|; 

$key = q/u_1(x_1,x_2,x_3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7026.png"
 ALT="$ u_1(x_1, x_2, x_3)$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter03slashfig6.09.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="371" BORDER="0"
 SRC="|."$dir".q|img1312.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter03/fig6.09.eps}}
\end{figure}">|; 

$key = q/displaystyledelta{mitPhi}={mitPhi}_{rmext}+sum_{n=1,infty}{mitPhi}_n.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4438.png"
 ALT="$\displaystyle \delta{\mit\Phi}= {\mit\Phi}_{\rm ext} + \sum_{n=1,\infty}{\mit\Phi}_n.$">|; 

$key = q/displaystyleF_{21}=int_0^{h_2}(p_2+rho,v_2^{,2}),dz-int_0^{h_1}(p_1+rho,v_1^{,2}),dz.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1468.png"
 ALT="$\displaystyle F_{21} = \int_0^{h_2}(p_2+\rho v_2^{ 2}) dz -\int_0^{h_1}(p_1+\rho v_1^{ 2}) dz.$">|; 

$key = q/displaystyle{rmMa}^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6133.png"
 ALT="$\displaystyle {\rm Ma}^{ 2}$">|; 

$key = q/n,{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6243.png"
 ALT="$ n {\bf a}$">|; 

$key = q/displaystylepsi(r,theta)=frac{1}{4},V,a^2,sin^2thetaleft(frac{mu}{mu+overline{mu)left(frac{r}{a}right)^2left[1-left(frac{r}{a}right)^2right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3857.png"
 ALT="$\displaystyle \psi(r,\theta) = \frac{1}{4} V a^2 \sin^2\theta\left(\frac{\mu...
...mu}}\right)\left(\frac{r}{a}\right)^2\left[1-\left(\frac{r}{a}\right)^2\right],$">|; 

$key = q/displaystylev,dp+p,dv={calR},dT.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5309.png"
 ALT="$\displaystyle v dp + p dv = {\cal R} dT.$">|; 

$key = q/displaystylephantom{=}+rho,a,frac{dV_x}{dr},costheta+rho,a,frac{dV_y}{dr},sintheta-rho,g,a,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1838.png"
 ALT="$\displaystyle \phantom{=}+ \rho a \frac{dV_x}{dr} \cos\theta+\rho a \frac{dV_y}{dr} \sin\theta -\rho g a \sin\theta$">|; 

$key = q/{calH}={calE}+pslashrho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5461.png"
 ALT="$ {\cal H}={\cal E}+p/\rho$">|; 

$key = q/nabla^{,2}psi_3=nabla^{,2}psi_1+nabla^{,2}psi_2=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2016.png"
 ALT="$ \nabla^{ 2}\psi_3=\nabla^{ 2}\psi_1+\nabla^{ 2}\psi_2=0$">|; 

$key = q/displaystyleds=(dx^{,2}+dy^{,2})^{1slash2}=frac{1}{2},h(theta),dtheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3356.png"
 ALT="$\displaystyle ds = (dx^{ 2}+dy^{ 2})^{1/2} = \frac{1}{2} h(\theta) d\theta,$">|; 

$key = q/{bfe}_zequiv(0,,0,,1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6266.png"
 ALT="$ {\bf e}_z\equiv
(0, 0, 1)$">|; 

$key = q/displaystyleomega^{,2}=g,k,tanh(k,d).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3923.png"
 ALT="$\displaystyle \omega^{ 2} = g k \tanh(k d).$">|; 

$key = q/displaystyle=frac{p_2}{p_1}left[frac{2,gamma_1+(gamma_1-1),(p_2slashp_1-1)}{2,gamma_1+(gamma_1+1),(p_2slashp_1-1)}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5648.png"
 ALT="$\displaystyle = \frac{p_2}{p_1}\left[\frac{2 \gamma_1+(\gamma_1-1) (p_2/p_1-1)}{2 \gamma_1+(\gamma_1+1) (p_2/p_1-1)}\right],$">|; 

$key = q/dFslashdzeta=dFslashdz=V,{rme}^{,{rmi},alpha};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3316.png"
 ALT="$ dF/d\zeta= dF/dz=V {\rm e}^{ {\rm i} \alpha}$">|; 

$key = q/{mitLambda}<-12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3176.png"
 ALT="$ {\mit\Lambda}&lt;-12$">|; 

$key = q/alpha=beta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2170.png"
 ALT="$ \alpha=\beta=0$">|; 

$key = q/displaystylefrac{1}{a_i},frac{partialJ}{partiala_i}=ointfrac{2,n_i^{,2}slasha_i^}{left(sum_{i=1,3}n_i^{,2}slasha_i^{,2}right)^2},d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="79" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7254.png"
 ALT="$\displaystyle \frac{1}{a_i} \frac{\partial J}{\partial a_i} = \oint\frac{2 n_...
...,2}/a_i^{ 4}}{\left(\sum_{i=1,3} n_i^{ 2}/a_i^{ 2}\right)^2} d{\mit\Omega}.$">|; 

$key = q/omega={bfe}_zcdotnablatimes{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5073.png"
 ALT="$ \omega = {\bf e}_z\cdot\nabla \times {\bf v}$">|; 

$key = q/displaystylefrac{d}{dx}left({calH}+frac{1}{2},u^{,2}right)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5460.png"
 ALT="$\displaystyle \frac{d}{dx}\left({\cal H} +\frac{1}{2} u^{ 2}\right)=0,$">|; 

$key = q/displaystyle=left[1-frac{1}{2},(gamma-1),frac{V_p}{c_1}right]^{,2slash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5561.png"
 ALT="$\displaystyle = \left[1-\frac{1}{2} (\gamma-1) \frac{V_p}{c_1}\right]^{ 2/(\gamma-1)},$">|; 

$key = q/displaystyle=sum_{i=1,3}nablacdot(A_i,{bfe}_i)=sum_{i=1,3}nablacdot(h_i,A_i,nabla!cdot!left(frac{h}{h_i},A_i,frac{h_i^{,2}}{h},nablau_iright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7070.png"
 ALT="$\displaystyle =\sum_{i=1,3}\nabla\cdot(A_i {\bf e}_i) =\sum_{i=1,3}\nabla\cdot...
...\nabla\!\cdot\!\left(\frac{h}{h_i} A_i \frac{h_i^{ 2}}{h} \nabla u_i\right)$">|; 

$key = q/lambda(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7332.png"
 ALT="$ \lambda(x)$">|; 

$key = q/displaystyleK_{rmfluid}=frac{1}{4},m',V^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2679.png"
 ALT="$\displaystyle K_{\rm fluid} = \frac{1}{4} m' V^{ 2}.$">|; 

$key = q/displaystylecos^2(beta-theta)simeqfrac{{rmMa}_1^{,2}-1}{{rmMa}_1^{,2}}left(1-frac{2,(epsilon-theta)}{sqrt{{rmMa}_1^{,2}-1}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5741.png"
 ALT="$\displaystyle \cos^2(\beta-\theta) \simeq \frac{{\rm Ma}_1^{ 2}-1}{{\rm Ma}_1^{ 2}}\left(1-\frac{2 (\epsilon-\theta)}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right).$">|; 

$key = q/b<a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1048.png"
 ALT="$ b&lt;a$">|; 

$key = q/displaystyleepsilonsimeqleft(frac{gamma+1}{4}right)left(frac{{rmMa}_1^{,2}}{{rmMa}_1^{,2}-1}right)theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5738.png"
 ALT="$\displaystyle \epsilon\simeq \left(\frac{\gamma+1}{4}\right)\left(\frac{{\rm Ma}_1^{ 2}}{{\rm Ma}_1^{ 2}-1}\right)\theta.$">|; 

$key = q/eta=2,zslashb;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3502.png"
 ALT="$ \eta=2 z/b$">|; 

$key = q/displaystyledH;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5331.png"
 ALT="$\displaystyle dH$">|; 

$key = q/displaystylefrac{1}{gamma-1}left(frac{Drho}{Dt}-frac{gamma,p}{rho},frac{Drho}{Dt}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="$\displaystyle \frac{1}{\gamma-1}\left(\frac{D\rho}{Dt} - \frac{\gamma p}{\rho} \frac{D\rho}{Dt}\right)$">|; 

$key = q/displaystyle=frac{(p_L-p_U),c,cosalpha_0}{q_1,c}=(C_{pL}-C_{pU}),cosalpha_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5800.png"
 ALT="$\displaystyle = \frac{(p_L-p_U) c \cos \alpha_0}{q_1 c}= (C_{pL}-C_{pU}) \cos\alpha_0,$">|; 

$key = q/bar{u}=-partialskew{3}bar{phi}slashpartialxi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6041.png"
 ALT="$ \bar{u}= -\partial\skew{3}\bar{\phi}/\partial\xi$">|; 

$key = q/VrightarrowV_z(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2743.png"
 ALT="$ V\rightarrow V_z(t)$">|; 

$key = q/displaystyle{bfA}({bfr})=(3,x,y^{,2},z^{,2}-y^{,2},,-y^{,3},z^{,2}+x^{,2},y,,3,x^{,2}-x^{,2},z).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6747.png"
 ALT="$\displaystyle {\bf A}({\bf r}) = (3 x y^{ 2} z^{ 2}-y^{ 2}, -y^{ 3} z^{ 2}+x^{ 2} y, 3 x^{ 2}-x^{ 2} z).
$">|; 

$key = q/displaystylesigma_{ij}=-p,delta_{ij}+2,muleft(e_{ij}-frac{1}{3},e_{kk},delta_{ij}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="$\displaystyle \sigma_{ij} = - p \delta_{ij} +2 \mu\left(e_{ij} - \frac{1}{3} e_{kk} \delta_{ij}\right),$">|; 

$key = q/displaystylem=frac{omega^{,2},a_0}{g_0}=frac{3}{2},widehat{omega}^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img908.png"
 ALT="$\displaystyle m = \frac{\omega^{ 2} a_0}{g_0} =\frac{3}{2} \widehat{\omega}^{ 2},$">|; 

$key = q/displaystyle=-frac{{mitDelta}P}{l},frac{pi}{8,mu,langlea^{,-4}rangle}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3682.png"
 ALT="$\displaystyle = -\frac{{\mit\Delta} P}{l} \frac{\pi}{8 \mu \langle a^{ -4}\rangle}.$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashpohl.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img3180.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/pohl.eps}}
\end{figure}">|; 

$key = q/lambda(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$ \lambda (\theta )$">|; 

$key = q/displaystyle=-left(frac{partial{calG}}{partialT}right)_p,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5591.png"
 ALT="$\displaystyle = -\left(\frac{\partial {\cal G}}{\partial T}\right)_p,$">|; 

$key = q/displaystyle=frac{pi+2,sinzeta}{pi+2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2233.png"
 ALT="$\displaystyle =\frac{\pi+2 \sin\zeta}{\pi+2},$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter03slashfig6.08.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="371" BORDER="0"
 SRC="|."$dir".q|img1287.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter03/fig6.08.eps}}
\end{figure}">|; 

$key = q/displaystyle{rmFr}_{rmc}=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1450.png"
 ALT="$\displaystyle {\rm Fr}_{\rm c} = 1.$">|; 

$key = q/displaystylephi(r,theta)=-V_0,r,cos(theta-theta_0).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1668.png"
 ALT="$\displaystyle \phi(r,\theta) = -V_0 r \cos(\theta-\theta_0).$">|; 

$key = q/displaystyleW_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img582.png"
 ALT="$\displaystyle W_z$">|; 

$key = q/{mitDelta}p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1109.png"
 ALT="$ {\mit\Delta} p$">|; 

$key = q/displaystyleB_n,mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4428.png"
 ALT="$\displaystyle B_n \mu$">|; 

$key = q/s<0.25;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img770.png"
 ALT="$ s&lt;0.25$">|; 

$key = q/Q,Q'<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2436.png"
 ALT="$ Q Q'&lt;0$">|; 

$key = q/displaystyle=frac{1}{m},frac{partialskew{3}bar{phi}}{partialxi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6017.png"
 ALT="$\displaystyle =\frac{1}{m} \frac{\partial\skew{3}\bar{\phi}}{\partial \xi},$">|; 

$key = q/displaystyle=mu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2587.png"
 ALT="$\displaystyle =\mu,$">|; 

$key = q/A'B'C';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2190.png"
 ALT="$ A'B'C'$">|; 

$key = q/a_1=a_0,(cosgamma)^{-1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img889.png"
 ALT="$ a_1=a_0 (\cos\gamma)^{-1/3}$">|; 

$key = q/p=p(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img564.png"
 ALT="$ p=p(z)$">|; 

$key = q/displaystyle=-V_0left(r+frac{a^{,2}}{r}right)costheta-frac{mitGamma}{2pi},theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1989.png"
 ALT="$\displaystyle =-V_0\left(r+\frac{a^{ 2}}{r}\right)\cos\theta-\frac{\mit\Gamma}{2\pi} \theta,$">|; 

$key = q/{bfb}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6436.png"
 ALT="$ {\bf b}(t)$">|; 

$key = q/displaystylem,frac{dV_z}{dt}=F_z-m,g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2672.png"
 ALT="$\displaystyle m \frac{dV_z}{dt} = F_z -m g,$">|; 

$key = q/q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5296.png"
 ALT="$ q$">|; 

$key = q/displaystyle=2,l,cosh[ln(rslashl)],costheta=(r+l^{,2}slashr),costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2122.png"
 ALT="$\displaystyle = 2 l \cosh[\ln(r/l)] \cos\theta = (r+l^{ 2}/r) \cos\theta,$">|; 

$key = q/w,{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6728.png"
 ALT="$ w {\bf c}$">|; 

$key = q/widetilde{psi}(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2090.png"
 ALT="$ \widetilde{\psi}(x,y)$">|; 

$key = q/displaystylex_1=c_1,t.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5565.png"
 ALT="$\displaystyle x_1= c_1 t.$">|; 

$key = q/mu,m_pslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5187.png"
 ALT="$ \mu m_p/k$">|; 

$key = q/displaystyle=frac{1}{4},(c+delta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3306.png"
 ALT="$\displaystyle = \frac{1}{4} (c+\delta),$">|; 

$key = q/mull1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1025.png"
 ALT="$ \mu\ll1$">|; 

$key = q/F(infty)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3239.png"
 ALT="$ F(\infty)=1$">|; 

$key = q/C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img967.png"
 ALT="$ C$">|; 

$key = q/displaystylenu({rmMa})equivint_1^{rmMa}sqrt{{rmMa}^{,2}-1},frac{dw}{w};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6210.png"
 ALT="$\displaystyle \nu({\rm Ma})\equiv \int_1^{\rm Ma}\sqrt{{\rm Ma}^{ 2}-1} \frac{dw}{w}$">|; 

$key = q/displaystyle(widetilde{nabla{bfA}})_{rphi}=(widetilde{nabla{bfA}})_{phir};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7191.png"
 ALT="$\displaystyle (\widetilde{\nabla {\bf A}})_{r\phi}=(\widetilde{\nabla {\bf A}})_{\phi r}$">|; 

$key = q/displaystylefrac{partial(r,v_r)}{partialr}+frac{partialv_theta}{partialtheta}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1918.png"
 ALT="$\displaystyle \frac{\partial (r v_r)}{\partial r} + \frac{\partial v_\theta}{\partial\theta}=0,
$">|; 

$key = q/X_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4220.png"
 ALT="$ X_1$">|; 

$key = q/displaystyle=frac{gamma-1+(gamma+1),(p_2slashp_1)}{2,gamma},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5490.png"
 ALT="$\displaystyle = \frac{\gamma-1+(\gamma+1) (p_2/p_1)}{2 \gamma},$">|; 

$key = q/displaystyle{bfD};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3440.png"
 ALT="$\displaystyle {\bf D}$">|; 

$key = q/displaystyleomega=0.5805,(G,Mslasha_0^{,3})^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1077.png"
 ALT="$\displaystyle \omega = 0.5805 (G M/a_0^{ 3})^{1/2},
$">|; 

$key = q/overline{rho}llrho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3867.png"
 ALT="$ \overline{\rho}\ll \rho$">|; 

$key = q/displaystyler=a+zeta_b(theta,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4340.png"
 ALT="$\displaystyle r = a + \zeta_b(\theta,\phi),$">|; 

$key = q/p_rrightarrow{rmi}^{,n},p_n^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4992.png"
 ALT="$ p_r\rightarrow {\rm i}^{ n} p_n^{ m}$">|; 

$key = q/displaystylegamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6975.png"
 ALT="$\displaystyle \gamma$">|; 

$key = q/displaystyled{mitPsi}=-G,rholeft(int_0^{r'}r,dr+int_0^{r''}r,drright)d{mitOmega}=-frac{1}{2},G,rho,(r'^{,2}+r''^{,2}),d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7235.png"
 ALT="$\displaystyle d{\mit\Psi} = -G \rho\left(\int_0^{r'} r dr + \int_0^{r''} r d...
...ght)d{\mit\Omega} =- \frac{1}{2} G \rho (r'^{ 2}+r''^{ 2}) d{\mit\Omega}.$">|; 

$key = q/27.555;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4615.png"
 ALT="$ 27.555$">|; 

$key = q/a_{sskl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6947.png"
 ALT="$ a_{sskl}$">|; 

$key = q/{bfe}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7036.png"
 ALT="$ {\bf e}_i$">|; 

$key = q/displaystylefrac{1}{6},a^4,rho,omega^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1068.png"
 ALT="$\displaystyle \frac{1}{6} a^4 \rho \omega^{ 2},
$">|; 

$key = q/displaystyle=nablacdotnablatimes{bfv}equiv0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1545.png"
 ALT="$\displaystyle =\nabla\cdot\nabla\times {\bf v} \equiv 0$">|; 

$key = q/-7.12times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4619.png"
 ALT="$ -7.12\times 10^{-2}$">|; 

$key = q/displaystylesimeq-frac{theta}{sqrt{{rmMa}_1^{,2}-1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5745.png"
 ALT="$\displaystyle \simeq -\frac{\theta}{\sqrt{{\rm Ma}_1^{ 2}-1}},$">|; 

$key = q/M=rho,D,V_0,sin(alpha-theta_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2453.png"
 ALT="$ M= \rho D V_0 \sin(\alpha-\theta_0)$">|; 

$key = q/(muslashrho)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7025.png"
 ALT="$ (\mu/\rho)^{1/2}$">|; 

$key = q/displaystyle=d-frac{2}{pi},frac{skew{3}bar{psi}_1}{V}left(1+sinthetaright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2224.png"
 ALT="$\displaystyle =d-\frac{2}{\pi} \frac{\skew{3}\bar{\psi}_1}{V}\left(1+\sin\theta\right).$">|; 

$key = q/displaystylezeta_{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4578.png"
 ALT="$\displaystyle \zeta_{\mit\Omega}$">|; 

$key = q/zeta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5085.png"
 ALT="$ \zeta_0$">|; 

$key = q/{bfr}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6658.png"
 ALT="$ {\bf r}_0$">|; 

$key = q/displaystylep=-frac{1}{3},sigma_{ii}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="$\displaystyle p= -\frac{1}{3} \sigma_{ii}.$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter02slashfig5.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img918.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter02/fig5.02.eps}}
\end{figure}">|; 

$key = q/U_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2875.png"
 ALT="$ U_0$">|; 

$key = q/%latex2htmlidmarker87767-(sintheta)^{-1},(partialslashpartialtheta),[sintheta,(rf{e129c})]-(sintheta)^{-1},partial(ref{e130c})slashpartialphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4670.png"
 ALT="% latex2html id marker 87767
$ -(\sin\theta)^{-1} (\partial/\partial\theta) [\sin\theta (\ref{e129c})]
-(\sin\theta)^{-1} \partial(\ref{e130c})/\partial\phi$">|; 

$key = q/displaystylefrac{partialpsi(x,infty)}{partialy}=-U(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2908.png"
 ALT="$\displaystyle \frac{\partial\psi(x,\infty)}{\partial y}=-U(x),$">|; 

$key = q/yslashdeltarightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2869.png"
 ALT="$ y/\delta\rightarrow \infty$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashforce.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="339" BORDER="0"
 SRC="|."$dir".q|img2376.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/force.eps}}
\end{figure}">|; 

$key = q/displaystylez;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2724.png"
 ALT="$\displaystyle z$">|; 

$key = q/displaystylefrac{beta_{n,,-n'}^{,m,,-m'}}{c_n^{,m},c_{n'}^{,m'}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4976.png"
 ALT="$\displaystyle \frac{\beta_{n, -n'}^{ m, -m'}}{c_n^{ m} c_{n'}^{ m'}}$">|; 

$key = q/displaystylef(eta)=left{{array}{lcl}a+b,eta+c,eta^{,2}+d,eta^{,3}+e,eta^{,4}&mbox{hspace{1cm}}&0leqetaleq1[0.5ex]1&&eta>1{array}right.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="481" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3155.png"
 ALT="$\displaystyle f(\eta)=\left\{\begin{array}{lcl} a+b \eta+c \eta^{ 2}+d \eta...
... 4}&amp;\mbox{\hspace{1cm}}&amp;0\leq\eta\leq 1\ [0.5ex] 1&amp;&amp;\eta&gt;1\end{array}\right.,$">|; 

$key = q/displaystylesimeqfrac{sqrt{6}}{5},widehat{omega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img906.png"
 ALT="$\displaystyle \simeq \frac{\sqrt{6}}{5} \widehat{\omega},$">|; 

$key = q/sigma_{22}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="$ \sigma_{22}'$">|; 

$key = q/displaystyle{bfg}_{,rmcentripetal}=omega_1^{,2},{bfr}''=-nabla{mitPhi}_{rmcentripetal}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4232.png"
 ALT="$\displaystyle {\bf g}_{ \rm centripetal} =\omega_1^{ 2} {\bf r}''=-\nabla{\mit\Phi}_{\rm centripetal}.$">|; 

$key = q/v_theta=a,r^{,n},exp[-k,(n+1),theta];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1920.png"
 ALT="$ v_\theta=a r^{ n} \exp[-k (n+1) \theta]$">|; 

$key = q/oint_S{bfA}cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6575.png"
 ALT="$ \oint_S {\bf A}\cdot
d{\bf S}$">|; 

$key = q/deltay(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7303.png"
 ALT="$ \delta y(x)$">|; 

$key = q/deltaslashc=0.125;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3372.png"
 ALT="$ \delta/c=0.125$">|; 

$key = q/partialzetaslashpartialt=-(partialphislashpartialz)_{z=0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4144.png"
 ALT="$ \partial\zeta/\partial t = -(\partial\phi/\partial z)_{z=0}$">|; 

$key = q/dV=r^{,2},dr,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7231.png"
 ALT="$ dV = r^{ 2} dr d{\mit\Omega}$">|; 

$key = q/zeta_{mitOmega}+zeta_M+zeta_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4643.png"
 ALT="$ \zeta_{\mit\Omega}+\zeta_M+\zeta_E$">|; 

$key = q/displaystyle{bfA}=frac{x,{bfe}_x+y,{bfe}_y}{sqrt{x^{,2}+y^{,2}}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6742.png"
 ALT="$\displaystyle {\bf A} = \frac{x {\bf e}_x + y {\bf e}_y}{\sqrt{x^{ 2} + y^{ 2}}}
$">|; 

$key = q/displaystyle=mu,nablaphitimesnabla(omega_phi,r,sintheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3785.png"
 ALT="$\displaystyle = \mu \nabla\phi\times \nabla(\omega_\phi r \sin\theta).$">|; 

$key = q/pi^{,4},rho,V_0^{,2}slash32;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2500.png"
 ALT="$ \pi^{ 4} \rho V_0^{ 2}/32$">|; 

$key = q/a>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5673.png"
 ALT="$ a&gt;0$">|; 

$key = q/displaystylenablacdot{bfv}=nabla^{,2}phi=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1797.png"
 ALT="$\displaystyle \nabla\cdot{\bf v}= \nabla^{ 2}\phi = 0.$">|; 

$key = q/r=r_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3625.png"
 ALT="$ r=r_2$">|; 

$key = q/displaystyle{bfcurl},{bfA}=nablatimes{bfA}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6646.png"
 ALT="$\displaystyle {\bf curl} {\bf A} = \nabla\times {\bf A}.$">|; 

$key = q/{bfA}({bfr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6473.png"
 ALT="$ {\bf A}({\bf r})$">|; 

$key = q/{mitGamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1702.png"
 ALT="$ {\mit\Gamma}$">|; 

$key = q/displaystyle=intx_2,x_3,dV=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img851.png"
 ALT="$\displaystyle = \int x_2 x_3 dV= 0,$">|; 

$key = q/CC'=R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img970.png"
 ALT="$ CC'=R$">|; 

$key = q/displaystyleleft|frac{dF}{dz}right|^{,2}=v_x^{,2}+v_y^{,2}=v^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1993.png"
 ALT="$\displaystyle \left\vert\frac{dF}{dz}\right\vert^{ 2} = v_x^{ 2} + v_y^{ 2} =v^{ 2},$">|; 

$key = q/phi_+=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4933.png"
 ALT="$ \phi_+=\pi$">|; 

$key = q/displaystyleF(gamma,alpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img945.png"
 ALT="$\displaystyle F(\gamma,\alpha)$">|; 

$key = q/displaystylenu({rmMa});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5771.png"
 ALT="$\displaystyle \nu({\rm Ma})$">|; 

$key = q/{mitPi}(A)=(A_{ii},A_{jj}-A_{ij},A_{ji})slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7008.png"
 ALT="$ {\mit\Pi}(A)= (A_{ii} A_{jj}-A_{ij} A_{ji})/2$">|; 

$key = q/displaystylefrac{dF}{dz}=-frac{Q}{pi},frac{z}{z^{,2}+a^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2027.png"
 ALT="$\displaystyle \frac{dF}{dz} = -\frac{Q}{\pi} \frac{z}{z^{ 2}+a^{ 2}}.$">|; 

$key = q/B=-(3slash4),V,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3780.png"
 ALT="$ B=-(3/4) V a$">|; 

$key = q/displaystylea_{iss}=a_{sis}=a_{ssi}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6932.png"
 ALT="$\displaystyle a_{iss}=a_{sis}= a_{ssi} = 0.$">|; 

$key = q/{mitPhi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4794.png"
 ALT="$ {\mit\Phi}$">|; 

$key = q/displaystylenabla^{,2}psi=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1642.png"
 ALT="$\displaystyle \nabla^{ 2}\psi = 0.$">|; 

$key = q/displaystylefrac{d^{,2}v_x}{dy^{,2}}=-frac{G}{mu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3580.png"
 ALT="$\displaystyle \frac{d^{ 2} v_x}{dy^{ 2}} = - \frac{G}{\mu}.$">|; 

$key = q/displaystyle{rmdet}(C)={rmdet}(A),{rmdet}(B).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7001.png"
 ALT="$\displaystyle {\rm det}(C) = {\rm det}(A) {\rm det}(B).
$">|; 

$key = q/displaystyleu(x,y)simequ_0(y)+A,frac{du_0}{dy}sqrt{x_0-x}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3133.png"
 ALT="$\displaystyle u(x,y) \simeq u_0(y) + A \frac{du_0}{dy}\sqrt{x_0-x}.$">|; 

$key = q/displaystylez=zeta+frac{l^{,2}}{zeta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2502.png"
 ALT="$\displaystyle z = \zeta+ \frac{l^{ 2}}{\zeta}.
$">|; 

$key = q/displaystyleskew{5}bar{zeta}'=(1+k_2-h_2),skew{5}bar{zeta}_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5056.png"
 ALT="$\displaystyle \skew{5}\bar{\zeta}'=(1+k_2-h_2) \skew{5}\bar{\zeta}_2.
$">|; 

$key = q/g_0=G,M^{,2}slasha_0^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img909.png"
 ALT="$ g_0=G M^{ 2}/a_0^{ 2}$">|; 

$key = q/u_1slashc_1=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5413.png"
 ALT="$ u_1/c_1=1$">|; 

$key = q/displaystylef(x)=frac{U,epsilon}{lambda},sin(alpha,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5993.png"
 ALT="$\displaystyle f(x) = \frac{U \epsilon}{\lambda} \sin(\alpha x),$">|; 

$key = q/displaystylenablacdotleft(frac{h_i^{,2}}{h},nablau_iright)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7057.png"
 ALT="$\displaystyle \nabla\cdot \left(\frac{h_i^{ 2}}{h} \nabla u_i\right) = 0.$">|; 

$key = q/displaystyle=-int_Sleft(frac{partialpsi}{partialx}+frac{partialphi}{partialy}right)dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2360.png"
 ALT="$\displaystyle =-\int_S\left(\frac{\partial\psi}{\partial x} + \frac{\partial\phi}{\partial y}\right)dS,$">|; 

$key = q/displaystyle=F_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1843.png"
 ALT="$\displaystyle = F_x,$">|; 

$key = q/displaystyle=x^{,2}-y^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1954.png"
 ALT="$\displaystyle = x^{ 2} - y^{ 2},$">|; 

$key = q/displaystylesimeqmu_2-epsilon,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6120.png"
 ALT="$\displaystyle \simeq \mu_2-\epsilon,$">|; 

$key = q/pslashp_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$ p/p_1$">|; 

$key = q/v_x=v_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6763.png"
 ALT="$ v_x= v_1$">|; 

$key = q/C=b>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1254.png"
 ALT="$ C=b&gt;0$">|; 

$key = q/P=u+2,cslash(gamma-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5660.png"
 ALT="$ P=u+2 c/(\gamma-1)$">|; 

$key = q/displaystylep({bfr},t)=p_0-rho,g,z+p_1({bfr},t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3897.png"
 ALT="$\displaystyle p({\bf r},t) = p_0-\rho g z+p_1({\bf r},t),$">|; 

$key = q/C_{ij}=A_{ik},B_{kj};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7000.png"
 ALT="$ C_{ij} = A_{ik} B_{kj}$">|; 

$key = q/displaystyleH^{,2},V^{,2}=(H+h-d)^{,2},(V^{,2}-2,g,h),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1457.png"
 ALT="$\displaystyle H^{ 2} V^{ 2} = (H+h-d)^{ 2} (V^{ 2}-2 g h),$">|; 

$key = q/B_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4413.png"
 ALT="$ B_n$">|; 

$key = q/displaystylep(x,z,t)=p_0+rho,g,h(x,z,t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3949.png"
 ALT="$\displaystyle p(x,z,t) = p_0 + \rho g h(x,z,t).$">|; 

$key = q/5,V^{,2}slash(8,g);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2848.png"
 ALT="$ 5 V^{ 2}/(8 g)$">|; 

$key = q/0.99;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$ 0.99$">|; 

$key = q/displaystyle=frac{partial^{,2}v_x}{partialy^{,2}}left[1+{calO}left(frac{d}{l}right)^2right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3657.png"
 ALT="$\displaystyle = \frac{\partial^{ 2} v_x}{\partial y^{ 2}}\left[1+{\cal O}\left(\frac{d}{l}\right)^2\right].$">|; 

$key = q/displaystyle(1-{rmMa}_infty^{,2}),frac{partialu'}{partialx}+frac{partialv'}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5903.png"
 ALT="$\displaystyle (1-{\rm Ma}_\infty^{ 2}) \frac{\partial u'}{\partial x} + \frac{\partial v'}{\partial y}$">|; 

$key = q/displaystyleV=frac{a^{,2},g}{3,nu},left(1-frac{overline{rho}}{rho}right)left[frac{mu+overline{mu}}{mu+(3slash2),overline{mu}}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3864.png"
 ALT="$\displaystyle V = \frac{a^{ 2} g}{3 \nu} \left(1-\frac{\overline{\rho}}{\rho}\right)\left[\frac{\mu+\overline{\mu}}{\mu+(3/2) \overline{\mu}}\right],$">|; 

$key = q/{bfd}=lambda,{bfa}+mu,{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6715.png"
 ALT="$ {\bf d} = \lambda {\bf a} + \mu {\bf b}$">|; 

$key = q/displaystylephi(xi_0,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2747.png"
 ALT="$\displaystyle \phi(\xi_0,\eta)$">|; 

$key = q/displaystylefrac{rho_3}{rho_4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5649.png"
 ALT="$\displaystyle \frac{\rho_3}{\rho_4}$">|; 

$key = q/{rmMa}_1rightarrow{rmMa}_1,sinbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5686.png"
 ALT="$ {\rm Ma}_1\rightarrow
{\rm Ma}_1 \sin\beta$">|; 

$key = q/displaystyleM=rhoint_{-infty}^{infty}u^{,2},dy,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3247.png"
 ALT="$\displaystyle M = \rho\int_{-\infty}^{\infty}u^{ 2} dy,
$">|; 

$key = q/displaystyleA,kleft(1-{rme}^{-2,k,d}right)cos(omega,t-k,x)=A,frac{omega^{,2}}{g}left(1+{rme}^{-2,k,d}right)cos(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="528" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3922.png"
 ALT="$\displaystyle A k\left(1-{\rm e}^{-2 k d}\right)\cos(\omega t-k x) = A \frac{\omega^{ 2}}{g}\left(1+{\rm e}^{-2 k d}\right)\cos(\omega t-k x).$">|; 

$key = q/displaystylefrac{D{calE}}{Dt}=-frac{p}{rho},frac{partialv_i}{partialx_i}+frac{1}artialx_j}!left(kappa,frac{partialT}{partialx_j}right)right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="$\displaystyle \frac{D{\cal E}}{Dt} = - \frac{p}{\rho} \frac{\partial v_i}{\par...
...l}{\partial x_j}\!\left( \kappa \frac{\partial T}{\partial x_j}\right)\right].$">|; 

$key = q/k,dgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3935.png"
 ALT="$ k d\gg 1$">|; 

$key = q/360^circslashomega_M=27.322;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4588.png"
 ALT="$ 360^\circ/\omega_M=27.322$">|; 

$key = q/displaystylefrac{1}{gamma-1}left[frac{Doverline{p}}{Doverline{t}}right.!!;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="$\displaystyle \frac{1}{\gamma-1}\left[\frac{D\overline{p}}{D\overline{t}}\right.\!\!$">|; 

$key = q/sintheta_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2424.png"
 ALT="$ \sin\theta_0)$">|; 

$key = q/F=sqrt{1+y'^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7326.png"
 ALT="$ F = \sqrt{1+y'^{ 2}}$">|; 

$key = q/u_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7016.png"
 ALT="$ u_i$">|; 

$key = q/displaystylesigma_{rvarphi}=sigma_{varphir};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img498.png"
 ALT="$\displaystyle \sigma_{r\varphi}=\sigma_{\varphi r}$">|; 

$key = q/3^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6769.png"
 ALT="$ 3^n$">|; 

$key = q/displaystylemu=frac{M'}{M},frac{a_0^{,3}}{R^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1013.png"
 ALT="$\displaystyle \mu = \frac{M'}{M}  \frac{a_0^{ 3}}{R^{ 3}},$">|; 

$key = q/displaystyle{calE}=gamma,S-p_0,V,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1244.png"
 ALT="$\displaystyle {\cal E} = \gamma S - p_0 V,$">|; 

$key = q/displaystylesigma_{rtheta}(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3837.png"
 ALT="$\displaystyle \sigma_{r\theta}(r,\theta)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter07slashdisk.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img2786.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter07/disk.eps}}
\end{figure}">|; 

$key = q/displaystylephantom{=}+(-a_x,sintheta+a_y,costheta),(-b_x,sintheta+b_y,costheta)+a_z,b_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6304.png"
 ALT="$\displaystyle \phantom{=}+(-a_x \sin\theta + a_y \cos\theta) (-b_x \sin \theta + b_y \cos\theta) +a_z  b_z$">|; 

$key = q/displaystyle{bfa}times({bfb}times{bfc})equiv({bfa}cdot{bfc}),{bfb}-({bfa}cdot{bfb}),{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6421.png"
 ALT="$\displaystyle {\bf a}\times({\bf b} \times {\bf c}) \equiv ({\bf a}\cdot {\bf c})  {\bf b} - ({\bf a}\cdot {\bf b}) {\bf c}$">|; 

$key = q/displaystyle=-frac{partialP}{partialtheta}-frac{1}{sintheta},frac{partialQ}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4739.png"
 ALT="$\displaystyle = -\frac{\partial P}{\partial\theta} - \frac{1}{\sin\theta} \frac{\partial Q}{\partial\phi},$">|; 

$key = q/displaystylem=4pi,rho_c,a^{,3},y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5218.png"
 ALT="$\displaystyle m = 4\pi \rho_c a^{ 3} y,$">|; 

$key = q/{bfa}times{bfb}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6379.png"
 ALT="$ {\bf a}\times{\bf b} = {\bf0}$">|; 

$key = q/(partialxslashpartialu)_{u=u_0}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3114.png"
 ALT="$ (\partial x/\partial u)_{u=u_0}=0$">|; 

$key = q/l=sqrt{d,g}slash|f|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5102.png"
 ALT="$ l=\sqrt{d g}/\vert f\vert$">|; 

$key = q/displaystyleP_3(mu);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2590.png"
 ALT="$\displaystyle P_3(\mu)$">|; 

$key = q/{rmFr}>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1444.png"
 ALT="$ {\rm Fr}&gt;1$">|; 

$key = q/displaystyleint_{-h}^h,u(y),dy=2,int_{-l}^lv(x),dxsimeqQ,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3052.png"
 ALT="$\displaystyle \int_{-h}^h u(y) dy=2 \int_{-l}^lv(x) dx \simeq Q,$">|; 

$key = q/R=8.315,{rmJ,mol^{-1},K^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5129.png"
 ALT="$ R= 8.315 {\rm J mol^{-1} K^{-1}}$">|; 

$key = q/displaystyle=-frac{1}{a,sintheta},frac{partial}{partialphi}left(frac{p}{skew{3}barrho}+{mitPhi}+chiright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4464.png"
 ALT="$\displaystyle = - \frac{1}{a \sin\theta} \frac{\partial}{\partial\phi}\left(\frac{p}{\skew{3}\bar\rho}+ {\mit\Phi} +\chi\right),$">|; 

$key = q/displaystylesigma_{rr}(a_+,theta)-sigma_{rr}(a_-,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3859.png"
 ALT="$\displaystyle \sigma_{rr}(a_+,\theta)-\sigma_{rr}(a_-,\theta)$">|; 

$key = q/m<m_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2974.png"
 ALT="$ m&lt;m_\ast$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter06slashsource.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img2026.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter06/source.eps}}
\end{figure}">|; 

$key = q/{bfA}=(phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2352.png"
 ALT="$ {\bf A}=(\phi$">|; 

$key = q/zeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$ \zeta $">|; 

$key = q/displaystylefrac{1}{a,sintheta},frac{partial}{partialtheta},(sintheta,u)+frac{1}a},frac{partialv}{partialphi}+frac{partialw}{partialz}simeq0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4460.png"
 ALT="$\displaystyle \frac{1}{a \sin\theta} \frac{\partial}{\partial\theta} (\sin\t...
...heta} \frac{\partial v}{\partial\phi}+ \frac{\partial w}{\partial z} \simeq 0,$">|; 

$key = q/+y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1786.png"
 ALT="$ +y$">|; 

$key = q/y_U(x_L)=y_C(x_L)=y_L(x_L);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5821.png"
 ALT="$ y_U(x_L)=y_C(x_L)=y_L(x_L)$">|; 

$key = q/displaystylerightarrowU(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2866.png"
 ALT="$\displaystyle \rightarrow U(x),$">|; 

$key = q/displaystyleM;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2392.png"
 ALT="$\displaystyle M$">|; 

$key = q/displaystyle=nabla{mitOmega}_varphitimesnablavarphi+omega_varphi,{bfe}_varphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2564.png"
 ALT="$\displaystyle = \nabla{\mit\Omega}_\varphi\times \nabla\varphi + \omega_\varphi {\bf e}_\varphi,$">|; 

$key = q/10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$ 10$">|; 

$key = q/displaystylev_r(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1694.png"
 ALT="$\displaystyle v_r(r,\theta)$">|; 

$key = q/displaystyle=-frac{{mitOmega}^{,2},a^{,2}}{3}left(frac{r}{a}right)^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4380.png"
 ALT="$\displaystyle =- \frac{{\mit\Omega}^{ 2} a^{ 2}}{3}\left(\frac{r}{a}\right)^2,$">|; 

$key = q/displaystyle+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6804.png"
 ALT="$\displaystyle +1$">|; 

$key = q/n+n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5014.png"
 ALT="$ n+n'$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashwedge1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1876.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/wedge1.eps}}
\end{figure}">|; 

$key = q/displaystyle=-frac{g}{a,sintheta},frac{partial}{partialphi},(zeta-skew{5}bar{zeta}'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5055.png"
 ALT="$\displaystyle = - \frac{g}{a \sin\theta} \frac{\partial}{\partial\phi} (\zeta-\skew{5}\bar{\zeta}'),$">|; 

$key = q/(1-k^{,2})^{1slash2}|leqrslash|alpha|leqalpha|,(1-k^{,2})^{1slash4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1330.png"
 ALT="$ (1-k^{ 2})^{1/2}\vert\leq r/\vert\alpha\vert\leq \alpha\vert (1-k^{ 2})^{1/4}$">|; 

$key = q/dz'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2078.png"
 ALT="$ dz''$">|; 

$key = q/displaystylev=(2,g,h)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1410.png"
 ALT="$\displaystyle v = (2 g h)^{1/2}.$">|; 

$key = q/displaystyledV=r,dr,dtheta,dz.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7127.png"
 ALT="$\displaystyle dV = r dr d\theta dz.$">|; 

$key = q/displaystyle=-kappa,nablaT,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img521.png"
 ALT="$\displaystyle =-\kappa \nabla T,$">|; 

$key = q/dslashall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4353.png"
 ALT="$ d/a\ll 1$">|; 

$key = q/p=p_0+(1slash2),rho,V^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2822.png"
 ALT="$ p=p_0+(1/2) \rho V^{ 2}$">|; 

$key = q/displaystyleskew{5}hat{zeta}_j(vartheta)=left[frac{1+k_2-h_2}{1-(1+k_2'-h_2'),alpha_2}right]skew{5}tilde{zeta}_j,P_2^{,m_j}(sinvartheta).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4719.png"
 ALT="$\displaystyle \skew{5}\hat{\zeta}_j(\vartheta)= \left[\frac{1+k_2-h_2}{1-(1+k_2'-h_2') \alpha_2}\right]\skew{5}\tilde{\zeta}_j P_2^{ m_j}(\sin\vartheta).$">|; 

$key = q/displaystyle{bfv}=v_x(y),{bfe}_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3590.png"
 ALT="$\displaystyle {\bf v} = v_x(y) {\bf e}_x,$">|; 

$key = q/displaystyleF(v)=frac{Q}{v}+frac{v^{,2}}{2,g}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1429.png"
 ALT="$\displaystyle F(v) = \frac{Q}{v} + \frac{v^{ 2}}{2 g}.$">|; 

$key = q/partialvslashpartialy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3107.png"
 ALT="$ \partial v/\partial y$">|; 

$key = q/displaystyle-theta=nu({rmMa})-nu({rmMa}_1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5777.png"
 ALT="$\displaystyle -\theta=\nu({\rm Ma})-\nu({\rm Ma}_1),$">|; 

$key = q/displaystyleu,frac{partialu}{partialx}+v,frac{partialu}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5875.png"
 ALT="$\displaystyle u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}$">|; 

$key = q/displaystylecdotnablaphi=-phi_Q+phi_P=int_{P}^Q{bfv}cdotd{bfr}simeq{bfv}cdot;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1582.png"
 ALT="$\displaystyle \cdot\nabla\phi = -\phi_Q+\phi_P = \int_{P}^Q {\bf v}\cdot d{\bf r} \simeq {\bf v}\cdot$">|; 

$key = q/displaystyle=a_{sskl},delta_{ij}+a_{isks},delta_{jl}+a_{issl},delta_{jk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6941.png"
 ALT="$\displaystyle = a_{sskl} \delta_{ij} +a_{isks} \delta_{jl}+ a_{issl} \delta_{jk},$">|; 

$key = q/displaystylexi_z(x,z,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3939.png"
 ALT="$\displaystyle \xi_z(x,z,t)$">|; 

$key = q/rho_2=(p_2slashp_1)^{1slashgamma},rho_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5163.png"
 ALT="$ \rho_2= (p_2/p_1)^{1/\gamma} \rho_1$">|; 

$key = q/displaystyle=-xsintheta+ycostheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6278.png"
 ALT="$\displaystyle = -x\sin\theta + y\cos\theta,$">|; 

$key = q/displaystyled_{ij}=A_{ijkl},frac{partialv_k}{partialx_l}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="$\displaystyle d_{ij} = A_{ijkl} \frac{\partial v_k}{\partial x_l}.$">|; 

$key = q/-aslash2<y<aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2100.png"
 ALT="$ -a/2&lt;y&lt;a/2$">|; 

$key = q/{bfF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1093.png"
 ALT="$ {\bf F}$">|; 

$key = q/alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$ \alpha $">|; 

$key = q/(1+m),(1-alphaslash2)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1871.png"
 ALT="$ (1+m) (1-\alpha/2) = 1$">|; 

$key = q/displaystyle={mitPhi}_r',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4838.png"
 ALT="$\displaystyle ={\mit\Phi}_r',$">|; 

$key = q/{bfa}cdot{bfb}times{bfc}neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6418.png"
 ALT="$ {\bf a}\cdot{\bf b} \times{\bf c} \neq 0$">|; 

$key = q/{calA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6082.png"
 ALT="$ {\cal A}$">|; 

$key = q/displaystyle=-frac{U_0,x^{,m-1}}{2},[2,m,f'+(m-1),eta,f''],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2933.png"
 ALT="$\displaystyle = -\frac{U_0 x^{ m-1}}{2} [2 m f'+(m-1) \eta f''],$">|; 

$key = q/z>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img557.png"
 ALT="$ z&gt;0$">|; 

$key = q/rho(x,t)=rho_0+rho_1(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5353.png"
 ALT="$ \rho(x,t)=\rho_0+\rho_1(x,t)$">|; 

$key = q/v_p,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4014.png"
 ALT="$ v_p t$">|; 

$key = q/Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2382.png"
 ALT="$ Y$">|; 

$key = q/hggdelta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3050.png"
 ALT="$ h\gg \delta$">|; 

$key = q/displaystyle({bfa}times{bfb})_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6782.png"
 ALT="$\displaystyle ({\bf a}\times {\bf b})_2$">|; 

$key = q/OBP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1573.png"
 ALT="$ OBP$">|; 

$key = q/A_{ij}=lambda_i,delta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7015.png"
 ALT="$ A_{ij}=\lambda_i \delta_{ij}$">|; 

$key = q/displaystyleT,d{calS}=d{calE}-frac{p}{rho^{,2}},drho,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img522.png"
 ALT="$\displaystyle T d{\cal S} = d{\cal E}- \frac{p}{\rho^{ 2}} d\rho,
$">|; 

$key = q/U_0,delta(x)slashnu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3068.png"
 ALT="$ U_0 \delta(x)/\nu$">|; 

$key = q/displaystylesimfrac{{mitDelta}(delta{mitPhi})}{a,{mitOmega}}simfrac{d}{a},frac{delta{mitPhi}}{a,{mitOmega}}simfrac{d}{a},ullu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4500.png"
 ALT="$\displaystyle \sim \frac{{\mit\Delta} (\delta{\mit\Phi})}{a {\mit\Omega}}\sim \frac{d}{a} \frac{\delta {\mit\Phi}}{a {\mit\Omega}}\sim \frac{d}{a} u\ll u,$">|; 

$key = q/omega_2={mitPi}_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4601.png"
 ALT="$ \omega_2={\mit\Pi}_E$">|; 

$key = q/displaystylealpha_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1002.png"
 ALT="$\displaystyle \alpha_2$">|; 

$key = q/displaystylesigma_{xy}=sigma_{yx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="$\displaystyle \sigma_{xy}=\sigma_{yx}$">|; 

$key = q/displaystyle{mitPsi}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img983.png"
 ALT="$\displaystyle {\mit\Psi}'$">|; 

$key = q/displaystyle=-gleft(frac{3}{2,n+1}right)left(zeta_{b,n}+frac{skew{3}bar{rho}}{rho},zeta_nright)left(frac{r}{a}right)^{-(n+1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4369.png"
 ALT="$\displaystyle =-g\left(\frac{3}{2 n+1}\right)\left(\zeta_{b n} + \frac{\skew{3}\bar{\rho}}{\rho} \zeta_n\right)\left(\frac{r}{a}\right)^{-(n+1)},$">|; 

$key = q/displaystyletau_i=int_Vepsilon_{ijk},x_j,F_k,dV+oint_Sepsilon_{ijk},x_j,sigma_{kl},dS_l,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$\displaystyle \tau_i =\int_V \epsilon_{ijk} x_j F_k dV + \oint_S\epsilon_{ijk} x_j \sigma_{kl} dS_l,$">|; 

$key = q/displaystyleI_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7283.png"
 ALT="$\displaystyle I_3$">|; 

$key = q/{rmdet}(A);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6999.png"
 ALT="$ {\rm det}(A)$">|; 

$key = q/displaystyle={bfA}cdotnablaA_r-frac{A_theta^{,2}}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7136.png"
 ALT="$\displaystyle ={\bf A}\cdot\nabla A_r - \frac{A_\theta^{ 2}}{r},$">|; 

$key = q/e_{13}=7.50times10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1030.png"
 ALT="$ e_{13}=7.50\times 10^{-3}$">|; 

$key = q/displaystylefrac{b}{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6078.png"
 ALT="$\displaystyle \frac{b}{c}$">|; 

$key = q/displaystyleC_D=frac{(epsilon,alpha)^{,2}}{sqrt{{rmMa}_infty^{,2}-1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6003.png"
 ALT="$\displaystyle C_D = \frac{(\epsilon \alpha)^{ 2}}{\sqrt{{\rm Ma}_\infty^{ 2}-1}}.$">|; 

$key = q/|V|=|c|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4132.png"
 ALT="$ \vert V\vert=\vert c\vert$">|; 

$key = q/displaystyle_S,dS.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3444.png"
 ALT="$\displaystyle _S dS.$">|; 

$key = q/vartheta=pislash2-theta=sin^{-1}(mu);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4710.png"
 ALT="$ \vartheta=\pi/2-\theta=\sin^{-1}(\mu)$">|; 

$key = q/Drightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3995.png"
 ALT="$ D\rightarrow 0$">|; 

$key = q/2,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3597.png"
 ALT="$ 2 h$">|; 

$key = q/{bfj};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3387.png"
 ALT="$ {\bf j}$">|; 

$key = q/beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$ \beta $">|; 

$key = q/displaystylev_t(r)=A,r^{-m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1883.png"
 ALT="$\displaystyle v_t(r) = A r^{-m},$">|; 

$key = q/displaystyleF(zeta)={rmi},frac{mitGamma}{2pi},lnleft(frac{zeta-{rmi}}{zeta+{rmi}}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2105.png"
 ALT="$\displaystyle F(\zeta) = {\rm i} \frac{\mit\Gamma}{2\pi} \ln\left(\frac{\zeta-{\rm i}}{\zeta+{\rm i}}\right)$">|; 

$key = q/displaystyle{bfa}times({bfb}times{bfc})=({bfa}cdot{bfc}),{bfb}-({bfa}cdot{bfb}),{bfc}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6816.png"
 ALT="$\displaystyle {\bf a}\times ({\bf b}\times {\bf c}) = ({\bf a}\cdot{\bf c}) {\bf b} - ({\bf a}\cdot{\bf b}) {\bf c}.$">|; 

$key = q/cproptoT^{,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5563.png"
 ALT="$ c\propto T^{ 1/2}$">|; 

$key = q/displaystyle=-{rmi},frac{partialphi}{partialy}+frac{partialpsi}{partialy}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1966.png"
 ALT="$\displaystyle = -{\rm i} \frac{\partial \phi}{\partial y} + \frac{\partial \psi} {\partial y}.$">|; 

$key = q/hat{rho}=rhoslashrho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5138.png"
 ALT="$ \hat{\rho}=\rho/\rho_0$">|; 

$key = q/displaystyle=frac{c,sinxi}{cosheta-cosxi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2462.png"
 ALT="$\displaystyle = \frac{c \sin\xi}{\cosh \eta-\cos\xi}.$">|; 

$key = q/12.000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4636.png"
 ALT="$ 12.000$">|; 

$key = q/{rmMa}_p=V_pslashc_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5537.png"
 ALT="$ {\rm Ma}_p=V_p/c_1$">|; 

$key = q/gamma=4slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5194.png"
 ALT="$ \gamma=4/3$">|; 

$key = q/e_{13}=0.81267;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img955.png"
 ALT="$ e_{13}=0.81267$">|; 

$key = q/I_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7287.png"
 ALT="$ I_3$">|; 

$key = q/displaystylefrac{epsilon,alpha}{sqrt{1-{rmMa}_infty^{,2}}}ll1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5980.png"
 ALT="$\displaystyle \frac{\epsilon \alpha}{\sqrt{1-{\rm Ma}_\infty^{ 2}}}\ll 1.$">|; 

$key = q/widehat{U}(X)=UslashU_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2897.png"
 ALT="$ \widehat{U}(X)=U/U_0$">|; 

$key = q/|zeta|slashskew{5}tilde{zeta}_8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5049.png"
 ALT="$ \vert\zeta\vert/\skew{5}\tilde{\zeta}_8$">|; 

$key = q/dslashb=u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1269.png"
 ALT="$ d/b=u$">|; 

$key = q/omega^{,2},a^{,2}slashg<4,(H-h);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1061.png"
 ALT="$ \omega^{ 2} a^{ 2}/g&lt; 4 (H-h)$">|; 

$key = q/displaystyleJ=2pi,a_1,a_2,a_3int_0^inftyfrac{du}{mitDelta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7265.png"
 ALT="$\displaystyle J = 2\pi a_1 a_2 a_3\int_0^\infty \frac{du}{\mit\Delta},$">|; 

$key = q/displaystylef_r(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3796.png"
 ALT="$\displaystyle f_r(\theta)$">|; 

$key = q/n=5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5282.png"
 ALT="$ n=5$">|; 

$key = q/v_p=V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3982.png"
 ALT="$ v_p=V$">|; 

$key = q/displaystylefrac{z'}{k}simeqzeta^{,-2}-frac{2,{rmi}}{zeta,sinalpha}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2324.png"
 ALT="$\displaystyle \frac{z'}{k}\simeq \zeta^{ -2}-\frac{2 {\rm i}}{\zeta \sin\alpha}.$">|; 

$key = q/a_3=a_1,cosgamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img997.png"
 ALT="$ a_3=a_1 \cos\gamma$">|; 

$key = q/displaystylefrac{u}{u_1}={rmMa}left[1+left(frac{gamma-1}{gamma+1}right)left({rmMa}^{,2}-1right)right]^{-1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5416.png"
 ALT="$\displaystyle \frac{u}{u_1}= {\rm Ma}\left[1+\left(\frac{\gamma-1}{\gamma+1}\right)\left({\rm Ma}^{ 2}-1\right)\right]^{-1/2}.$">|; 

$key = q/displaystylea_{ijkl}+a_{iklj}+a_{iljk}=a_{ijlk}+a_{ikjl}+a_{ilkj}=frac{1}{5},(la{ij},delta_{kl}+delta_{ik},delta_{jl}+delta_{il},delta_{jk}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="652" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6971.png"
 ALT="$\displaystyle a_{ijkl}+a_{iklj}+a_{iljk}= a_{ijlk}+a_{ikjl}+a_{ilkj} = \frac{1}...
...,(\delta_{ij} \delta_{kl} +\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}).$">|; 

$key = q/displaystylebeta_{n,,-n'}^{,m,,-m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4952.png"
 ALT="$\displaystyle \beta_{n, -n'}^{ m, -m'}$">|; 

$key = q/-|theta_{rmmax}|leqetaleqeta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6154.png"
 ALT="$ -\vert\theta_{\rm max}\vert\leq \eta\leq \eta_1$">|; 

$key = q/lambda';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7354.png"
 ALT="$ \lambda'$">|; 

$key = q/displaystylealphasimeqfrac{W}{L_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3528.png"
 ALT="$\displaystyle \alpha\simeq \frac{W}{L_0},$">|; 

$key = q/displaystyle{mitDelta}J_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2370.png"
 ALT="$\displaystyle {\mit\Delta} J_i$">|; 

$key = q/displaystyle=int_0^inftyfrac{du}{(a_i^{,2}+u),{mitDelta}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img861.png"
 ALT="$\displaystyle =\int_0^\infty \frac{du}{(a_i^{ 2}+u) {\mit\Delta}},$">|; 

$key = q/sinhxi_0=(1-e^{,2})^{1slash2}slashe;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2770.png"
 ALT="$ \sinh\xi_0=(1-e^{ 2})^{1/2}/e$">|; 

$key = q/c_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="$ c_p$">|; 

$key = q/{rmMa}_pll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5547.png"
 ALT="$ {\rm Ma}_p\ll 1$">|; 

$key = q/displaystylea_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img928.png"
 ALT="$\displaystyle a_3$">|; 

$key = q/displaystyle{mitPhi}_{rmtide}({bfr})=-frac{G,m'}{r'}left[1+left(frac{r}{r'}right)^2P_2(cosgamma)+{calO}left(frac{r}{r'}right)^3right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="424" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4249.png"
 ALT="$\displaystyle {\mit\Phi}_{\rm tide}({\bf r}) = - \frac{G m'}{r'}\left[1+\left(\frac{r}{r'}\right)^2 P_2(\cos\gamma)+ {\cal O}\left(\frac{r}{r'}\right)^3\right].$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{Chapter06slashsource3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="221" BORDER="0"
 SRC="|."$dir".q|img2169.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{Chapter06/source3.eps}}
\end{figure}">|; 

$key = q/displaystylechi_0(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4379.png"
 ALT="$\displaystyle \chi_0(r)$">|; 

$key = q/{calE}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5445.png"
 ALT="$ {\cal E}_1$">|; 

$key = q/displaystylecostheta=frac{gamma_{23}-gamma_{13}}{gamma_{12}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1129.png"
 ALT="$\displaystyle \cos\theta = \frac{\gamma_{23}-\gamma_{13}}{\gamma_{12}}.$">|; 

$key = q/displaystylex=int_z^hfrac{df}{f_x}=lint_{zslash2l}^{hslash2l}frac{1-2,y^{,2}}{y,(1-y^{,2})^{1slash2}},dy,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1225.png"
 ALT="$\displaystyle x = \int_z^h \frac{df}{f_x}= l \int_{z/2l}^{h/2l}\frac{1-2 y^{ 2}}{y (1-y^{ 2})^{1/2}} dy,$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partial{mitPhi}_r''(theta,phi_pm)}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4857.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial {\mit\Phi}_r''(\theta,\phi_\pm)}{\partial\phi}$">|; 

$key = q/xi=rslasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5254.png"
 ALT="$ \xi=r/a$">|; 

$key = q/displaystyleP(theta,phi,t)=({mitPhi}+chi)_{z=d+zeta_b+zeta}simeqg,(zeta_b+zeta)+delta{mitPhi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4486.png"
 ALT="$\displaystyle P(\theta,\phi,t) = ({\mit\Phi}+\chi)_{z=d+\zeta_b+\zeta} \simeq g (\zeta_b+\zeta) + \delta{\mit\Phi},$">|; 

$key = q/displaystylex_0-x=f(y),(u-u_0)^2+{calO}left[(u-u_0)^3right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3115.png"
 ALT="$\displaystyle x_0-x = f(y) (u-u_0)^2 + {\cal O}\left[(u-u_0)^3\right],$">|; 

$key = q/displaystyle=lim_{epsilonrightarrow0}left{left(frac{1}{sinphi},lnleft[frac{sin,(}{sin,(1slash2),(phi-phi')}right]right)_0^{phi-epsilon}right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3497.png"
 ALT="$\displaystyle =\lim_{\epsilon\rightarrow 0}\left\{\left(\frac{1}{\sin\phi} \ln...
...\phi+\phi')} {\sin (1/2) (\phi-\phi')}\right]\right)_0^{\phi-\epsilon}\right.$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{AppendixAslashfigA.07.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="169" BORDER="0"
 SRC="|."$dir".q|img6345.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{AppendixA/figA.07.eps}}
\end{figure}">|; 

$key = q/AP_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1622.png"
 ALT="$ AP_1$">|; 

$key = q/{calA}_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6085.png"
 ALT="$ {\cal A}_e$">|; 

$key = q/{rmPr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="$ {\rm Pr}$">|; 

$key = q/nabla{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7091.png"
 ALT="$ \nabla{\bf A}$">|; 

$key = q/n,{bfa}=|n|,(-{bfa});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6244.png"
 ALT="$ n {\bf a} = \vert n\vert (-{\bf a})$">|; 

$key = q/a=c+l^{,2}slashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2504.png"
 ALT="$ a=c+l^{ 2}/c$">|; 

$key = q/displaystyle2,a,C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1291.png"
 ALT="$\displaystyle 2 a C$">|; 

$key = q/l=z,tanphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3473.png"
 ALT="$ l=z \tan\phi$">|; 

$key = q/bar{F};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2186.png"
 ALT="$ \bar{F}$">|; 

$key = q/displaystyle=muleft(frac{partialv_y}{partialz}+frac{partialv_z}{partialy}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="$\displaystyle = \mu\left(\frac{\partial v_y}{\partial z}+\frac{\partial v_z}{\partial y}\right),$">|; 

$key = q/v_p=v_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4165.png"
 ALT="$ v_p=v_c$">|; 

$key = q/displaystylehsimeqfrac{2,gamma,costheta}{rho,g,a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1142.png"
 ALT="$\displaystyle h \simeq \frac{2 \gamma \cos\theta}{\rho g a}.$">|; 

$key = q/partialuslashpartialy<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3141.png"
 ALT="$ \partial u/\partial y&lt;0$">|; 

$key = q/displaystyle=-g,nabla(zeta-skew{5}bar{zeta}'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5071.png"
 ALT="$\displaystyle =-g \nabla(\zeta-\skew{5}\bar{\zeta}'),$">|; 

$key = q/displaystyle=a,omega,{rme}^{,k,z},cos(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3944.png"
 ALT="$\displaystyle = a \omega {\rm e}^{ k z} \cos(\omega t-k x),$">|; 

$key = q/nablacdot{bfF}=r^{,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7206.png"
 ALT="$ \nabla\cdot {\bf F} = r^{ m}$">|; 

$key = q/displaystyleVll(g,d)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4063.png"
 ALT="$\displaystyle V \ll (g d)^{1/2}.$">|; 

$key = q/{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1096.png"
 ALT="$ {\bf b}$">|; 

$key = q/2,{mitOmega}-3,omega_M+{mitPi}_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4562.png"
 ALT="$ 2 {\mit\Omega}-3 \omega_M+{\mit\Pi}_M$">|; 

$key = q/c>c_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img680.png"
 ALT="$ c&gt;c_0$">|; 

$key = q/displaystyler=a+d+zeta_t(theta,phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4344.png"
 ALT="$\displaystyle r = a+ d+\zeta_t(\theta,\phi),$">|; 

$key = q/displaystylek_0=frac{g}{V^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4027.png"
 ALT="$\displaystyle k_0 = \frac{g}{V^{ 2}}$">|; 

$key = q/{bfv}=V_0tmspace+thinmuskip{.1667em}{rme}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$ {\bf v}= V_0\tmspace +\thinmuskip {.1667em}{\rm e}_x$">|; 

$key = q/A=A(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5407.png"
 ALT="$ A=A(x)$">|; 

$key = q/C_L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6037.png"
 ALT="$ C_L$">|; 

$key = q/{bfk}cdot{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6749.png"
 ALT="$ {\bf k}\cdot{\bf r}$">|; 

$key = q/displaystyleBP';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img775.png"
 ALT="$\displaystyle BP'$">|; 

$key = q/displaystyle=(4,lambda-mu-nu)slash10,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6973.png"
 ALT="$\displaystyle = (4 \lambda-\mu-\nu)/10,$">|; 

$key = q/displaystyleU=-frac{26}{35},frac{G,M_ast^{,2}}{R}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5287.png"
 ALT="$\displaystyle U = - \frac{26}{35} \frac{G M_\ast^{ 2}}{R}.
$">|; 

$key = q/p_{rmatm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4398.png"
 ALT="$ p_{\rm atm}$">|; 

$key = q/displaystylechi_2(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4381.png"
 ALT="$\displaystyle \chi_2(r,\theta)$">|; 

$key = q/displaystyle=-{calL}^{,2}(psi),nablavarphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3743.png"
 ALT="$\displaystyle = - {\cal L}^{ 2}(\psi) \nabla\varphi.$">|; 

$key = q/displaystyle{bfv}(P_-);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3422.png"
 ALT="$\displaystyle {\bf v}(P_-)$">|; 

$key = q/displaystyle=r,dr,dtheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7126.png"
 ALT="$\displaystyle = r dr d\theta,$">|; 

$key = q/dz={rmi},epsilon,{rme}^{,{rmi},theta},dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2418.png"
 ALT="$ dz = {\rm i} \epsilon {\rm e}^{ {\rm i} \theta} d\theta$">|; 

$key = q/nablacdot(phi,{bfa})=phi,nablacdot{bfa}+{bfa}cdotnablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6987.png"
 ALT="$ \nabla\cdot(\phi {\bf a}) = \phi \nabla\cdot {\bf a} + {\bf a}\cdot\nabla \phi$">|; 

$key = q/displaystylefrac{partial{bfv}}{partialt}+2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4448.png"
 ALT="$\displaystyle \frac{\partial{\bf v}}{\partial t} + 2 $">|; 

$key = q/|{bfr}|^{,n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6750.png"
 ALT="$ \vert{\bf r}\vert^{ n},$">|; 

$key = q/varpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2524.png"
 ALT="$ \varpi$">|; 

$key = q/LsimeqL_0,alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3522.png"
 ALT="$ L\simeq L_0 \alpha$">|; 

$key = q/displaystylealeft[left(1-frac{c^{,2}}{V^{,2}}right)cosh(k,d)right]^{,-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4207.png"
 ALT="$\displaystyle a\left[\left(1-\frac{c^{ 2}}{V^{ 2}}\right)\cosh (k d)\right]^{ -1},
$">|; 

$key = q/{calH}=Hslashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5317.png"
 ALT="$ {\cal H} = H/m$">|; 

$key = q/displaystyle=y_C(x)-h(x)=y_C(x_L)-alpha_0,(x-x_L)-alpha_C(x)-h(x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5836.png"
 ALT="$\displaystyle =y_C(x)- h(x) = y_C(x_L) -\alpha_0 (x-x_L) -\alpha_C(x)-h(x).$">|; 

$key = q/tan^{-1}(1slash!sqrt{8})=19.47^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4061.png"
 ALT="$ \tan^{-1}(1/\!\sqrt{8})= 19.47^\circ$">|; 

$key = q/a_i,b_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6779.png"
 ALT="$ a_i b_j$">|; 

$key = q/displaystyle|u_2|=left(frac{2}{gamma_4-2}right)c_4.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5655.png"
 ALT="$\displaystyle \vert u_2\vert = \left(\frac{2}{\gamma_4-2}\right)c_4.
$">|; 

$key = q/displaystyle{calR}_n(r,theta,varphi)proptor^{,n},{calS}_n(theta,varphi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4323.png"
 ALT="$\displaystyle {\cal R}_n(r,\theta,\varphi) \propto r^{ n} {\cal S}_n(\theta,\varphi).$">|; 

$key = q/dy=a,sintheta,dphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5060.png"
 ALT="$ dy=a \sin\theta d\phi$">|; 

$key = q/displaystylesigma_{rtheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3892.png"
 ALT="$\displaystyle \sigma_{r\theta}$">|; 

$key = q/displaystyledelta_{ij}=left{{array}{lll}1&mbox{hspace{0.5cm}}&mbox{ifi=j}[0.5ex]0&&mbox{otherwise}{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6790.png"
 ALT="$\displaystyle \delta_{ij} = \left\{ \begin{array}{lll} 1&amp;\mbox{\hspace{0.5cm}}&amp; \mbox{if $i=j$}\ [0.5ex] 0&amp;&amp;\mbox{otherwise} \end{array}\right. .$">|; 

$key = q/D=Q,d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1692.png"
 ALT="$ D=Q d$">|; 

$key = q/|zeta_t|lldlla;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4350.png"
 ALT="$ \vert\zeta_t\vert \ll d\ll a$">|; 

$key = q/displaystylefrac{p_2}{p_1}=sqrt{frac{T_2}{T_1}},frac{{rmMa}_1}{{rmMa}_2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5476.png"
 ALT="$\displaystyle \frac{p_2}{p_1} = \sqrt{\frac{T_2}{T_1}} \frac{{\rm Ma}_1}{{\rm Ma}_2},$">|; 

$key = q/displaystyle=left[V,sin(alpha+theta)+frac{{mitGamma}}{pi,(c+delta)}right]frac{(cdelta)}{(c^{,2},sin^2theta+delta^{,2},cos^2theta)^{1slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3335.png"
 ALT="$\displaystyle = \left[V \sin(\alpha+\theta)+\frac{{\mit\Gamma}}{\pi (c+\delta...
...ht]\frac{(c+\delta)}{(c^{ 2} \sin^2\theta+\delta^{ 2} \cos^2\theta)^{1/2}}.$">|; 

$key = q/betarightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img950.png"
 ALT="$ \beta\rightarrow 0$">|; 

$key = q/u_1-u_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5520.png"
 ALT="$ u_1-u_2$">|; 

$key = q/displaystylephantom{=}+left(frac{1}{m'+1}right)I_{n,,n'}^{,m,,m'+2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4975.png"
 ALT="$\displaystyle \phantom{=} + \left(\frac{1}{m'+1}\right)I_{n, n'}^{ m, m'+2},$">|; 

$key = q/displaystyle=-gleft(frac{3}{2,n+1}right)left(zeta_{b,n}+frac{skew{3}bar{rho}}{rho},zeta_nright)left(frac{r}{a}right)^n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4367.png"
 ALT="$\displaystyle = -g\left(\frac{3}{2 n+1}\right)\left(\zeta_{b n} + \frac{\skew{3}\bar{\rho}}{\rho} \zeta_n\right)\left(\frac{r}{a}\right)^n,$">|; 

$key = q/mu=sin^{-1}(1slash{rmMa}_infty);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6063.png"
 ALT="$ \mu=\sin^{-1}(1/{\rm Ma}_\infty)$">|; 

$key = q/displaystyle{bfW}=-M_0,g,{bfe}_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img589.png"
 ALT="$\displaystyle {\bf W} = -M_0 g {\bf e}_z.$">|; 

$key = q/i=0,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img991.png"
 ALT="$ i=0,3$">|; 

$key = q/displaystyle=U+u',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5896.png"
 ALT="$\displaystyle = U + u',$">|; 

$key = q/displaystylefrac{partialphi}{partialeta}=-frac{1}{c,sinhxi,sineta},frac{partialpsi}{partialxi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2805.png"
 ALT="$\displaystyle \frac{\partial\phi}{\partial\eta} = -\frac{1}{c \sinh\xi \sin\eta} \frac{\partial\psi}{\partial\xi},$">|; 

$key = q/displaystyle=nabla{calH}_0-T,nabla{calS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5870.png"
 ALT="$\displaystyle = \nabla {\cal H}_0 - T \nabla{\cal S}.$">|; 

$key = q/y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$ y$">|; 

$key = q/displaystylesigma_{zz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="$\displaystyle \sigma_{zz}$">|; 

$key = q/displaystylecostheta=frac{{bfa}cdot{bfb}}{|{bfa}|,|{bfb}|}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6322.png"
 ALT="$\displaystyle \cos\theta = \frac{{\bf a} \cdot {\bf b}}{\vert{\bf a}\vert  \vert{\bf b}\vert }.$">|; 

$key = q/displaystyle=nabla^{,2}A_theta+frac{2}{r^{,2}},frac{partialA_r}{partialtheta}-frac{A_theta}{r^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7162.png"
 ALT="$\displaystyle = \nabla^{ 2} A_\theta + \frac{2}{r^{ 2}} \frac{\partial A_r}{\partial \theta}-\frac{A_\theta}{r^{ 2}},$">|; 

$key = q/displaystyledS_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7121.png"
 ALT="$\displaystyle dS_r$">|; 

$key = q/z^{,2}=tantheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img778.png"
 ALT="$ z^{ 2}=\tan\theta$">|; 

$key = q/phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2355.png"
 ALT="$ \phi)$">|; 

$key = q/0<d_2<d_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3718.png"
 ALT="$ 0&lt;d_2&lt;d_1$">|; 

$key = q/displaystyle{bfU}times({bfn}times{bfU})=U^{,2},{bfn}-({bfn}cdot{bfU}),{bfU}=U^{,2},{bfn},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="345" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3433.png"
 ALT="$\displaystyle {\bf U}\times ({\bf n}\times {\bf U}) = U^{ 2} {\bf n}- ({\bf n}\cdot{\bf U}) {\bf U} = U^{ 2} {\bf n},$">|; 

$key = q/eta>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2098.png"
 ALT="$ \eta&gt;0$">|; 

$key = q/a_0<a_0';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1022.png"
 ALT="$ a_0&lt;a_0'$">|; 

$key = q/displaystyle-A_y(z+dz),dy+A_y(z),dy=-frac{partialA_y}{partialy},dy,dz.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6632.png"
 ALT="$\displaystyle -A_y(z+dz) dy + A_y(z) dy = -\frac{\partial A_y}{\partial y} dy dz.$">|; 

$key = q/p=-sigma_{ii}slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="$ p=-\sigma_{ii}/3$">|; 

$key = q/D,V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3988.png"
 ALT="$ D V$">|; 

$key = q/1.6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$ 1.6$">|; 

$key = q/[A,,B]equiv{bfe}_zcdotnablaAtimesnablaB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img540.png"
 ALT="$ [A, B]\equiv {\bf e}_z\cdot\nabla A\times \nabla B$">|; 

$key = q/B={rmi},pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2200.png"
 ALT="$ B={\rm i} \pi/2$">|; 

$key = q/displaystyledelta(l)=frac{d}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3006.png"
 ALT="$\displaystyle \delta(l) = \frac{d}{2}.$">|; 

$key = q/displaystylefrac{DV}{Dt}=oint_S{bfv}cdotd{bfS}=oint_Sv_i,dS_i=int_Vfrac{partialv_i}{partialx_i},dV=int_Vnabla!cdot!{bfv},dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="453" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="$\displaystyle \frac{DV}{Dt} = \oint_S{\bf v}\cdot d{\bf S}=\oint_S v_i dS_i = ...
..._V \frac{\partial v_i}{\partial x_i} dV = \int_V \nabla\!\cdot \! {\bf v} dV,$">|; 

$key = q/f_zequivdfslashdz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1234.png"
 ALT="$ f_z\equiv df/dz$">|; 

$key = q/{bfb}equiv(1,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6295.png"
 ALT="$ {\bf b} \equiv (1, 0, 0)$">|; 

$key = q/{rmFr}={rmFr}_{rmc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1449.png"
 ALT="$ {\rm Fr}={\rm Fr}_{\rm c}$">|; 

$key = q/displaystyledF(x,y)=f(x,y),dx+g(x,y),dy.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6468.png"
 ALT="$\displaystyle dF(x,y) = f(x,y) dx + g(x,y) dy.$">|; 

$key = q/displaystyle=int_lambda^inftyfrac{du}{(a_i^{,2}+u),{mitDelta}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7295.png"
 ALT="$\displaystyle =\int_\lambda^\infty \frac{du}{(a_i^{ 2}+u) {\mit\Delta}},$">|; 

$key = q/deltap;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img712.png"
 ALT="$ \delta p$">|; 

$key = q/widehat{omega}=7.27times10^{-5},{rmrad.,s^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img912.png"
 ALT="$ \widehat{\omega}=7.27\times 10^{-5} {\rm rad. s^{-1}}$">|; 

$key = q/0leqyleqxslash({rmMa}_infty^{,2}-1)^{,1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6068.png"
 ALT="$ 0\leq y\leq x/({\rm Ma}_\infty^{ 2}-1)^{ 1/2}$">|; 

$key = q/displaystylef_x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3712.png"
 ALT="$\displaystyle f_x'$">|; 

$key = q/displaystylesinbeta=frac{v_p}{V}=left(frac{g}{k,V^{,2}}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4026.png"
 ALT="$\displaystyle \sin\beta =\frac{v_p}{V}=\left( \frac{g}{k V^{ 2}}\right)^{1/2}.$">|; 

$key = q/displaystylephi(r,theta)=-V,r,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2603.png"
 ALT="$\displaystyle \phi(r,\theta) = -V r \cos\theta.$">|; 

$key = q/displaystyle=nablaomegacdotnablapsi+omega^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1927.png"
 ALT="$\displaystyle =\nabla\omega\cdot\nabla\psi + \omega^{ 2},$">|; 

$key = q/deltatheta_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6397.png"
 ALT="$ \delta \theta_x$">|; 

$key = q/displaystylenablaphicdotnablapsi=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1662.png"
 ALT="$\displaystyle \nabla\phi\cdot\nabla\psi = 0.$">|; 

$key = q/displaystylez';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6279.png"
 ALT="$\displaystyle z'$">|; 

$key = q/X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2381.png"
 ALT="$ X$">|; 

$key = q/skew{3}bar{psi}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2210.png"
 ALT="$ \skew{3}\bar{\psi}=0$">|; 

$key = q/1.0times10^{-3},{rmkgslash(m,s)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="$ 1.0\times 10^{-3}  {\rm kg/(m s)}$">|; 

$key = q/displaystylev_x(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2865.png"
 ALT="$\displaystyle v_x(x,y)$">|; 

$key = q/displaystyle=p_0-left[rho,g-frac{rho,k,(V-c)^2}{tanh(k,d)}right]zeta_0,sin(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4101.png"
 ALT="$\displaystyle =p_0-\left[\rho g-\frac{\rho k (V-c)^2}{\tanh(k d)}\right]\zeta_0 \sin(\omega t-k x).$">|; 

$key = q/displaystyledelta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3266.png"
 ALT="$\displaystyle \delta_2$">|; 

$key = q/displaystylenabla^{,2}(x_i,{calR}_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4332.png"
 ALT="$\displaystyle \nabla^{ 2}(x_i {\cal R}_n)$">|; 

$key = q/displaystylef'(infty);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2943.png"
 ALT="$\displaystyle f'(\infty)$">|; 

$key = q/displaystylev_t(r)=A,r^{,m},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1873.png"
 ALT="$\displaystyle v_t(r) = A r^{ m},$">|; 

$key = q/partial^{,2}Tslashpartialy^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6619.png"
 ALT="$ \partial^{ 2} T/\partial y^{ 2}$">|; 

$key = q/displaystyles'=frac{V_1}{V}=frac{V-V_0}{V}=1-s.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img605.png"
 ALT="$\displaystyle s' = \frac{V_1}{V}=\frac{V-V_0}{V} =1-s.$">|; 

$key = q/displaystylenablaequivleft(frac{partial}{partialx},,frac{partial}{partialy},,frac{partial}{partialz}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6562.png"
 ALT="$\displaystyle \nabla \equiv \left( \frac{\partial}{\partial x},  \frac{\partial}{\partial y},  \frac{\partial }{\partial z}\right),$">|; 

$key = q/displaystyle=frac{L,V_0}{nu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="$\displaystyle = \frac{L V_0}{\nu},$">|; 

$key = q/displaystylefrac{partialv_y}{partialx}-frac{partialv_x}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1916.png"
 ALT="$\displaystyle \frac{\partial v_y}{\partial x}- \frac{\partial v_x}{\partial y}
$">|; 

$key = q/displaystylesum_{n=1,infty}(1+k_n'-h_n'),alpha_n,zeta_n=int_{-pi}^piint_0^piG_F(,phi;theta',phi'),zeta(theta',phi'),,sintheta',dtheta',dphi',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="577" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4514.png"
 ALT="$\displaystyle \sum_{n=1,\infty} (1+k_n'-h_n') \alpha_n \zeta_n = \int_{-\pi}^...
...eta,\phi;\theta',\phi') \zeta(\theta',\phi')  \sin\theta' d\theta' d\phi',$">|; 

$key = q/displaystylefrac{Drho}{Dt}=-rho,nablacdot{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5436.png"
 ALT="$\displaystyle \frac{D\rho}{Dt} = -\rho \nabla\cdot{\bf v},$">|; 

$key = q/{bfa}-3,{bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6700.png"
 ALT="$ {\bf a}-3 {\bf b}$">|; 

$key = q/Q+deltaQ-V,(h+deltah);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3605.png"
 ALT="$ Q+\delta Q - V (h+\delta h)$">|; 

$key = q/W,z_{GH};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img706.png"
 ALT="$ W z_{GH}$">|; 

$key = q/displaystyle=deltatheta_x,{bfe}_x+deltatheta_y,{bfe}_y+deltatheta_z,{bfe}_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6402.png"
 ALT="$\displaystyle = \delta\theta_x  {\bf e}_x + \delta\theta_y  {\bf e}_y + \delta\theta_z  {\bf e}_z.$">|; 

$key = q/displaystyleoint_Sa_{ij},dS_i=int_Vfrac{partiala_{ij}}{partialx_i},dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6893.png"
 ALT="$\displaystyle \oint_S a_{ij} dS_i = \int_V \frac{\partial a_{ij}}{\partial x_i} dV,$">|; 

$key = q/displaystyletheta=1-frac{xi^{,2}}{6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5278.png"
 ALT="$\displaystyle \theta=1-\frac{\xi^{ 2}}{6}
$">|; 

$key = q/{calH}_1={calE}_1+p_1slashrho_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5468.png"
 ALT="$ {\cal H}_1={\cal E}_1+p_1/\rho_1$">|; 

$key = q/{bfq}cdotd{bfS}=q_i,dS_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="$ {\bf q}\cdot d{\bf S} = q_i dS_i$">|; 

$key = q/displaystyle=frac{V_0}{sqrt{gamma,p_0slashrho_0}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="$\displaystyle = \frac{V_0}{\sqrt{\gamma p_0/\rho_0}},$">|; 

$key = q/dFslashdz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1992.png"
 ALT="$ dF/dz$">|; 

$key = q/z=z+dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3464.png"
 ALT="$ z=z+dz$">|; 

$key = q/displaystylepsi(r,mu)=beta_0,mu+sum_{l=1,infty}left(frac{alpha_l}{l+1},r^{,l+1}-{beta_l}{l},r^{,-l}right)left(1-mu^{,2}right)frac{dP_l}{dmu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2597.png"
 ALT="$\displaystyle \psi(r,\mu)= \beta_0 \mu+\sum_{l=1,\infty}\left(\frac{\alpha_l}{...
...1}-\frac{\beta_l}{l} r^{ -l}\right)\left(1-\mu^{ 2}\right)\frac{dP_l}{d\mu},$">|; 

$key = q/displaystyle=frac{sin^2theta}{a_1^{,2}}+frac{cos^2theta}{a_3^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7261.png"
 ALT="$\displaystyle = \frac{\sin^2\theta}{a_1^{ 2}} + \frac{\cos^2\theta}{a_3^{ 2}},$">|; 

$key = q/displaystyle{mitPhi}_0(r>a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4364.png"
 ALT="$\displaystyle {\mit\Phi}_0(r&gt;a)$">|; 

$key = q/j=m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6810.png"
 ALT="$ j=m$">|; 

$key = q/DAE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4020.png"
 ALT="$ DAE$">|; 

$key = q/rho(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5158.png"
 ALT="$ \rho(z)$">|; 

$key = q/mu,l,Uslashd^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3708.png"
 ALT="$ \mu l U/d^{ 2}$">|; 

$key = q/displaystylev_xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2693.png"
 ALT="$\displaystyle v_\xi$">|; 

$key = q/displaystylemu_rint_{mitOmega}{mitPsi}_r,{mitPsi}_s,d{mitOmega}=mu_sint_{mitOmega}{mitPsi}_r,{mitPsi}_s,d{mitOmega}=delta_{r,s}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4831.png"
 ALT="$\displaystyle \mu_r\int_{\mit\Omega} {\mit\Psi}_r {\mit\Psi}_s d{\mit\Omega}=\mu_s\int_{\mit\Omega} {\mit\Psi}_r {\mit\Psi}_s d{\mit\Omega} = \delta_{r s}.$">|; 

$key = q/displaystyled{bfx}cdotd{bfx}=dr^{,2}+r^{,2},dtheta^{,2}+dz^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7115.png"
 ALT="$\displaystyle d{\bf x}\cdot d{\bf x} = dr^{ 2} + r^{ 2} d\theta^{ 2} + dz^{ 2}.$">|; 

$key = q/displaystyleleft.frac{partialphi}{partialr}right|_{r=a}=-V_x,costheta-V_y,sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1829.png"
 ALT="$\displaystyle \left.\frac{\partial\phi}{\partial r}\right\vert _{r=a} = - V_x \cos\theta- V_y \sin\theta.$">|; 

$key = q/k,lll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4157.png"
 ALT="$ k l\ll 1$">|; 

$key = q/displaystyle=int_0^inftyleft(U,frac{dU}{dx}-u,frac{partialu}{partialx}-v,frac{partialu}{partialy}right)dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3080.png"
 ALT="$\displaystyle =\int_0^\infty\left(U \frac{dU}{dx} - u \frac{\partial u}{\partial x}-v \frac{\partial u}{\partial y}\right)dy$">|; 

$key = q/U,dUslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3124.png"
 ALT="$ U dU/dx$">|; 

$key = q/displaystyle=frac{sigma_j}{2,{mitOmega}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4687.png"
 ALT="$\displaystyle = \frac{\sigma_j}{2 {\mit\Omega}},$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{Chapter04slashbump.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="506" HEIGHT="198" BORDER="0"
 SRC="|."$dir".q|img1451.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{Chapter04/bump.eps}}
\end{figure}">|; 

$key = q/displaystyleC_psimeqfrac{2}{U}left.frac{partialphi}{partialx}right|_{y=0}=frac{2,epsilon,alpha}{sqrt{{rmMa}_infty^{,2}-1}},cos(alpha,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5997.png"
 ALT="$\displaystyle C_p \simeq \frac{2}{U}\left.\frac{\partial\phi}{\partial x}\right...
... = \frac{2 \epsilon \alpha}{\sqrt{{\rm Ma}_\infty^{ 2}-1}} \cos(\alpha x).$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter09slashdownwash.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3506.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter09/downwash.eps}}
\end{figure}">|; 

$key = q/displaystyleP_n(cosgamma);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4257.png"
 ALT="$\displaystyle P_n(\cos\gamma)$">|; 

$key = q/displaystylesimeq-frac{G,M'}{R}left[1+frac{x_1}{R}+frac{x_1^{,2}-(1slash2),x_2^{,2}-(1slash2),x_3^{,2}}{R^{,2}}+cdotsright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img985.png"
 ALT="$\displaystyle \simeq -\frac{G M'}{R}\left[1+\frac{x_1}{R}+ \frac{x_1^{ 2}-(1/2) x_2^{ 2}-(1/2) x_3^{ 2}}{R^{ 2}}+\cdots\right],$">|; 

$key = q/g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="$ g$">|; 

$key = q/M=(4pislash3),a^{,3},overline{rho};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3811.png"
 ALT="$ M=(4\pi/3) a^{ 3} \overline{\rho}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter11slashkelvin.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img4064.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter11/kelvin.eps}}
\end{figure}">|; 

$key = q/(zeta-a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2148.png"
 ALT="$ (\zeta-a)$">|; 

$key = q/displaystylenabla({bfA}cdot{bfB});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6683.png"
 ALT="$\displaystyle \nabla({\bf A}\cdot{\bf B})$">|; 

$key = q/displaystyle{bfn}=(sintheta,cosphi,,sintheta,sinphi,,costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7257.png"
 ALT="$\displaystyle {\bf n} = (\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta),$">|; 

$key = q/pminfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3051.png"
 ALT="$ \pm \infty$">|; 

$key = q/displaystyleleft(frac{partial}{partialt},D+2,{mitOmega},frac{partial}{partialphia}right)skew{5}dot{mitPsi}=frac{g,d}{a^{,2}},D,(zeta-{calF}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="553" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4671.png"
 ALT="$\displaystyle \left(\frac{\partial}{\partial t} D +2 {\mit\Omega} \frac{\par...
...theta}\right)\skew{5}\dot{\mit\Psi} =\frac{g d}{a^{ 2}} D (\zeta-{\cal F}).$">|; 

$key = q/z=-d+a,sin(k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4118.png"
 ALT="$ z=-d+ a \sin(k x)$">|; 

$key = q/p(r,z)=p_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img821.png"
 ALT="$ p(r,z)=p_0$">|; 

$key = q/displaystyleV^{,2}left[(H+h-d)^2-H^{,2}right]=2,g,h,(H+h-d)^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1458.png"
 ALT="$\displaystyle V^{ 2}\left[(H+h-d)^2-H^{ 2}\right]=2 g h (H+h-d)^2,$">|; 

$key = q/theta=180^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$ \theta =180^\circ $">|; 

$key = q/displaystyle=frac{cosgamma}{sin^2gamma}left[(1+2,cos^2gamma),frac{gamma}{singamma}-3,cosgammaright],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img896.png"
 ALT="$\displaystyle =\frac{\cos\gamma}{\sin^2\gamma}\left[(1+2 \cos^2\gamma) \frac{\gamma}{\sin\gamma}-3 \cos\gamma\right],$">|; 

$key = q/displaystyle=-frac{2,costheta,f(r)}{r^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3766.png"
 ALT="$\displaystyle = -\frac{2 \cos\theta f(r)}{r^{ 2}},$">|; 

$key = q/displaystyle=r,sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1942.png"
 ALT="$\displaystyle =r \sin\theta.$">|; 

$key = q/gamma_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5632.png"
 ALT="$ \gamma_4$">|; 

$key = q/displaystylefrac{u(x,y)}{U(x)}=Fleft[left(frac{U_0}{nu}right)^{1slash2},frac{y}{x}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3235.png"
 ALT="$\displaystyle \frac{u(x,y)}{U(x)} = F\left[\left(\frac{U_0}{\nu}\right)^{1/2} \frac{y}{x}\right],
$">|; 

$key = q/z'=(sinalpha+{rmi},cosalpha),z=x'+{rmi},y';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2321.png"
 ALT="$ z'=(\sin\alpha+{\rm i} \cos\alpha) z=x'+{\rm i} y'$">|; 

$key = q/(sigma_{11}'+sigma_{22}'+sigma_{33}')slash3=sigma_{ii}slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="$ (\sigma_{11}'+\sigma_{22}'+\sigma_{33}')/3= \sigma_{ii}/3$">|; 

$key = q/displaystylerho(r)=rho_c,(1-rslashR).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5285.png"
 ALT="$\displaystyle \rho(r)=\rho_c (1-r/R).
$">|; 

$key = q/displaystyledeltah=left(frac{dh}{dQ}right)deltaQ.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3603.png"
 ALT="$\displaystyle \delta h = \left(\frac{dh}{dQ}\right)\delta Q.$">|; 

$key = q/K';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2149.png"
 ALT="$ K'$">|; 

$key = q/hat{p}=pslashp_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5137.png"
 ALT="$ \hat{p}=p/p_0$">|; 

$key = q/displaystylechi_w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6107.png"
 ALT="$\displaystyle \chi_w$">|; 

$key = q/rho_0,V_0,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img623.png"
 ALT="$ \rho_0 V_0 g$">|; 

$key = q/displaystylec=sqrt{frac{gamma,p}{rho}}=sqrt{gamma,{calR},T},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5363.png"
 ALT="$\displaystyle c =\sqrt{\frac{\gamma p}{\rho}}=\sqrt{\gamma {\cal R} T},$">|; 

$key = q/displaystylea_{isks};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6957.png"
 ALT="$\displaystyle a_{isks}$">|; 

$key = q/-bslasha<z<bslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3505.png"
 ALT="$ -b/a&lt;z&lt;b/2$">|; 

$key = q/lambda(x_s)equivlambda_s=-0.1567;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3210.png"
 ALT="$ \lambda(x_s)\equiv\lambda_s= -0.1567$">|; 

$key = q/displaystyleleft.sigma_{xy}right|_{y=0}=frac{1}{2},rho,U^{,2}left(frac{nu}{U,x}right)^{1slash2}(m+1)^{1slash2},sqrt{2},f''(0).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2946.png"
 ALT="$\displaystyle \left.\sigma_{xy}\right\vert _{y=0} =\frac{1}{2} \rho U^{ 2}\left(\frac{\nu}{U x}\right)^{1/2}(m+1)^{1/2} \sqrt{2} f''(0).$">|; 

$key = q/displaystyle=frac{1}{2pi},frac{x}{sqrt{{rmMa}_infty^{,2}-1}}int_{-1}^1,cos^{-1}(k),dk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6072.png"
 ALT="$\displaystyle =\frac{1}{2\pi} \frac{x}{\sqrt{{\rm Ma}_\infty^{ 2}-1}}\int_{-1}^1 \cos^{-1}(k) dk$">|; 

$key = q/displaystyle{bfa}+{bf0}={bfa},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6237.png"
 ALT="$\displaystyle {\bf a} + {\bf0} = {\bf a},$">|; 

$key = q/displaystyle{bf0}=nablap+rho,nabla{mitPsi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img559.png"
 ALT="$\displaystyle {\bf0} = \nabla p + \rho \nabla{\mit\Psi},$">|; 

$key = q/displaystylec=c_0+left(frac{gamma-1}{2}right)u,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5664.png"
 ALT="$\displaystyle c= c_0 + \left(\frac{\gamma-1}{2}\right)u,
$">|; 

$key = q/c(z>bslash2)=delta(z>bslash2)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3399.png"
 ALT="$ c(z&gt;b/2)=\delta(z&gt;b/2)=0$">|; 

$key = q/k=i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6925.png"
 ALT="$ k=i$">|; 

$key = q/displaystylec^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4209.png"
 ALT="$\displaystyle c^{ 2}$">|; 

$key = q/dx=x-x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6533.png"
 ALT="$ dx=x-x_0$">|; 

$key = q/displaystylezeta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4542.png"
 ALT="$\displaystyle \zeta_2$">|; 

$key = q/x=c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$ x=c$">|; 

$key = q/lambda<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2299.png"
 ALT="$ \lambda&lt;0$">|; 

$key = q/0.47-6,lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3204.png"
 ALT="$ 0.47-6 \lambda$">|; 

$key = q/displaystyle(widetilde{nabla{bfA}})_{rr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7147.png"
 ALT="$\displaystyle (\widetilde{\nabla {\bf A}})_{rr}$">|; 

$key = q/(X,,Y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1851.png"
 ALT="$ (X, Y)$">|; 

$key = q/theta=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2096.png"
 ALT="$ \theta=\pi$">|; 

$key = q/displaystylelambda=frac{kappa_y^{,2},A_0}{V_0}-z_{GH},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img655.png"
 ALT="$\displaystyle \lambda = \frac{\kappa_y^{ 2} A_0}{V_0}-z_{GH},$">|; 

$key = q/displaystyle=frac{sqrt{6}}{10},frac{1+cos^2beta}{(cosbeta,cosgamma)^{2slash3}},widehat{omega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img948.png"
 ALT="$\displaystyle = \frac{\sqrt{6}}{10} \frac{1+\cos^2\beta}{(\cos\beta \cos\gamma)^{2/3}} \widehat{\omega},$">|; 

$key = q/|zeta|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2325.png"
 ALT="$ \vert\zeta\vert$">|; 

$key = q/displaystyleBQ'-AP'=2,a,tantheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img736.png"
 ALT="$\displaystyle BQ' - AP' = 2 a \tan\theta.$">|; 

$key = q/2.7times10^{-3},{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1121.png"
 ALT="$ 2.7\times 10^{-3} {\rm m}$">|; 

$key = q/{multline}sum_{n=1,infty}left{z_nleft[frac{n,(n+1)}{beta},(1-[1+k_n'-h_n'],alphac{6}{beta},(1+k_2-h_2),skew{5}tilde{zeta}_j,P_2^{,m_j}(mu),{multline};LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="598" HEIGHT="147" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4685.png"
 ALT="\begin{multline}
\sum_{n=1,\infty}\left\{z_n\left[\frac{n (n+1)}{\beta} (1-[1+...
...}{\beta} (1+k_2-h_2) \skew{5}\tilde{\zeta}_j P_2^{ m_j}(\mu),
\end{multline}">|; 

$key = q/displaystyleoint_Srho,{bfv}cdotd{bfS}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3042.png"
 ALT="$\displaystyle \oint_S \rho {\bf v}\cdot d{\bf S}=0,$">|; 

$key = q/partial{calR}_nslashpartialx_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4327.png"
 ALT="$ \partial {\cal R}_n/\partial x_i$">|; 

$key = q/Y(theta,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4732.png"
 ALT="$ Y(\theta,\phi,t)$">|; 

$key = q/p+q=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3249.png"
 ALT="$ p+q=1$">|; 

$key = q/psi(x,y)=-U(x),delta(x),f[yslashdelta(x)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2916.png"
 ALT="$ \psi(x,y)=-U(x) \delta(x) f[y/\delta(x)]$">|; 

$key = q/F(lambda);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$ F(\lambda )$">|; 

$key = q/displaystyle=frac{4,q_1}{sqrt{{rmMa}_1^{,2}-1}}int_{x_L}^{x_R}alpha(x),dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5847.png"
 ALT="$\displaystyle = \frac{4 q_1}{\sqrt{{\rm Ma}_1^{ 2}-1}}\int_{x_L}^{x_R} \alpha(x) dx,$">|; 

$key = q/displaystyleQ=-frac{{mitDelta}P}{l},frac{1}{12,mu,langled^{,-3}rangle}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3674.png"
 ALT="$\displaystyle Q = -\frac{{\mit\Delta} P}{l} \frac{1}{12 \mu \langle d^{ -3}\rangle}.$">|; 

$key = q/thetasimeqHH'slashMH';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img643.png"
 ALT="$ \theta \simeq HH'/MH'$">|; 

$key = q/d{bfr}={bft},dr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1090.png"
 ALT="$ d{\bf r}= {\bf t} dr$">|; 

$key = q/displaystyle=-frac{partialu}{partialx}-frac{partialv}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5884.png"
 ALT="$\displaystyle =-\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y},$">|; 

$key = q/U_0,delta(L)slashnu<600;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3000.png"
 ALT="$ U_0 \delta(L)/\nu&lt;600$">|; 

$key = q/displaystyle=frac{V_0}{(g,L)^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="$\displaystyle = \frac{V_0}{(g L)^{1/2}},$">|; 

$key = q/({bfa}times{bfb})times{bfc}+({bfb}times{bfc})times{bfa}+({bfc}times{bfa})times{bfb}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="366" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3460.png"
 ALT="$ ({\bf a}\times{\bf b})\times {\bf c} + ({\bf b}\times{\bf c})\times{\bf a} + ({\bf c}\times {\bf a})\times {\bf b} = {\bf0}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter08slashvx.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img2986.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter08/vx.eps}}
\end{figure}">|; 

$key = q/displaystyleT(z)=T_0left(1-frac{gamma-1}{gamma},frac{z}{H}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5146.png"
 ALT="$\displaystyle T(z) = T_0\left(1- \frac{\gamma-1}{\gamma} \frac{z}{H}\right).$">|; 

$key = q/Q,Q'>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2435.png"
 ALT="$ Q Q'&gt;0$">|; 

$key = q/drslashdp>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img724.png"
 ALT="$ dr/dp&gt;0$">|; 

$key = q/displaystylefrac{partial}{partialtheta},(sintheta,X')+frac{partialY'}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4746.png"
 ALT="$\displaystyle \frac{\partial}{\partial\theta} (\sin\theta X') +\frac{\partial Y'}{\partial\phi}$">|; 

$key = q/displaystylevarpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2726.png"
 ALT="$\displaystyle \varpi$">|; 

$key = q/displaystyle{mitGamma}_C=oint_C{bfv}cdotd{bfr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1553.png"
 ALT="$\displaystyle {\mit\Gamma}_C = \oint_C {\bf v}\cdot d{\bf r}.$">|; 

$key = q/displaystyle|V_s|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5645.png"
 ALT="$\displaystyle \vert V_s\vert$">|; 

$key = q/600;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2998.png"
 ALT="$ 600$">|; 

$key = q/bar{F}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2264.png"
 ALT="$ \bar{F}=0$">|; 

$key = q/displaystyleu(x,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3149.png"
 ALT="$\displaystyle u(x,0)$">|; 

$key = q/m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1368.png"
 ALT="$ m$">|; 

$key = q/displaystyleD,Q=frac{1}{sintheta}left[frac{partial}{partialtheta},(sintheta,Y)-frac{partialX}{partialphi}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4737.png"
 ALT="$\displaystyle D Q =\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta} (\sin\theta Y) - \frac{\partial X}{\partial\phi}\right],$">|; 

$key = q/displaystyle=a_{ssi}+a_{sis},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6927.png"
 ALT="$\displaystyle =a_{ssi}+a_{sis},$">|; 

$key = q/beta_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5015.png"
 ALT="$ \beta_{n, n'}^{ m, m'}$">|; 

$key = q/deltal;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7304.png"
 ALT="$ \delta l$">|; 

$key = q/displaystylezeta(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4640.png"
 ALT="$\displaystyle \zeta(\theta)$">|; 

$key = q/displaystyle=frac{xi_1^{,4}}{16pi,y_1^{,2}},frac{G,M_ast^{,2}}{R^{,4}}=1.24times10^{16},{rmN,m^{-2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5245.png"
 ALT="$\displaystyle = \frac{\xi_1^{ 4}}{16\pi y_1^{ 2}} \frac{G M_\ast^{ 2}}{R^{ 4}}= 1.24\times 10^{16} {\rm N m^{-2}},$">|; 

$key = q/k,d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3992.png"
 ALT="$ k d$">|; 

$key = q/displaystylephi(xi,eta)=-V,a,eleft[frac{1-sinhxi,tan^{-1}(1slashsinhxi)}{e,(1-e^{,2})^{1slash2}-sin^{-1}e}right]coseta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2777.png"
 ALT="$\displaystyle \phi(\xi,\eta) =-V a e\left[\frac{1 -\sinh\xi \tan^{-1}(1/\sinh\xi)} {e (1-e^{ 2})^{1/2}-\sin^{-1}e}\right]\cos\eta.$">|; 

$key = q/r<;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4322.png"
 ALT="$ r&lt;$">|; 

$key = q/displaystylev'(x,0)=Uleft(frac{dy}{dx}right)_{rmbody},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5941.png"
 ALT="$\displaystyle v'(x,0)= U\left(\frac{dy}{dx}\right)_{\rm body},$">|; 

$key = q/{bfF}=-{bfr}slash|r^{,3}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6477.png"
 ALT="$ {\bf F} = - {\bf r}/ \vert r^{ 3}\vert$">|; 

$key = q/displaystylefrac{1}{V},frac{partialskew{3}bar{phi}}{partialy}=-lambdapmsqrt{lambda^{,2}-1}=frac{-1}{lambdapmsqrt{lambda^{,2}-1}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2298.png"
 ALT="$\displaystyle \frac{1}{V} \frac{\partial\skew{3}\bar{\phi}}{\partial y}=-\lambda\pm \sqrt{\lambda^{ 2}-1}=\frac{-1}{\lambda\pm\sqrt{\lambda^{ 2}-1}},$">|; 

$key = q/displaystyleH=frac{R,T_0}{g,M}=frac{p_0}{g,rho_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5142.png"
 ALT="$\displaystyle H = \frac{R T_0}{g M}=\frac{p_0}{g \rho_0}.$">|; 

$key = q/{mitDelta}w=w_2-w_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5746.png"
 ALT="$ {\mit\Delta}w=w_2-w_1$">|; 

$key = q/displaystylefrac{dv_x}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3556.png"
 ALT="$\displaystyle \frac{dv_x}{dt}$">|; 

$key = q/displaystylephantom{-frac{G,m',a^{2}}{r'^{,3}}}+frac{1}{3},P_2^{,1},(costheta'),cosvarphi',{calR}_2^{,(-1)}(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4281.png"
 ALT="$\displaystyle \phantom{ - \frac{G m' a^{2}}{r'^{ 3}}} +\frac{1}{3} P_2^{ 1} (\cos\theta') \cos\varphi' {\cal R}_2^{ (-1)}(r,\theta,\varphi)$">|; 

$key = q/drightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2449.png"
 ALT="$ d\rightarrow 0$">|; 

$key = q/2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4453.png"
 ALT="$ 2 $">|; 

$key = q/displaystyleleft.frac{partialphi}{partialz}right|_{z=-d}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3908.png"
 ALT="$\displaystyle \left.\frac{\partial \phi}{\partial z}\right\vert _{z=-d} = 0.$">|; 

$key = q/-3,e,zeta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4527.png"
 ALT="$ -3 e \zeta_2$">|; 

$key = q/displaystyledelta_1=frac{(pislash2-1),nu}{v_s}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3270.png"
 ALT="$\displaystyle \delta_1 = \frac{(\pi/2-1) \nu}{v_s}.
$">|; 

$key = q/f_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4712.png"
 ALT="$ f_j$">|; 

$key = q/C';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img968.png"
 ALT="$ C'$">|; 

$key = q/displaystyleV_0=int_{A_0}int_0^deltadx,dy,dz=int_{A_0},delta(x,y),dx,dy.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img631.png"
 ALT="$\displaystyle V_0 =\int_{A_0}\int_0^\delta dx dy dz= \int_{A_0} \delta (x,y) dx dy.$">|; 

$key = q/displaystylefrac{dhat{p}}{dhat{z}}=-hat{p}^{,1slashgamma},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5143.png"
 ALT="$\displaystyle \frac{d\hat{p}}{d\hat{z}} = - \hat{p}^{ 1/\gamma},$">|; 

$key = q/displaystyle=left(frac{rho}{rho_0}right)^gamma,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5879.png"
 ALT="$\displaystyle = \left(\frac{\rho}{\rho_0}\right)^\gamma,$">|; 

$key = q/displaystyleC_p=frac{2}{sqrt{{rmMa}_infty^{,2}-1}},frac{dy}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6000.png"
 ALT="$\displaystyle C_p =\frac{2}{\sqrt{{\rm Ma}_\infty^{ 2}-1}} \frac{dy}{dx},$">|; 

$key = q/displaystylefrac{1}{V},frac{dbar{F}}{dz}={rmi},lambda+sqrt{1-lambda^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2287.png"
 ALT="$\displaystyle \frac{1}{V} \frac{d\bar{F}}{dz}= {\rm i} \lambda+\sqrt{1-\lambda^{ 2}}.$">|; 

$key = q/displaystyle=c,coshxi,sineta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2754.png"
 ALT="$\displaystyle = c \cosh\xi \sin\eta.$">|; 

$key = q/displaystyleleft.frac{partialu}{partialy}right|_{y=0}=frac{U}{delta},f'(0)=frac{U}{delta}left(2+frac{{mitLambda}}{6}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3185.png"
 ALT="$\displaystyle \left.\frac{\partial u}{\partial y}\right\vert _{y=0} = \frac{U}{\delta} f'(0) = \frac{U}{\delta}\left(2 + \frac{{\mit\Lambda}}{6}\right).$">|; 

$key = q/P_n^{,m}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4319.png"
 ALT="$ P_n^{ m}(x)$">|; 

$key = q/displaystylec_p=M,{calC}_p=c_V+R.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5315.png"
 ALT="$\displaystyle c_p = M {\cal C}_p = c_V + R.$">|; 

$key = q/displaystyle=a_{sils}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6960.png"
 ALT="$\displaystyle = a_{sils}.$">|; 

$key = q/{bfe}_theta=nablathetaslash|nablatheta|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7110.png"
 ALT="$ {\bf e}_\theta = \nabla\theta/\vert\nabla\theta\vert$">|; 

$key = q/displaystylep=p_0-frac{1}{2},rholeft|frac{dF}{dz}right|^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2007.png"
 ALT="$\displaystyle p = p_0 -\frac{1}{2} \rho\left\vert\frac{dF}{dz}\right\vert^{ 2}.$">|; 

$key = q/xrightarrowx_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3123.png"
 ALT="$ x\rightarrow x_0$">|; 

$key = q/u_1slashc_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5684.png"
 ALT="$ u_1/c_1$">|; 

$key = q/bfb;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6321.png"
 ALT="$ \bf b$">|; 

$key = q/R_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img719.png"
 ALT="$ R_0$">|; 

$key = q/displaystyleint_a^bleft[frac{partialF}{partialy}-frac{d}{dx}!left(frac{partialF}]deltay,dx+left[frac{partialF}{partialy'},deltayright]_a^b=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7313.png"
 ALT="$\displaystyle \int_a^b\left[\frac{\partial F}{\partial y}- \frac{d}{dx}\!\left(...
...ight]\delta y dx +\left[\frac{\partial F}{\partial y'} \delta y\right]_a^b=0.$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{AppendixAslashfigA.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="198" BORDER="0"
 SRC="|."$dir".q|img6241.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.03.eps}}
\end{figure}">|; 

$key = q/overline{t}=(V_0slashL),t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="$ \overline{t} = (V_0/L) t$">|; 

$key = q/P_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1618.png"
 ALT="$ P_1$">|; 

$key = q/sigma_{21}=-(;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$ \sigma_{21}=-($">|; 

$key = q/n,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3875.png"
 ALT="$ n a$">|; 

$key = q/displaystyle=rho,V,{mitGamma}left(sinalpha,{bfe}_xi+cosalpha,{bfe}_etaright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3300.png"
 ALT="$\displaystyle = \rho V {\mit\Gamma}\left(\sin\alpha {\bf e}_\xi+\cos\alpha {\bf e}_\eta\right),$">|; 

$key = q/theta_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5013.png"
 ALT="$ \theta_{n, n'}^{ m, m'}$">|; 

$key = q/displaystylev_x(x,y)=frac{G(x)}{2,mu},y,[d(x)-y].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3665.png"
 ALT="$\displaystyle v_x(x,y) = \frac{G(x)}{2 \mu} y [d(x)-y].$">|; 

$key = q/displaystylefrac{v_g}{v_p}=frac{1}{2}left[1+frac{2,k,d}{sinh(2,k,d)}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3965.png"
 ALT="$\displaystyle \frac{v_g}{v_p} =\frac{1}{2}\left[1+\frac{2 k d}{\sinh(2 k d)}\right].$">|; 

$key = q/displaystylefrac{rho,v_x,d^{,2}}{mu,l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3661.png"
 ALT="$\displaystyle \frac{\rho v_x d^{ 2}}{\mu l}$">|; 

$key = q/displaystylenu,frac{partial^{,3}psi}{partialy^{,3}}-frac{partialpsi}{partialx},fialpsi}{partialy},frac{partial^{,2}psi}{partialx,partialy}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3242.png"
 ALT="$\displaystyle \nu \frac{\partial^{ 3}\psi}{\partial y^{ 3}}-\frac{\partial\p...
...rtial\psi}{\partial y} \frac{\partial^{ 2}\psi}{\partial x \partial y} = 0.
$">|; 

$key = q/20^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1146.png"
 ALT="$ 20^\circ$">|; 

$key = q/{rmM}_{,e}<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5423.png"
 ALT="$ {\rm M}_{ e}&lt;1$">|; 

$key = q/displaystyle=a_{sjs},delta_{ik}+a_{iss},delta_{jk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6920.png"
 ALT="$\displaystyle =a_{sjs} \delta_{ik} + a_{iss} \delta_{jk},$">|; 

$key = q/displaystyle=frac{partialxi}{partialx},frac{partial}{partialxi}+frac{partialeta}partialx},frac{partial}{partialeta}=frac{partial}{partialxi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6015.png"
 ALT="$\displaystyle = \frac{\partial\xi}{\partial x} \frac{\partial}{\partial\xi} +\...
...\eta}{\partial x} \frac{\partial}{\partial\eta} =\frac{\partial}{\partial\xi},$">|; 

$key = q/displaystyle=nabla,(nablacdot{bfA})-nabla^{,2}{bfA},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6694.png"
 ALT="$\displaystyle = \nabla (\nabla\cdot{\bf A})- \nabla^{ 2}{\bf A},$">|; 

$key = q/f'(eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2960.png"
 ALT="$ f'(\eta)$">|; 

$key = q/displaystyle=int_{mitOmega}left(frac{partial{mitPhi}'}{partialtheta},frac{partiaialphi},frac{partial{mitPhi}_r}{partialphi}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4898.png"
 ALT="$\displaystyle =\int_{\mit\Omega} \left(\frac{\partial {\mit\Phi}'}{\partial\the...
...}{\partial\phi} \frac{\partial{\mit\Phi}_r}{\partial\phi}\right)d{\mit\Omega},$">|; 

$key = q/displaystyle{bff}(theta)=-frac{3,mu,V}{2,a},{bfe}_z+left(-p_0+rho,g,a,costhetaright){bfe}_r.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3805.png"
 ALT="$\displaystyle {\bf f}(\theta) = -\frac{3 \mu V}{2 a} {\bf e}_z + \left(-p_0+\rho g a \cos\theta\right){\bf e}_r.$">|; 

$key = q/displaystylev_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1646.png"
 ALT="$\displaystyle v_\theta$">|; 

$key = q/displaystylec^{,2}=c_infty^{,2}-frac{1}{2},(gamma-1)left(2,U,u'+u'^{,2}+v'^{,2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5902.png"
 ALT="$\displaystyle c^{ 2} = c_\infty^{ 2}-\frac{1}{2} (\gamma-1)\left(2 U u'+u'^{ 2}+v'^{ 2}\right).$">|; 

$key = q/mu,epsilon_{ijk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6897.png"
 ALT="$ \mu \epsilon_{ijk}$">|; 

$key = q/u(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5449.png"
 ALT="$ u(x)$">|; 

$key = q/displaystyle=({bfa}cdot{bfc}),({bfb}cdot{bfd})-({bfa}cdot{bfd}),({bfb}cdot{bfc}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6678.png"
 ALT="$\displaystyle = ({\bf a}\cdot{\bf c}) ({\bf b}\cdot{\bf d}) - ({\bf a}\cdot{\bf d}) ({\bf b}\cdot{\bf c}),$">|; 

$key = q/D<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6098.png"
 ALT="$ D&lt;0$">|; 

$key = q/c_fproptof''(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2970.png"
 ALT="$ c_f\propto f''(0)$">|; 

$key = q/displaystylelambda(x)=frac{delta_2^{,2}}{nu},frac{dU}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3187.png"
 ALT="$\displaystyle \lambda(x) = \frac{\delta_2^{ 2}}{\nu} \frac{dU}{dx},$">|; 

$key = q/displaystyle{rmFr}^{,2}left[left(1+frac{h-d}{H}right)^2-1right]=2,frac{h}{H}left(1+frac{h-d}{H}right)^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1459.png"
 ALT="$\displaystyle {\rm Fr}^{ 2}\left[\left(1+\frac{h-d}{H}\right)^2-1\right] = 2 \frac{h}{H}\left(1+\frac{h-d}{H}\right)^{ 2}.$">|; 

$key = q/v_x=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2978.png"
 ALT="$ v_x=0$">|; 

$key = q/psi(a,theta)=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2045.png"
 ALT="$ \psi(a,\theta)=$">|; 

$key = q/displaystylelambda=frac{kappa_x^{,2},A_0}{V_0}-z_{GH},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img652.png"
 ALT="$\displaystyle \lambda = \frac{\kappa_x^{ 2} A_0}{V_0}-z_{GH},$">|; 

$key = q/displaystyle=left(frac{2}{gamma+1}right),frac{c_0}{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5676.png"
 ALT="$\displaystyle = \left(\frac{2}{\gamma+1}\right) \frac{c_0}{a},$">|; 

$key = q/displaystylefrac{partialeta}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6012.png"
 ALT="$\displaystyle \frac{\partial\eta}{\partial x}$">|; 

$key = q/M_ast=m(R);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5225.png"
 ALT="$ M_\ast= m(R)$">|; 

$key = q/z=bslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3401.png"
 ALT="$ z=b/2$">|; 

$key = q/displaystylephantom{=}left.+frac{sinalpha,sinbeta,cosbeta,cosgamma}{cos^2alpha}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1009.png"
 ALT="$\displaystyle \phantom{=}\left.+ \frac{\sin\alpha \sin\beta \cos\beta \cos\gamma}{\cos^2\alpha}\right],$">|; 

$key = q/displaystylea_{ij}=frac{a_{ss}}{3},delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6906.png"
 ALT="$\displaystyle a_{ij} = \frac{a_{ss}}{3} \delta_{ij},$">|; 

$key = q/displaystylep_{rmmax}=left{{array}{lll}(n-m)slash2&mbox{hspace{0.5cm}}&mbox{n-me](n-m-1)slash2&mbox{hspace{0.5cm}}&mbox{n-modd}{array}right.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="371" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4972.png"
 ALT="$\displaystyle p_{\rm max} = \left\{ \begin{array}{lll} (n-m)/2&amp;\mbox{\hspace{0....
... [0.5ex] (n-m-1)/2&amp;\mbox{\hspace{0.5cm}}&amp;\mbox{$n-m$ odd} \end{array}\right.,$">|; 

$key = q/360^circslashomega_1=365.24,{rmdays};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4603.png"
 ALT="$ 360^\circ/\omega_1= 365.24 {\rm days}$">|; 

$key = q/displaystyleint_{-bslash2}^{bslash2}{mitGamma}(z),dz=frac{{mitGamma}_0,b}{2}int_0^pisin^2phi,dphi=frac{pi}{4},{mitGamma}_0,b.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3507.png"
 ALT="$\displaystyle \int_{-b/2}^{b/2} {\mit\Gamma}(z) dz = \frac{{\mit\Gamma}_0 b}{2}\int_0^\pi\sin^2\phi d\phi = \frac{\pi}{4} {\mit\Gamma}_0 b.$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter04slashborda.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="281" BORDER="0"
 SRC="|."$dir".q|img1420.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter04/borda.eps}}
\end{figure}">|; 

$key = q/displaystylep_{1,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5469.png"
 ALT="$\displaystyle p_{1,2}$">|; 

$key = q/a_{ij},b_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6776.png"
 ALT="$ a_{ij} b_j$">|; 

$key = q/displaystyle-m,k,B_1left[A_1,sin(k,x)+A_2,cos(k,x)right]=-U,epsilon,alpha,cos(alpha,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5971.png"
 ALT="$\displaystyle -m k B_1\left[A_1 \sin(k x) +A_2 \cos(k x)\right] = -U \epsilon \alpha \cos(\alpha x).$">|; 

$key = q/displaystylepsi(xi_0,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2733.png"
 ALT="$\displaystyle \psi(\xi_0,\eta)$">|; 

$key = q/displaystyle{rmFr}=frac{V}{sqrt{g,H}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1460.png"
 ALT="$\displaystyle {\rm Fr} = \frac{V}{\sqrt{g H}}$">|; 

$key = q/Q=(a,0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6480.png"
 ALT="$ Q=(a,0,0)$">|; 

$key = q/displaystylesimfrac{1}{x},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3015.png"
 ALT="$\displaystyle \sim \frac{1}{x},$">|; 

$key = q/displaystyleD=rho,U_0^{,2}left(frac{nu}{U_0}right)^{1slash2}sqrt{2},f''(0)int_0^,U_0^{,2}left(frac{nu,L}{U_0}right)^{1slash2}2sqrt{2},f''(0).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="567" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2994.png"
 ALT="$\displaystyle D = \rho U_0^{ 2}\left(\frac{\nu}{U_0}\right)^{1/2}\sqrt{2} f'...
.../2} dx= \rho U_0^{ 2}\left(\frac{\nu L}{U_0}\right)^{1/2}2\sqrt{2} f''(0).$">|; 

$key = q/widehat{L}=Lslash(G,M^{,3},a_0)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img893.png"
 ALT="$ \widehat{L}=L/(G M^{ 3} a_0)^{1/2}$">|; 

$key = q/U_0-u(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3044.png"
 ALT="$ U_0-u(y)$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter04slashvortex.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="283" BORDER="0"
 SRC="|."$dir".q|img1536.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter04/vortex.eps}}
\end{figure}">|; 

$key = q/displaystyleoint{bfA}cdotd{bfr}={bfcurl},{bfA}cdotd{bfS}=|{bfcurl},{bfA}|,|d{bfS}|,costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6644.png"
 ALT="$\displaystyle \oint {\bf A}\cdot d{\bf r} = {\bf curl} {\bf A} \cdot d{\bf S} = \vert{\bf curl} {\bf A}\vert \vert d {\bf S}\vert \cos\theta,$">|; 

$key = q/deltaI=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7336.png"
 ALT="$ \delta I = 0$">|; 

$key = q/displaystyle=-frac{nablap}{rho}-nabla{mitPsi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img549.png"
 ALT="$\displaystyle = - \frac{\nabla p}{\rho} - \nabla{\mit\Psi},$">|; 

$key = q/v=Vslashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5288.png"
 ALT="$ v=V/m$">|; 

$key = q/displaystyle=-p,dv+T,d{calS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5334.png"
 ALT="$\displaystyle =-p dv+T d{\cal S},$">|; 

$key = q/displaystyle=l,exp(cosh^{-1}p),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2137.png"
 ALT="$\displaystyle = l \exp(\cosh^{-1}p),$">|; 

$key = q/displaystyle=left[1-frac{1}{2}(gamma-1)left(frac{v}{c_0}right)^2right]^{,1slash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5619.png"
 ALT="$\displaystyle = \left[1-\frac{1}{2}(\gamma-1)\left(\frac{v}{c_0}\right)^2\right]^{ 1/(\gamma-1)},$">|; 

$key = q/displaystyle{bfF}=-rho,Q,{bfV}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2426.png"
 ALT="$\displaystyle {\bf F} =- \rho Q {\bf V}.$">|; 

$key = q/displaystyleF=-V_0left(zeta+frac{c^{,2}}{zeta}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2131.png"
 ALT="$\displaystyle F = - V_0\left(\zeta+\frac{c^{ 2}}{\zeta}\right),$">|; 

$key = q/dleq0.6627,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1284.png"
 ALT="$ d\leq 0.6627 c$">|; 

$key = q/displaystyleleft(frac{dF}{dz}right)^{,2}=V^{,2},{rme}^{,2,{rmi},alpha}+{rmi},fra,{rmi},alpha}-{mitGamma}^{,2}}{4pi^2,z^{,2}}+{calO}(z^{,-3}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="513" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3285.png"
 ALT="$\displaystyle \left(\frac{dF}{dz}\right)^{ 2} = V^{ 2} {\rm e}^{ 2 {\rm i}...
...^{ {\rm i} \alpha}-{\mit\Gamma}^{ 2}}{4\pi^2 z^{ 2}} + {\cal O}(z^{ -3}).$">|; 

$key = q/displaystyle=-int_{S}left(frac{partialpsi}{partialx}+frac{partialphi}{partialy}right)dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2369.png"
 ALT="$\displaystyle =-\int_{S}\left(\frac{\partial\psi}{\partial x} + \frac{\partial\phi}{\partial y}\right)dS,$">|; 

$key = q/W,r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img716.png"
 ALT="$ W r$">|; 

$key = q/displaystyle=-frac{1}{3},b-sqrt{-Q}left(cosphi-sqrt{3},sinphiright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6108.png"
 ALT="$\displaystyle = -\frac{1}{3} b-\sqrt{-Q}\left(\cos\phi-\sqrt{3} \sin\phi\right),$">|; 

$key = q/displaystyle{calF}(theta,phi,t)=(1+k_2-h_2),skew{5}bar{zeta}_2(theta,phi,t)+ointG_F(theta,phi;theta',phi'),zeta(theta',phi',t),d{mitOmega}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="564" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4729.png"
 ALT="$\displaystyle {\cal F}(\theta,\phi,t)=(1+k_2-h_2) \skew{5}\bar{\zeta}_2(\theta...
...)+\oint G_F(\theta,\phi;\theta',\phi') \zeta(\theta',\phi',t) d{\mit\Omega}'.$">|; 

$key = q/dyslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5938.png"
 ALT="$ dy/dx$">|; 

$key = q/{bfr}equiv,stackrel{displaystylerightarrow}{OP};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="30" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6281.png"
 ALT="$ {\bf r} \equiv  \stackrel{\displaystyle \rightarrow}{OP}$">|; 

$key = q/-0.0904;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$ -0.0904$">|; 

$key = q/displaystyleleft.frac{partialphi}{partialr}right|_{r=a}=-V_x,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1802.png"
 ALT="$\displaystyle \left.\frac{\partial\phi}{\partial r}\right\vert _{r=a} = - V_x \cos\theta.$">|; 

$key = q/z<f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1155.png"
 ALT="$ z&lt;f(x)$">|; 

$key = q/dV=h,du_1,du_2,du_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7054.png"
 ALT="$ dV = h du_1 du_2 du_3$">|; 

$key = q/displaystyle=chi+frac{kappa,M}{R},nabla^{,2}left(frac{p}{rho}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="$\displaystyle =\chi + \frac{\kappa M}{R} \nabla^{ 2}\left(\frac{p}{\rho}\right),$">|; 

$key = q/displaystyle{bfv}=V_0,costheta_0,{bfe}_x+V_0,sintheta_0,{bfe}_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1663.png"
 ALT="$\displaystyle {\bf v} = V_0 \cos\theta_0 {\bf e}_x + V_0 \sin\theta_0 {\bf e}_y,$">|; 

$key = q/zeta=c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2156.png"
 ALT="$ \zeta=c$">|; 

$key = q/G(1)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5497.png"
 ALT="$ G(1)=1$">|; 

$key = q/displaystyleD=f,sinalpha=left(frac{2pi,sin^2alpha}{pi,sinalpha+4}right)rho,V^{,2},b.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2337.png"
 ALT="$\displaystyle D = f \sin\alpha = \left(\frac{2\pi \sin^2\alpha}{\pi \sin\alpha+4}\right)\rho V^{ 2} b.$">|; 

$key = q/E(phi,k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1309.png"
 ALT="$ E(\phi,k)$">|; 

$key = q/displaystyle=epsilon_{lmn}-deltatheta_aleft(delta_{ma},delta_{nl}-delta_{ml},del,delta_{la}+delta_{la},delta_{mn}-delta_{ln},delta_{ma}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="557" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6838.png"
 ALT="$\displaystyle = \epsilon_{lmn}-\delta\theta_a\left(\delta_{ma} \delta_{nl}-\de...
...elta_{nm} \delta_{la}+\delta_{la} \delta_{mn}-\delta_{ln} \delta_{ma}\right)$">|; 

$key = q/dbar{F}=dphi-{rmi},dpsi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2404.png"
 ALT="$ d\bar{F}=d\phi-{\rm i} d\psi$">|; 

$key = q/sgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1860.png"
 ALT="$ s\gg 1$">|; 

$key = q/displaystylephi(r,theta,t)rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1799.png"
 ALT="$\displaystyle \phi(r,\theta,t)\rightarrow 0$">|; 

$key = q/displaystyle=left(frac{1}{r},frac{partialA_z}{partialtheta}-frac{partialA_theta}heta)-frac{1}{r},frac{partialA_r}{partialtheta}right){bfe}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="594" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7131.png"
 ALT="$\displaystyle = \left(\frac{1}{r} \frac{\partial A_z}{\partial \theta}-\frac{\...
... A_\theta) - \frac{1}{r} \frac{\partial A_r}{\partial\theta}\right){\bf e}_z,$">|; 

$key = q/r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="$ r$">|; 

$key = q/displaystylebeta_{r,,-s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4875.png"
 ALT="$\displaystyle \beta_{r, -s}$">|; 

$key = q/displaystyle=1leftslashleft(frac{partialY}{partialX}right)_Zright.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5574.png"
 ALT="$\displaystyle = 1\left/\left(\frac{\partial Y}{\partial X}\right)_Z\right.,$">|; 

$key = q/displaystyleG=rho,g,sinalpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3592.png"
 ALT="$\displaystyle G = \rho g \sin\alpha.$">|; 

$key = q/e_{12}=0.8594;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1034.png"
 ALT="$ e_{12}=0.8594$">|; 

$key = q/c=[(gslashk),tanh(k,d)]^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4129.png"
 ALT="$ c=[(g/k) \tanh(k d)]^{1/2}$">|; 

$key = q/displaystylesimeq-frac{3}{5}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img907.png"
 ALT="$\displaystyle \simeq -\frac{3}{5}.$">|; 

$key = q/xrightarrow-x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1192.png"
 ALT="$ x\rightarrow -x$">|; 

$key = q/displaystyleP_n(x)=frac{1}{2^{,n},n!},frac{d^{,n}}{dx^{,n}},[(x^{,2}-1)^n],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4237.png"
 ALT="$\displaystyle P_n(x)=\frac{1}{2^{ n} n!} \frac{d^{ n}}{dx^{ n}} [(x^{ 2}-1)^n],$">|; 

$key = q/displaystyleu({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5080.png"
 ALT="$\displaystyle u({\bf r},t)$">|; 

$key = q/displaystyley_C(x)=frac{y_U(x)+y_L(x)}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5823.png"
 ALT="$\displaystyle y_C(x)=\frac{y_U(x)+y_L(x)}{2},$">|; 

$key = q/displaystylefrac{d{bfa}}{dt}equivleft(frac{da_x}{dt},frac{da_y}{dt},frac{da_z}{dt}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6434.png"
 ALT="$\displaystyle \frac{d {\bf a}}{dt} \equiv \left(\frac{d a_x}{dt}, \frac{d a_y}{d t}, \frac{d a_z}{ d t}\right).$">|; 

$key = q/displaystylev_p=frac{omega}{k}=left(frac{g}{k}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3929.png"
 ALT="$\displaystyle v_p = \frac{\omega}{k} = \left(\frac{g}{k}\right)^{1/2}.$">|; 

$key = q/partialslashpartialxll1slashdelta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3154.png"
 ALT="$ \partial/\partial x\ll 1/\delta$">|; 

$key = q/4.63times10^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4620.png"
 ALT="$ 4.63\times 10^{-1}$">|; 

$key = q/displaystyle{mitOmega}(r_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3628.png"
 ALT="$\displaystyle {\mit\Omega}(r_1)$">|; 

$key = q/displaystyledh=frac{partialh}{partialx},dx+frac{partialh}{partialy},dy.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6535.png"
 ALT="$\displaystyle dh = \frac{\partial h}{\partial x}  dx +\frac{\partial h} {\partial y}  dy.$">|; 

$key = q/displaystylefrac{partialomega}{partialt}+[psi,omega];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1923.png"
 ALT="$\displaystyle \frac{\partial\omega}{\partial t} +[\psi,\omega]$">|; 

$key = q/displaystylealpha_{n,p}^{,m}=frac{(-1)^{m+p},(n+m)!}{2^{,m+2p},(m+p)!,p!,(n-m-2p)!},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4971.png"
 ALT="$\displaystyle \alpha_{n,p}^{ m} = \frac{(-1)^{m+p} (n+m)!}{2^{ m+2p} (m+p)! p! (n-m-2p)!},$">|; 

$key = q/displaystyleQ=int_0^d,v_x,dy=frac{G(x),d^{,3}(x)}{12,mu}+frac{1}{2},U,d(x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3698.png"
 ALT="$\displaystyle Q = \int_0^d v_x dy =\frac{G(x) d^{ 3}(x)}{12 \mu} + \frac{1}{2} U d(x).$">|; 

$key = q/displaystylek,dll1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3950.png"
 ALT="$\displaystyle k d\ll 1,$">|; 

$key = q/displaystylenabla^{,2}{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3633.png"
 ALT="$\displaystyle \nabla^{ 2} {\bf v}$">|; 

$key = q/v_-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1435.png"
 ALT="$ v_-$">|; 

$key = q/cosmu=({rmMa}_1^{,2}-1)^{1slash2}slash{rmMa}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5735.png"
 ALT="$ \cos\mu= ({\rm Ma}_1^{ 2}-1)^{1/2}/{\rm Ma}_1$">|; 

$key = q/d(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1526.png"
 ALT="$ d(0)$">|; 

$key = q/zeta=b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2155.png"
 ALT="$ \zeta=b$">|; 

$key = q/F(v);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1430.png"
 ALT="$ F(v)$">|; 

$key = q/{calE}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5442.png"
 ALT="$ {\cal E}(x)$">|; 

$key = q/displaystyle{rmMa}_1simeqsqrt{left(frac{gamma+1}{2,gamma}right)frac{{mitDelta}p}{p_1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5523.png"
 ALT="$\displaystyle {\rm Ma}_1\simeq \sqrt{\left(\frac{\gamma+1}{2 \gamma}\right)\frac{{\mit\Delta} p}{p_1}}.$">|; 

$key = q/displaystylefrac{dbar{F}}{dz}=frac{1}{v_x-{rmi},v_y}simeqfrac{1}{V}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2227.png"
 ALT="$\displaystyle \frac{d\bar{F}}{dz} =\frac{1}{v_x-{\rm i} v_y} \simeq \frac{1}{V}.$">|; 

$key = q/displaystyle=-frac{partial{mitPhi}_r''}{partialtheta}-frac{1}{sintheta},frac{partial{mitPsi}_r'}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4861.png"
 ALT="$\displaystyle = - \frac{\partial{\mit\Phi}_r''}{\partial\theta} -\frac{1}{\sin\theta} \frac{\partial{\mit\Psi}_r'}{\partial\phi},$">|; 

$key = q/displaystyle{mitPsi}(theta,phi,t)=sum_{r=1,infty}p_{-r}(t),{mitPsi}_r(theta,phi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4891.png"
 ALT="$\displaystyle {\mit\Psi}(\theta,\phi,t) = \sum_{r=1,\infty} p_{-r}(t) {\mit\Psi}_r(\theta,\phi).$">|; 

$key = q/{bfS}^T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7144.png"
 ALT="$ {\bf S}^T$">|; 

$key = q/displaystylec_2=c_1-frac{1}{2},(gamma-1),V_p.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5560.png"
 ALT="$\displaystyle c_2 = c_1-\frac{1}{2} (\gamma-1) V_p.$">|; 

$key = q/partialpslashpartialt=partial{mitPsi}slashpartialt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img550.png"
 ALT="$ \partial p/\partial t=
\partial{\mit\Psi}/\partial t=0$">|; 

$key = q/sigma_{22};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$ \sigma_{22}$">|; 

$key = q/displaystylesigma_{xx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="$\displaystyle \sigma_{xx}$">|; 

$key = q/(-);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$ (-)$">|; 

$key = q/displaystyle{bfS}={bfa}times{bfb},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6381.png"
 ALT="$\displaystyle {\bf S} = {\bf a}\times {\bf b},$">|; 

$key = q/displaystylefrac{dA}{A}=frac{du}{u},({rmMa}^{,2}-1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5625.png"
 ALT="$\displaystyle \frac{dA}{A} =\frac{du}{u} ({\rm Ma}^{ 2}-1).
$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter11slashfroude.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="342" BORDER="0"
 SRC="|."$dir".q|img4004.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter11/froude.eps}}
\end{figure}">|; 

$key = q/displaystyleu,frac{partialu}{partialx}+v,frac{partialu}{partialy}-U,frac{dU}{dx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3074.png"
 ALT="$\displaystyle u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} - U \frac{dU}{dx}$">|; 

$key = q/PQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img729.png"
 ALT="$ PQ$">|; 

$key = q/{bfa}times({bfb}times{bfc})neq({bfa}times{bfb})times{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6420.png"
 ALT="$ {\bf a}\times({\bf b} \times {\bf c}) \neq ({\bf a}\times{\bf b} )\times {\bf c}$">|; 

$key = q/displaystylee_{ij}=frac{1}{2}left(frac{partialv_i}{partialx_j}+frac{partialv_j}{partialx_i}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="$\displaystyle e_{ij} = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right)$">|; 

$key = q/displaystyle=-ointP,costheta,a,dtheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1782.png"
 ALT="$\displaystyle = -\oint P \cos\theta a d\theta,$">|; 

$key = q/displaystylekappa_x=left(frac{int_{A_0}y^2,dx,dy}{A_0}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img640.png"
 ALT="$\displaystyle \kappa_x = \left(\frac{\int_{A_0}y^2 dx dy}{A_0}\right)^{1/2},$">|; 

$key = q/h_eta=|nablaeta|^{,-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2692.png"
 ALT="$ h_\eta=\vert\nabla\eta\vert^{ -1}$">|; 

$key = q/displaystyle=-p+2,mu,frac{partialv_y}{partialy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="$\displaystyle = -p + 2 \mu \frac{\partial v_y}{\partial y},$">|; 

$key = q/hat{t}=tslasht_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3338.png"
 ALT="$ \hat{t}=t/t_0$">|; 

$key = q/displaystyle(mu,nabla^{,2}{bfv})_rsimmu,frac{partial^{,2}v_r}{partialr^{,2}}simfrac{mu,V,a}{r^{,3}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3830.png"
 ALT="$\displaystyle (\mu \nabla^{ 2}{\bf v})_r \sim \mu \frac{\partial^{ 2} v_r}{\partial r^{ 2}}\sim \frac{\mu V a}{r^{ 3}}.$">|; 

$key = q/displaystyle=frac{1}{r},frac{partial}{partialr},(r,A_r)+frac{1}{r},frac{partialA_theta}{partialtheta}+frac{partialA_z}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7130.png"
 ALT="$\displaystyle =\frac{1}{r} \frac{\partial}{\partial r} (r A_r) + \frac{1}{r} \frac{\partial A_\theta}{\partial\theta} + \frac{\partial A_z}{\partial z},$">|; 

$key = q/displaystyle+2int_{-l}^lrho,U_0,v(x),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3058.png"
 ALT="$\displaystyle + 2\int_{-l}^l \rho U_0 v(x) dx$">|; 

$key = q/zeta>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2003.png"
 ALT="$ \zeta&gt;1$">|; 

$key = q/displaystyley_L(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5835.png"
 ALT="$\displaystyle y_L(x)$">|; 

$key = q/displaystylef(a)=left.frac{df}{dr}right|_{r=a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3769.png"
 ALT="$\displaystyle f(a)=\left.\frac{df}{dr}\right\vert _{r=a}$">|; 

$key = q/1.8times10^{-5},{rmkgslash(m,s)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="$ 1.8 \times 10^{-5}  {\rm kg/(m s)}$">|; 

$key = q/displaystyleleft({array}{ccc}sigma_{11}'-frac{1}{3}sigma_{ii},&0,&0[0.5ex]0,&sig&0[0.5ex]0,&0,&sigma_{33}'-frac{1}{3}sigma_{ii}{array}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="98" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="$\displaystyle \left( \begin{array}{ccc} \sigma_{11}'-\frac{1}{3}\sigma_{ii},&amp; 0...
...}, &amp; 0 \ [0.5ex] 0,&amp;0,&amp;\sigma_{33}'-\frac{1}{3}\sigma_{ii} \end{array}\right).$">|; 

$key = q/displaystylefrac{Q}{U_0,delta}=0.664left(frac{L}{x}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3062.png"
 ALT="$\displaystyle \frac{Q}{U_0 \delta} = 0.664\left(\frac{L}{x}\right)^{1/2}.$">|; 

$key = q/+(3slash2),sin^2delta,zeta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4529.png"
 ALT="$ +(3/2) \sin^2\delta \zeta_2$">|; 

$key = q/vsimu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3100.png"
 ALT="$ v\sim u$">|; 

$key = q/displaystylezeta({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5078.png"
 ALT="$\displaystyle \zeta({\bf r},t)$">|; 

$key = q/displaystylefrac{dlnT}{dlnrho}<gamma-1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5170.png"
 ALT="$\displaystyle \frac{d\ln T}{d\ln\rho} &lt; \gamma-1.$">|; 

$key = q/displaystylefrac{T}{T_0}=left[1+frac{1}{2},(gamma-1),{rmMa}^{,2}right]^{-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5383.png"
 ALT="$\displaystyle \frac{T}{T_0}=\left[1+\frac{1}{2} (\gamma-1) {\rm Ma}^{ 2}\right]^{-1},$">|; 

$key = q/z=z_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1242.png"
 ALT="$ z=z_1$">|; 

$key = q/x=b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7315.png"
 ALT="$ x=b$">|; 

$key = q/displaystylefrac{partialu}{partialx}+frac{partialv}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3073.png"
 ALT="$\displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}$">|; 

$key = q/r=r_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3624.png"
 ALT="$ r=r_1$">|; 

$key = q/displaystylefrac{1}{r},frac{d}{dr}!left(r,frac{dv_z}{dr}right)=-frac{G}{mu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3612.png"
 ALT="$\displaystyle \frac{1}{r} \frac{d}{dr}\!\left(r \frac{d v_z}{dr}\right) = -\frac{G}{\mu},$">|; 

$key = q/C=(0,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img975.png"
 ALT="$ C=(0, 0, 0)$">|; 

$key = q/delta{bfv}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1613.png"
 ALT="$ \delta {\bf v}={\bf0}$">|; 

$key = q/s={rmsgn}(f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5103.png"
 ALT="$ s={\rm sgn}(f)$">|; 

$key = q/displaystyle{bfv}=v_x(x,y,t),{bfe}_x+v_y(x,y,t),{bfe}_y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img533.png"
 ALT="$\displaystyle {\bf v} = v_x(x,y,t) {\bf e}_x + v_y(x,y,t) {\bf e}_y.
$">|; 

$key = q/displaystyleh=cleft[frac{cos(pi,nslash2)+1}{sin(pi,nslash2)}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2484.png"
 ALT="$\displaystyle h = c\left[\frac{\cos(\pi n/2)+1}{\sin(\pi n/2)}\right],
$">|; 

$key = q/psi(r)=(1slash4),omega,(r^{,2}-a^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1913.png"
 ALT="$ \psi(r)=(1/4) \omega (r^{ 2}-a^{ 2})$">|; 

$key = q/displaystyleint_0^pifrac{dphi'}{cosphi'-cosphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3494.png"
 ALT="$\displaystyle \int_0^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}$">|; 

$key = q/displaystyleomega^{,2}=frac{W,lambda}{I}=frac{g}{s,c},frac{c_0^{,2}-4,s,(1-s),c^{,2}}{c_0^{,2}+(8slash3),[(1-s)^3+s^{,3}],c^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img686.png"
 ALT="$\displaystyle \omega^{ 2} = \frac{W \lambda}{I}= \frac{g}{s c} \frac{c_0^{ 2}-4 s (1-s) c^{ 2}}{c_0^{ 2}+(8/3) [(1-s)^3+s^{ 3}] c^{ 2}}.$">|; 

$key = q/zeta=xi+{rmi},eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2051.png"
 ALT="$ \zeta=\xi+{\rm i} \eta$">|; 

$key = q/{rmFr}={rmFr}_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4006.png"
 ALT="$ {\rm Fr}= {\rm Fr}_c$">|; 

$key = q/displaystylea_y=-left(frac{s-1}{s+1}right)g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1858.png"
 ALT="$\displaystyle a_y = -\left(\frac{s-1}{s+1}\right)g,$">|; 

$key = q/displaystyleint_{mitOmega}{mitPhi}_r,{mitPhi}_{r'},d{mitOmega}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4817.png"
 ALT="$\displaystyle \int_{\mit\Omega} {\mit\Phi}_r {\mit\Phi}_{r'} d{\mit\Omega} = 0.$">|; 

$key = q/displaystylefrac{dF}{dz}=~_{lim|deltaz|rightarrowinfty}frac{F(z+deltaz)-F(z)}{deltaz}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1957.png"
 ALT="$\displaystyle \frac{dF}{dz} =&nbsp;_{\lim \vert\delta z\vert\rightarrow\infty} \frac{F(z+\delta z) - F(z) }{\delta z}.$">|; 

$key = q/displaystyleC_{pL}=-C_{p0},pi^{-1},cos^{-1}left(1-2sqrt{{rmMa}_infty^{,2}-1},frac{y}{x}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6069.png"
 ALT="$\displaystyle C_{pL} =- C_{p0} \pi^{-1} \cos^{-1}\left(1-2\sqrt{{\rm Ma}_\infty^{ 2}-1} \frac{y}{x}\right)$">|; 

$key = q/displaystyleu(x,pminfty)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3023.png"
 ALT="$\displaystyle u(x,\pm\infty) = 0$">|; 

$key = q/displaystyle=-m^{,2},{mitTheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1735.png"
 ALT="$\displaystyle = -m^{ 2} {\mit\Theta},$">|; 

$key = q/displaystyle=-frac{g,d}{a^{,2},sintheta},frac{partial(zeta-{calF})}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4658.png"
 ALT="$\displaystyle =- \frac{g d}{a^{  2} \sin\theta} \frac{\partial (\zeta-{\cal F})}{\partial\phi}$">|; 

$key = q/displaystylev(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1530.png"
 ALT="$\displaystyle v(0)$">|; 

$key = q/a_y=-g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1861.png"
 ALT="$ a_y=-g$">|; 

$key = q/displaystylesimeqleft(frac{gamma+1}{2}right){rmMa}_p.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5546.png"
 ALT="$\displaystyle \simeq \left(\frac{\gamma+1}{2}\right){\rm Ma}_p.$">|; 

$key = q/displaystyle=int_0^a(a^{,2}-2,a,z+z^{,2}),dz=left[a^{,2},z-a,z^{,2}+z^{,3}slash3right]_0^a=frac{1}{3},a^{,3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="473" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6511.png"
 ALT="$\displaystyle =\int_0^a (a^{ 2}-2 a z+z^{ 2}) dz = \left[a^{ 2} z-a z^{ 2}+z^{ 3}/3\right]_0^a= \frac{1}{3} a^{ 3}.$">|; 

$key = q/displaystylenablacdot({bfA}times{bfB});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6690.png"
 ALT="$\displaystyle \nabla\cdot({\bf A}\times{\bf B})$">|; 

$key = q/T_c=T(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5198.png"
 ALT="$ T_c = T(0)$">|; 

$key = q/displaystylemsimeqfrac{1}{291}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img915.png"
 ALT="$\displaystyle m \simeq \frac{1}{291}.$">|; 

$key = q/displaystylep(z)=p_0,exp(-zslashH),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5122.png"
 ALT="$\displaystyle p(z) = p_0 \exp(-z/H),$">|; 

$key = q/displaystyleskew{3}bar{phi}(xi,eta)=m,phi(x,y).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6009.png"
 ALT="$\displaystyle \skew{3}\bar{\phi}(\xi,\eta) = m \phi(x,y).$">|; 

$key = q/displaystylefrac{d{mitPsi}}{dr}=frac{G,m}{r^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5174.png"
 ALT="$\displaystyle \frac{d{\mit\Psi}}{dr} = \frac{G m}{r^{ 2}},$">|; 

$key = q/displaystyletheta=frac{sinxi}{xi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5280.png"
 ALT="$\displaystyle \theta=\frac{\sin\xi}{\xi}
$">|; 

$key = q/displaystyle=lambda,delta_{ik},delta_{jl}+mu,delta_{il},delta_{jk}+nu,delta_{ij},delta_{kl},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6967.png"
 ALT="$\displaystyle =\lambda \delta_{ik} \delta_{jl} + \mu \delta_{il} \delta_{jk} +\nu \delta_{ij} \delta_{kl},$">|; 

$key = q/displaystyleF(z)=-Vleft(c^{,2},{rme}^{,{rmi},alpha}-l^{,2},{rme}^{-{rmi,alpha}}right),zeta^{,-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2513.png"
 ALT="$\displaystyle F(z)= -V\left(c^{ 2} {\rm e}^{ {\rm i} \alpha} -l^{ 2} {\rm e}^{-{\rm i \alpha}}\right) \zeta^{ -1}
$">|; 

$key = q/M_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$ M_2$">|; 

$key = q/{bfr}=-{bfd}slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1682.png"
 ALT="$ {\bf r}=-{\bf d}/2$">|; 

$key = q/displaystyleleft[frac{partial}{partialt}+(u-c),frac{partial}{partialx}right]left(u-frac{2,c}{gamma-1}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5659.png"
 ALT="$\displaystyle \left[\frac{\partial}{\partial t} + (u-c) \frac{\partial}{\partial x}\right]\left(u-\frac{2 c}{\gamma-1}\right)$">|; 

$key = q/r=p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img715.png"
 ALT="$ r=p$">|; 

$key = q/displaystyle|{bfa}+{bfb}|leq|{bfa}|+|{bfb}|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6707.png"
 ALT="$\displaystyle \vert{\bf a} + {\bf b}\vert\leq \vert{\bf a}\vert+\vert{\bf b}\vert.
$">|; 

$key = q/f'(z)=F(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2917.png"
 ALT="$ f'(z)=F(z)$">|; 

$key = q/sqrt{2}slash3<omega<1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img800.png"
 ALT="$ \sqrt{2}/3&lt;\omega&lt;1/2$">|; 

$key = q/r=f(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1232.png"
 ALT="$ r=f(z)$">|; 

$key = q/displaystyleV'<(1+s^{,-1})^{1slash2},c_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4200.png"
 ALT="$\displaystyle V' &lt; (1+s^{ -1})^{1/2} c_1$">|; 

$key = q/displaystylesigma_{rr}(a_+,theta)-sigma_{rr}(a_-,theta)=frac{2,gamma}{a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3862.png"
 ALT="$\displaystyle \sigma_{rr}(a_+,\theta)-\sigma_{rr}(a_-,\theta) = \frac{2 \gamma}{a},$">|; 

$key = q/5.00times10^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4626.png"
 ALT="$ 5.00\times 10^{-1}$">|; 

$key = q/6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4535.png"
 ALT="$ 6$">|; 

$key = q/displaystyleint_P^Qleft(3,x^{,2},y,dx+x^{,3},dyright)=left[x^{,3},yright]_P^Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6471.png"
 ALT="$\displaystyle \int_P^Q \left(3 x^{ 2} y dx + x^{ 3} dy\right) = \left[x^{ 3} y\right]_P^Q$">|; 

$key = q/displaystyle=p_1left(frac{cosz}{cosz_1}right)^{2,gammaslash(gamma-1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6137.png"
 ALT="$\displaystyle = p_1\left(\frac{\cos z}{\cos z_1}\right)^{2 \gamma/(\gamma-1)},$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashw1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img3034.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/w1.eps}}
\end{figure}">|; 

$key = q/v_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3260.png"
 ALT="$ v_s$">|; 

$key = q/eta_2leqetaleqeta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6142.png"
 ALT="$ \eta_2\leq \eta\leq \eta_1$">|; 

$key = q/displaystyle{mitPsi}(mu,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4677.png"
 ALT="$\displaystyle {\mit\Psi}(\mu,\phi,t)$">|; 

$key = q/T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5140.png"
 ALT="$ T_0$">|; 

$key = q/R=1.50times10^{11},{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4596.png"
 ALT="$ R=1.50\times 10^{11} {\rm m}$">|; 

$key = q/dl=sqrt{2},dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6447.png"
 ALT="$ dl= \sqrt{2}  dx$">|; 

$key = q/displaystyleintx_i^{,2},dA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img840.png"
 ALT="$\displaystyle \int x_i^{ 2} dA$">|; 

$key = q/epsilon_{mlt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6936.png"
 ALT="$ \epsilon_{mlt}$">|; 

$key = q/E,v_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3986.png"
 ALT="$ E v_g$">|; 

$key = q/omega^{,2},a^{,2}slashg<4,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1060.png"
 ALT="$ \omega^{ 2} a^{ 2}/g &lt; 4 h$">|; 

$key = q/displaystyle{rmRe}=frac{2,rho,V,a}{mu}=frac{4}{9},frac{a^{,3},g}{nu^{,2}}left|1-frac{overline{rho}}{rho}right|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3815.png"
 ALT="$\displaystyle {\rm Re} = \frac{2 \rho V a}{\mu} = \frac{4}{9} \frac{a^{ 3} g}{\nu^{ 2}}\left\vert 1-\frac{\overline{\rho}}{\rho}\right\vert.$">|; 

$key = q/displaystylell1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3688.png"
 ALT="$\displaystyle \ll 1.$">|; 

$key = q/c,{rme}^{,xi}rightarrowr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2744.png"
 ALT="$ c {\rm e}^{ \xi}\rightarrow r$">|; 

$key = q/zetaleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2000.png"
 ALT="$ \zeta \leq 1$">|; 

$key = q/u_0,sin(k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5667.png"
 ALT="$ u_0 \sin(k x)$">|; 

$key = q/ABP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1615.png"
 ALT="$ ABP$">|; 

$key = q/displaystyle=frac{kappa,M}{R,rho_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="$\displaystyle = \frac{\kappa M}{R \rho_0}.$">|; 

$key = q/displaystyle=frac{1-gamma+2,gamma,{rmMa}_1^{,2},sin^2beta}{gamma+1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5687.png"
 ALT="$\displaystyle = \frac{1-\gamma+2 \gamma {\rm Ma}_1^{ 2} \sin^2\beta}{\gamma+1},$">|; 

$key = q/displaystylex_i'={calR}_{ij},x_j.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6875.png"
 ALT="$\displaystyle x_i' = {\cal R}_{ij} x_j.$">|; 

$key = q/displaystylepsi(xi_0,eta)=-frac{1}{2},V,a^{,2},sin^2eta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2778.png"
 ALT="$\displaystyle \psi(\xi_0,\eta)= -\frac{1}{2} V a^{ 2} \sin^2\eta,$">|; 

$key = q/2,epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5809.png"
 ALT="$ 2 \epsilon$">|; 

$key = q/displaystylec'(x)=c(x)+u(x)=c_1+frac{1}{2},(gamma+1),u(x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5557.png"
 ALT="$\displaystyle c'(x)=c(x)+u(x)= c_1+\frac{1}{2} (\gamma+1) u(x).$">|; 

$key = q/displaystyle(nablatimesnablaphi)_i=epsilon_{ijk},frac{partial}{partialx_j}!left(epsilon_{ijk},frac{partial^{,2}phi}{partialx_j,partialx_k}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6885.png"
 ALT="$\displaystyle (\nabla\times \nabla\phi)_i=\epsilon_{ijk} \frac{\partial }{\par...
...ight)= \epsilon_{ijk} \frac{\partial^{ 2}\phi}{\partial x_j \partial x_k}=0,$">|; 

$key = q/displaystyle{rmRe}=frac{2,rho,V,a}{mu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3748.png"
 ALT="$\displaystyle {\rm Re} = \frac{2 \rho V a}{\mu},$">|; 

$key = q/(4slash3),pi,a^{,3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7201.png"
 ALT="$ (4/3) \pi a^{ 3}$">|; 

$key = q/overline{p}_0gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="$ \overline{p}_0\gg 1$">|; 

$key = q/displaystyle{bfx}'={bfx}+r',{bfn},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7238.png"
 ALT="$\displaystyle {\bf x}' = {\bf x} + r' {\bf n},$">|; 

$key = q/displaystyle{mitGamma}=-{rmRe}left({rmi}ointleft[V,R,{rme}^{,{rmi},(theta+alpha)}+A+{calO}(R^{,-1})right]dthetaright)=2pi,{rmIm}(A),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3281.png"
 ALT="$\displaystyle {\mit\Gamma}= -{\rm Re}\left({\rm i}\oint\left[V R {\rm e}^{ {...
...eta+\alpha)} + A + {\cal O}(R^{ -1})\right]d\theta\right) = 2\pi {\rm Im}(A),$">|; 

$key = q/omega=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2305.png"
 ALT="$ \omega=0$">|; 

$key = q/E=(h,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2842.png"
 ALT="$ E= (h, 0)$">|; 

$key = q/displaystylelim_{epsilonrightarrow0}left(int_0^{phi-epsilon}frac{dphi'}{cosphi'-osphi}+int_{phi+epsilon}^pifrac{dphi'}{cosphi'-cosphi}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3496.png"
 ALT="$\displaystyle \lim_{\epsilon\rightarrow 0}\left(\int_0^{\phi-\epsilon}\frac{d\p...
...hi'-\cos\phi}+\int_{\phi+\epsilon}^\pi\frac{d\phi'}{\cos\phi'-\cos\phi}\right).$">|; 

$key = q/displaystyle=frac{1}{2},V,a,coseta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2748.png"
 ALT="$\displaystyle = \frac{1}{2} V a \cos\eta.$">|; 

$key = q/r_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img718.png"
 ALT="$ r_0$">|; 

$key = q/PABQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img738.png"
 ALT="$ PABQ$">|; 

$key = q/displaystyles,h+frac{2,gamma}{a},costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1340.png"
 ALT="$\displaystyle s h + \frac{2 \gamma}{a} \cos\theta,
$">|; 

$key = q/w_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5789.png"
 ALT="$ w_1$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter14slashpiston.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img5531.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter14/piston.eps}}
\end{figure}">|; 

$key = q/B=A,exp(-2,k,d);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3920.png"
 ALT="$ B = A \exp(-2 k d)$">|; 

$key = q/theta_2rightarrowtheta_1rightarrowtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2627.png"
 ALT="$ \theta_2\rightarrow\theta_1\rightarrow \theta$">|; 

$key = q/mu=pm1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2582.png"
 ALT="$ \mu=\pm 1.$">|; 

$key = q/displaystylefrac{partialv}{partialy}=-frac{partialu}{partialx}simeqfrac{alpha(y)}{2sqrt{x_0-x}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3118.png"
 ALT="$\displaystyle \frac{\partial v}{\partial y} = -\frac{\partial u}{\partial x} \simeq \frac{\alpha(y)}{2\sqrt{x_0-x}}.$">|; 

$key = q/displaystylefrac{1}{2},h_2^{,2}-frac{1}{2},h_1^{,2}+frac{Q}{g},(v_2-v_1)=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1476.png"
 ALT="$\displaystyle \frac{1}{2} h_2^{ 2} -\frac{1}{2} h_1^{ 2}+\frac{Q}{g} (v_2-v_1)=0.$">|; 

$key = q/displaystylev_c=(Q,g)^{1slash3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1433.png"
 ALT="$\displaystyle v_c = (Q g)^{1/3}.$">|; 

$key = q/displaystyle={bfa}cdot{bfb}+{bfa}cdot{bfc}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6308.png"
 ALT="$\displaystyle = {\bf a} \cdot {\bf b} + {\bf a}\cdot {\bf c}.$">|; 

$key = q/nablatimes{bfA}={bf0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6661.png"
 ALT="$ \nabla\times{\bf A} ={\bf0}$">|; 

$key = q/displaystyle=U,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5895.png"
 ALT="$\displaystyle = U,$">|; 

$key = q/deltatheta_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6392.png"
 ALT="$ \delta \theta_z$">|; 

$key = q/displaystyleV(Q+dx)=V(Q)+int_Q^{Q+dx}{bfA}cdotd{bfr}=V(Q)+A_x,dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6555.png"
 ALT="$\displaystyle V(Q+dx) = V(Q) + \int_Q^{Q+dx} {\bf A}\cdot d{\bf r} = V(Q) + A_x dx.$">|; 

$key = q/displaystylefrac{partial,(r,xi_{r,n})}{partialx_i}=(n,A_n+B_n),r^{,2},frac{partilx_i}+2,(n,A_n+B_n),p_n,x_i+n,frac{partialphi_n}{partialx_i},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="509" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4416.png"
 ALT="$\displaystyle \frac{\partial (r \xi_{r n}) }{\partial x_i}= (n A_n+B_n) r^...
...tial x_i} + 2 (n A_n+B_n) p_n x_i + n \frac{\partial\phi_n}{\partial x_i},$">|; 

$key = q/displaystyle=left(frac{2}{gamma+1}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5396.png"
 ALT="$\displaystyle = \left(\frac{2}{\gamma+1}\right),$">|; 

$key = q/displaystylebeta_3simeqbeta_1+epsilon_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6184.png"
 ALT="$\displaystyle \beta_3\simeq \beta_1+\epsilon_2,
$">|; 

$key = q/s_-<s<s_+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img681.png"
 ALT="$ s_-&lt; s&lt; s_+$">|; 

$key = q/{mitOmega}^{,2},aslashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4490.png"
 ALT="$ {\mit\Omega}^{ 2} a/g$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter15slashturn.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="342" BORDER="0"
 SRC="|."$dir".q|img5747.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter15/turn.eps}}
\end{figure}">|; 

$key = q/nu=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5774.png"
 ALT="$ \nu=0$">|; 

$key = q/displaystyleleft[frac{n,(n+1)}{beta},(1-[1+k_n'-h_n'],alpha_n)+frac{m_j,f_j}{n,(n+1)}-f_j^{,2}right]z_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4694.png"
 ALT="$\displaystyle \left[\frac{n (n+1)}{\beta} (1-[1+k_n'-h_n'] \alpha_n)+\frac{m_j f_j}{n (n+1)}-f_j^{ 2}\right]z_n$">|; 

$key = q/({bfA}cdotnabla){bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7184.png"
 ALT="$ ({\bf A}\cdot\nabla){\bf A}$">|; 

$key = q/displaystyle=3,(1-x^{,2}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4270.png"
 ALT="$\displaystyle = 3 (1-x^{ 2}).$">|; 

$key = q/S,(p-p_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1421.png"
 ALT="$ S (p-p_0)$">|; 

$key = q/displaystylefrac{p_3-p_1}{p_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6194.png"
 ALT="$\displaystyle \frac{p_3-p_1}{p_1}$">|; 

$key = q/displaystylefrac{nu,(pislash2-1)^2}{delta_1}=U,(1-pislash4),frac{ddelta_1}{dx}+frac{dU}{dx},delta_1+(pislash2-1),v_s.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="455" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3268.png"
 ALT="$\displaystyle \frac{\nu (\pi/2-1)^2}{\delta_1} = U (1-\pi/4) \frac{d\delta_1}{dx} + \frac{dU}{dx} \delta_1+(\pi/2-1) v_s.
$">|; 

$key = q/displaystyle=-frac{g,d}{a^{,2},sintheta},frac{partial}{partialphi},(zeta-{calF}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4728.png"
 ALT="$\displaystyle =- \frac{g d}{a^{ 2} \sin\theta} \frac{\partial}{\partial\phi} (\zeta-{\cal F}),$">|; 

$key = q/displaystyle=X_0,sinbetaleft(1-frac{1}{2},sin^2betaright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4047.png"
 ALT="$\displaystyle = X_0 \sin\beta\left(1-\frac{1}{2} \sin^2\beta\right),$">|; 

$key = q/dA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img843.png"
 ALT="$ dA$">|; 

$key = q/delta(x,y)=delta_0=V_0slashA_0=s,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img666.png"
 ALT="$ \delta(x,y)=\delta_0=V_0/A_0=s c$">|; 

$key = q/{bfA}=nablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6624.png"
 ALT="$ {\bf A}= \nabla \phi$">|; 

$key = q/Omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3376.png"
 ALT="$ \Omega$">|; 

$key = q/F(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3238.png"
 ALT="$ F(0)=0$">|; 

$key = q/V<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3814.png"
 ALT="$ V&lt;0$">|; 

$key = q/displaystyle=-p+2,muleft(frac{1}{r,sintheta},frac{partialv_varphi}{partialvarphi}+frac{v_r}{r}+frac{cottheta,v_theta}{r}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img497.png"
 ALT="$\displaystyle =-p + 2 \mu\left(\frac{1}{r \sin\theta} \frac{\partial v_\varphi}{\partial \varphi}+\frac{v_r}{r}+\frac{\cot\theta v_\theta}{r}\right),$">|; 

$key = q/z_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1263.png"
 ALT="$ z_0=0$">|; 

$key = q/h_2<h_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1498.png"
 ALT="$ h_2&lt; h_1$">|; 

$key = q/displaystylefrac{D}{Dt}=frac{partial}{partialt}+{bfv}cdotnabla=frac{partial}{partialt}+v_i,frac{partial}{partialx_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="$\displaystyle \frac{D}{Dt}= \frac{\partial}{\partial t} + {\bf v}\cdot\nabla = \frac{\partial}{\partial t} + v_i \frac{\partial}{\partial x_i}.$">|; 

$key = q/displaystyle=X+frac{partialP}{partialtheta}+frac{1}{sintheta},frac{partialQ}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4742.png"
 ALT="$\displaystyle =X + \frac{\partial P}{\partial\theta} + \frac{1}{\sin\theta} \frac{\partial Q}{\partial\phi},$">|; 

$key = q/displaystyle=frac{1}{2},Bleft[sinhxi-cosh^2xi,tan^{-1}left(frac{1}{sinhxi}right)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2767.png"
 ALT="$\displaystyle = \frac{1}{2} B\left[\sinh\xi -\cosh^2\xi \tan^{-1}\left(\frac{1}{\sinh\xi}\right)\right],$">|; 

$key = q/displaystyle=frac{1}{2}left(frac{partialA_r}{partialz}+frac{partialA_z}{partialr}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7156.png"
 ALT="$\displaystyle =\frac{1}{2}\left(\frac{\partial A_r}{\partial z} + \frac{\partial A_z}{\partial r}\right),$">|; 

$key = q/I_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7286.png"
 ALT="$ I_2$">|; 

$key = q/nablapsicdotnablavarphi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2557.png"
 ALT="$ \nabla\psi\cdot\nabla\varphi=0$">|; 

$key = q/displaystyle{bfg}({bfr})=-nabla{mitPhi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4351.png"
 ALT="$\displaystyle {\bf g}({\bf r}) = -\nabla {\mit\Phi},$">|; 

$key = q/R_a=(1+e),R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4289.png"
 ALT="$ R_a=(1+e) R$">|; 

$key = q/displaystyleskew{3}hat{theta}_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5008.png"
 ALT="$\displaystyle \skew{3}\hat{\theta}_{n, n'}^{ m, m'}$">|; 

$key = q/b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img662.png"
 ALT="$ b$">|; 

$key = q/displaystyle{rmFr}=frac{V}{(g,l)^{1slash2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4003.png"
 ALT="$\displaystyle {\rm Fr} = \frac{V}{(g l)^{1/2}}$">|; 

$key = q/displaystylebar{F}=frac{k,V}{zeta^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2319.png"
 ALT="$\displaystyle \bar{F}=\frac{k V}{\zeta^{ 2}},$">|; 

$key = q/displaystylefrac{x}{d};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2230.png"
 ALT="$\displaystyle \frac{x}{d}$">|; 

$key = q/displaystylezeta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4540.png"
 ALT="$\displaystyle \zeta_1$">|; 

$key = q/displaystyleepsilon,alpha,sqrt{1-{rmMa}_infty^{,2}}ll1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5983.png"
 ALT="$\displaystyle \epsilon \alpha \sqrt{1-{\rm Ma}_\infty^{ 2}}\ll 1.$">|; 

$key = q/displaystyleV=frac{2}{9},frac{a^{,2},g}{nu}left(1-frac{overline{rho}}{rho}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3812.png"
 ALT="$\displaystyle V = \frac{2}{9} \frac{a^{ 2} g}{\nu}\left(1-\frac{\overline{\rho}}{\rho}\right),$">|; 

$key = q/displaystylephi(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6049.png"
 ALT="$\displaystyle \phi(x,t)$">|; 

$key = q/displaystyley(alpha)=frac{2,b}{pi,sinalpha+4}left[2,cosalpha,(1+sin^2alpha)+sinalphaleft(frac{pi}{2}-alpharight)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="469" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2316.png"
 ALT="$\displaystyle y(\alpha)= \frac{2 b}{\pi \sin\alpha+4}\left[2 \cos\alpha (1+\sin^2\alpha)+\sin\alpha\left(\frac{\pi}{2}-\alpha\right)\right]$">|; 

$key = q/60.7,{rmcm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5042.png"
 ALT="$ 60.7 {\rm cm}$">|; 

$key = q/displaystyleC_{pL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5839.png"
 ALT="$\displaystyle C_{pL}$">|; 

$key = q/displaystylepsi(r,theta)=sin^2theta,frac{V,(r-a)^2,(2,r+a)}{4,r}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3782.png"
 ALT="$\displaystyle \psi(r,\theta) = \sin^2\theta \frac{V (r-a)^2 (2 r+a)}{4 r}.$">|; 

$key = q/displaystyle_zcdotd{bfS}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$\displaystyle _z\cdot d{\bf S}).$">|; 

$key = q/displaystylepsi(r,theta)=frac{{mitGamma}}{2pi},lnr.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1708.png"
 ALT="$\displaystyle \psi(r,\theta) = \frac{{\mit\Gamma}}{2\pi} \ln r.$">|; 

$key = q/{bfv}rightarrow{bfV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3328.png"
 ALT="$ {\bf v}\rightarrow {\bf V}$">|; 

$key = q/T_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5892.png"
 ALT="$ T_\infty$">|; 

$key = q/{mitOmega}proptor^{,-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3640.png"
 ALT="$ {\mit\Omega}\propto r^{ -2}$">|; 

$key = q/displaystyleS=frac{pi}{4},b,c_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3511.png"
 ALT="$\displaystyle S = \frac{\pi}{4} b c_0.$">|; 

$key = q/partialphislashpartialxi=varpi^{,-1},partialpsislashpartialeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2720.png"
 ALT="$ \partial\phi/\partial\xi = \varpi^{ -1} \partial\psi/\partial\eta$">|; 

$key = q/v^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3900.png"
 ALT="$ v^{ 2}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter02slashfig5.05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img1023.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter02/fig5.05.eps}}
\end{figure}">|; 

$key = q/10^5,{rmN,m}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img571.png"
 ALT="$ 10^5 {\rm N m}^{-2}$">|; 

$key = q/h_1simeqh_2simeqh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1517.png"
 ALT="$ h_1\simeq h_2\simeq h$">|; 

$key = q/c_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4193.png"
 ALT="$ c_m$">|; 

$key = q/displaystylefrac{d}{dx}!left(frac{partialF}{partialy'}right)-frac{partialF}{partialy}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7317.png"
 ALT="$\displaystyle \frac{d}{dx}\!\left(\frac{\partial F}{\partial y'}\right)-\frac{\partial F}{\partial y} = 0.$">|; 

$key = q/displaystyle{bfF}=-rho,g,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img576.png"
 ALT="$\displaystyle {\bf F} =- \rho g {\bf e}_z$">|; 

$key = q/displaystyle={mitOmega},costheta,{bfe}_r-{mitOmega},sintheta,{bfe}_theta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4338.png"
 ALT="$\displaystyle = {\mit\Omega} \cos\theta {\bf e}_r - {\mit\Omega} \sin\theta {\bf e}_\theta.$">|; 

$key = q/displaystylec_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4176.png"
 ALT="$\displaystyle c_0$">|; 

$key = q/omega_M-{mitPi}_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4550.png"
 ALT="$ \omega_M-{\mit\Pi}_M$">|; 

$key = q/displaystylefrac{1}{r},frac{partialphi}{partialtheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1757.png"
 ALT="$\displaystyle \frac{1}{r} \frac{\partial\phi}{\partial\theta}$">|; 

$key = q/displaystylenablaqcdotnablaq=q,nabla^{,2}q,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1931.png"
 ALT="$\displaystyle \nabla q\cdot\nabla q = q \nabla^{ 2} q,
$">|; 

$key = q/displaystyle{calE}+frac{p}{rho}+frac{1}{2},v^{,2}+{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5376.png"
 ALT="$\displaystyle {\cal E} + \frac{p}{\rho} + \frac{1}{2} v^{ 2} +{\mit\Psi}$">|; 

$key = q/displaystylefrac{partialzeta}{partialt}=left.v_zright|_{z=0}=-left.frac{partialphi}{partialz}right|_{z=0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3909.png"
 ALT="$\displaystyle \frac{\partial \zeta}{\partial t} = \left.v_z\right\vert _{z=0}=-\left.\frac{\partial \phi}{\partial z}\right\vert _{z=0}.$">|; 

$key = q/2,{mitOmega}-2,omega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4537.png"
 ALT="$ 2 {\mit\Omega}-2 \omega_1$">|; 

$key = q/displaystylezeta(x,t)=a,sin(omega,t-k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3973.png"
 ALT="$\displaystyle \zeta(x,t) = a \sin(\omega t-k x)$">|; 

$key = q/dE;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5298.png"
 ALT="$ dE$">|; 

$key = q/displaystyleepsilon=frac{a_1-a_3}{a_0}simeqfrac{e_{13}^{,2}}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img910.png"
 ALT="$\displaystyle \epsilon = \frac{a_1-a_3}{a_0}\simeq \frac{e_{13}^{ 2}}{2}.$">|; 

$key = q/G(u_0)>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1274.png"
 ALT="$ G(u_0)&gt;0$">|; 

$key = q/displaystyle=v_0,{rme}^{,{rmi},(sigma,t-{bfk}cdot{bfr})}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5083.png"
 ALT="$\displaystyle = v_0 {\rm e}^{ {\rm i} (\sigma t-{\bf k}\cdot{\bf r})}.$">|; 

$key = q/sum_{j=1,3}sigma_{ij},dS_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$ \sum_{j=1,3}\sigma_{ij} dS_j$">|; 

$key = q/displaystylep_1(x,z,t)=rho,k,(V-c)left[A,cosh(k,z)+B,sinh(k,z)right]sin(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="531" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4080.png"
 ALT="$\displaystyle p_1(x,z,t) = \rho k (V-c)\left[A \cosh(k z) + B \sinh(k z)\right]\sin(\omega t-k x),$">|; 

$key = q/zeta=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2147.png"
 ALT="$ \zeta=a$">|; 

$key = q/displaystylefrac{D{bfv}}{Dt}=-nablaleft(frac{p}{rho}+{mitPsi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1557.png"
 ALT="$\displaystyle \frac{D{\bf v}}{Dt}= - \nabla\left(\frac{p}{\rho}+{\mit\Psi}\right),$">|; 

$key = q/displaystyle=-Vleft(1+frac{1}{2},frac{a^{,3}}{r^{,3}}right)sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2647.png"
 ALT="$\displaystyle = - V\left(1+\frac{1}{2} \frac{a^{ 3}}{r^{ 3}}\right)\sin\theta.$">|; 

$key = q/mu=qopnamerelaxo{sin}^{-1}(1slash{rmMa}_infty);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$ \mu =\qopname \relax o{sin}^{-1}(1/{\rm Ma}_\infty )$">|; 

$key = q/displaystyleD,{mitPhi}=zeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4666.png"
 ALT="$\displaystyle D  {\mit\Phi}=\zeta,$">|; 

$key = q/v_z|_{z=-d}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3907.png"
 ALT="$ v_z\vert _{z=-d}=0$">|; 

$key = q/varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="$ \varphi$">|; 

$key = q/{figure}{epsfysize{=1.25in{centerline{epsffile{Chapter09slashwing2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="140" BORDER="0"
 SRC="|."$dir".q|img3409.png"
 ALT="\begin{figure}
\epsfysize =1.25in
\centerline{\epsffile{Chapter09/wing2.eps}}
\end{figure}">|; 

$key = q/R=8.3145,{rmJ,K^{-1},mol^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="$ R=8.3145 {\rm J K^{-1} mol^{-1}}$">|; 

$key = q/exp(-k,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3918.png"
 ALT="$ \exp(-k z)$">|; 

$key = q/displaystyle-int_{-h}^hrho,U_0^{,2},dy+int_{-h}^hrholeft[U_0-u(y)right]^{,2},dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3057.png"
 ALT="$\displaystyle -\int_{-h}^h\rho U_0^{ 2} dy+\int_{-h}^h\rho\left[U_0-u(y)\right]^{ 2} dy$">|; 

$key = q/displaystyle=frac{1}{5},a_i^{,2},V,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img849.png"
 ALT="$\displaystyle = \frac{1}{5} a_i^{ 2} V,$">|; 

$key = q/E(gamma,alpha);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img952.png"
 ALT="$ E(\gamma,\alpha)$">|; 

$key = q/displaystyle=2,(k+s)^{1slash2},k^{1slash2}+frac{pi}{2},k.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2273.png"
 ALT="$\displaystyle = 2 (k+s)^{1/2} k^{1/2}+\frac{\pi}{2} k.$">|; 

$key = q/zeta={rmi},{rme}^{,pi,zslasha};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2110.png"
 ALT="$ \zeta={\rm i} {\rm e}^{ \pi z/a}$">|; 

$key = q/nablacdot(A,nablaA)equivnablaAcdotnablaA+A,nabla^{,2}A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1609.png"
 ALT="$ \nabla\cdot(A \nabla A)\equiv \nabla A\cdot \nabla A +A \nabla^{ 2} A$">|; 

$key = q/d_1>d_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3706.png"
 ALT="$ d_1&gt;d_2$">|; 

$key = q/displaystyletilde{F}(phi,k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1178.png"
 ALT="$\displaystyle \tilde{F}(\phi,k)$">|; 

$key = q/displaystylezeta=mu^{,2},frac{M_ast}{M_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5228.png"
 ALT="$\displaystyle \zeta = \mu^{ 2} \frac{M_\ast}{M_0},$">|; 

$key = q/(g,d)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3966.png"
 ALT="$ (g d)^{1/2}$">|; 

$key = q/Uneq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3201.png"
 ALT="$ U\neq 0$">|; 

$key = q/{calS}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5693.png"
 ALT="$ {\cal S}_2$">|; 

$key = q/partialpsislashpartialvarphi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2558.png"
 ALT="$ \partial\psi/\partial\varphi=0$">|; 

$key = q/1slashN^{,1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1339.png"
 ALT="$ 1/N^{ 1/3}$">|; 

$key = q/displaystyleH_1=h_1+frac{1}{2},frac{v_1^{,2}}{g},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1499.png"
 ALT="$\displaystyle H_1 = h_1+\frac{1}{2} \frac{v_1^{ 2}}{g},$">|; 

$key = q/displaystylesigma_{ji}=sigma_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="$\displaystyle \sigma_{ji} = \sigma_{ij}.$">|; 

$key = q/e=0.09262;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1073.png"
 ALT="$ e=0.09262$">|; 

$key = q/skew{5}tilde{zeta}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4523.png"
 ALT="$ \skew{5}\tilde{\zeta}_j$">|; 

$key = q/displaystylefrac{|{bfa}times{bfb}+{bfb}times{bfc}+{bfc}times{bfa}|}{|{bfb}-{bfc}|}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6720.png"
 ALT="$\displaystyle \frac{\vert{\bf a}\times {\bf b} + {\bf b}\times {\bf c} + {\bf c}\times {\bf a}\vert}{\vert{\bf b}-{\bf c}\vert}.
$">|; 

$key = q/{multline}G_F(theta,phi;theta',phi')=sum_{n=1,infty}sum_{m=0,n}frac{(2,n+1)}{4pitheta),P_n^{,m}(costheta'),{{rm{e}^{,{{rm{i},m,(phi-phi')}.{multline};LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="537" HEIGHT="100" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4515.png"
 ALT="\begin{multline}
G_F(\theta,\phi;\theta',\phi')=\sum_{n=1,\infty}\sum_{m=0,n}\fr...
...
) P_n^{ m}(\cos\theta') {\rm e}^{ {\rm i} m (\phi-\phi')}.
\end{multline}">|; 

$key = q/displaystyleleft(m+frac{1}{2},m'right)frac{dV_z}{dt}=-m,g+m',g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2673.png"
 ALT="$\displaystyle \left(m+\frac{1}{2} m'\right)\frac{dV_z}{dt} = -m g+m' g.$">|; 

$key = q/displaystylepsi=-frac{Q}{pi},{rmarg}(zeta)=-frac{Q}{pi},tan^{-1}left(frac{eta}{xi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2168.png"
 ALT="$\displaystyle \psi = -\frac{Q}{\pi} {\rm arg}(\zeta) =- \frac{Q}{\pi} \tan^{-1}\left(\frac{\eta}{\xi}\right),$">|; 

$key = q/C''=([M'slash(M+M')],R,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img977.png"
 ALT="$ C''=([M'/(M+M')] R, 0, 0)$">|; 

$key = q/displaystyle(nablatimes{bfA})_i=sum_{j,k=1,3}epsilon_{ijk},frac{h_i}{h},frac{partial(h_k,A_k)}{partialu_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7077.png"
 ALT="$\displaystyle (\nabla\times{\bf A})_i = \sum_{j,k=1,3}\epsilon_{ijk} \frac{h_i}{h} \frac{\partial (h_k A_k)}{\partial u_j}.$">|; 

$key = q/displaystyle=frac{D}{2pi},frac{sin(theta-theta_0)}{r^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1697.png"
 ALT="$\displaystyle =\frac{D}{2\pi} \frac{\sin(\theta-\theta_0)}{r^{ 2}},$">|; 

$key = q/displaystyleoverline{alpha}=alpha_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5852.png"
 ALT="$\displaystyle \overline{\alpha}= \alpha_0.$">|; 

$key = q/displaystyleepsilon_{ijk},{calR}_{li},{calR}_{mj},{calR}_{nk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6834.png"
 ALT="$\displaystyle \epsilon_{ijk} {\cal R}_{li} {\cal R}_{mj} {\cal R}_{nk}$">|; 

$key = q/6,mu,Aslasha^{,4}=6,overline{mu},overline{D},a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3852.png"
 ALT="$ 6 \mu A/a^{ 4} = 6 \overline{\mu} \overline{D} a$">|; 

$key = q/times|{bfr}-{bfr}'|={mitGamma},z,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3476.png"
 ALT="$ \times \vert{\bf r}-{\bf r}'\vert = {\mit\Gamma} z {\bf e}_y$">|; 

$key = q/x=X_0slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4054.png"
 ALT="$ x=X_0/2$">|; 

$key = q/displaystyle=frac{1}{2},rhooint_Cleft|frac{dF}{dz}right|^{,2}dy,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2394.png"
 ALT="$\displaystyle =\frac{1}{2} \rho\oint_C \left\vert\frac{dF}{dz}\right\vert^{ 2}dy,$">|; 

$key = q/displaystyleleft(frac{partial{calS}}{partialp}right)_{T}=-left(frac{partialv}{partialT}right)_p.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5593.png"
 ALT="$\displaystyle \left(\frac{\partial {\cal S}}{\partial p}\right)_{T} = -\left(\frac{\partial v}{\partial T}\right)_p.
$">|; 

$key = q/displaystyleH_L=frac{1}{4}frac{h_1}{h_2}left(frac{h_2}{h_1}-1right)left[-4,frac{h_2}{h_1}+left(frac{h_2}{h_1}+1right)^2right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1506.png"
 ALT="$\displaystyle H_L = \frac{1}{4}\frac{h_1}{h_2}\left(\frac{h_2}{h_1}-1\right)\left[-4 \frac{h_2}{h_1}+ \left(\frac{h_2}{h_1}+1\right)^2\right],$">|; 

$key = q/displaystyle=lambda,delta_{ij},delta_{kl}+mu,delta_{ik},delta_{jl}+nu,delta_{il},delta_{jk}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6963.png"
 ALT="$\displaystyle =\lambda \delta_{ij} \delta_{kl} + \mu \delta_{ik} \delta_{jl} +\nu \delta_{il} \delta_{jk}.$">|; 

$key = q/skew{2}bar{z}=x-{rmi},y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2397.png"
 ALT="$ \skew{2}\bar{z}=x-{\rm i} y$">|; 

$key = q/{mitOmega}(r)=v_theta(r)slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3627.png"
 ALT="$ {\mit\Omega}(r)=v_\theta(r)/r$">|; 

$key = q/displaystyled{calH};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5335.png"
 ALT="$\displaystyle d{\cal H}$">|; 

$key = q/displaystylefrac{partialB}{partialtheta}-frac{1}{sintheta},frac{partialA}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4781.png"
 ALT="$\displaystyle \frac{\partial B}{\partial\theta} -\frac{1}{\sin\theta} \frac{\partial A}{\partial\phi}$">|; 

$key = q/displaystylez=a,cos(omega,t),cos(k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4202.png"
 ALT="$\displaystyle z = a \cos(\omega t) \cos(k x).
$">|; 

$key = q/dW;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6327.png"
 ALT="$ dW$">|; 

$key = q/displaystyle=-ointp(a,theta,varphi),a^{,2},costheta,d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2657.png"
 ALT="$\displaystyle = -\oint p(a,\theta,\varphi) a^{ 2}  \cos\theta d{\mit\Omega},$">|; 

$key = q/displaystylesim({mitDelta}theta)^3.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5751.png"
 ALT="$\displaystyle \sim ({\mit\Delta}\theta)^3.$">|; 

$key = q/displaystylefrac{l}{d}=11.8,{rmRe},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3007.png"
 ALT="$\displaystyle \frac{l}{d} = 11.8 {\rm Re},$">|; 

$key = q/displaystylep(a,theta,t)=p_0-left.frac{1}{2},rho,v^{,2}right|_{r=a}-rho,g,a,cost{partialt}right|_{r=a}-rholeft.{bfV}cdotnablaphiright|_{r=a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="550" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2666.png"
 ALT="$\displaystyle p(a,\theta,t) = p_0 -\left.\frac{1}{2} \rho v^{ 2}\right\vert ...
...ial t}\right\vert _{r=a} - \rho\left.{\bf V}\cdot \nabla\phi\right\vert _{r=a}.$">|; 

$key = q/displaystyle=oint_CP,dx,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2391.png"
 ALT="$\displaystyle =\oint_C P dx,$">|; 

$key = q/displaystylev_perp=frac{1}{varpi},frac{partialpsi}{partials}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2541.png"
 ALT="$\displaystyle v_\perp = \frac{1}{\varpi} \frac{\partial\psi}{\partial s}.$">|; 

$key = q/y=w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$ y=w$">|; 

$key = q/zeta_{mitOmega}+zeta_M+zeta_E=7.32times10^3,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4642.png"
 ALT="$ \zeta_{\mit\Omega}+\zeta_M+\zeta_E =7.32\times 10^3 {\rm m}$">|; 

$key = q/A_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1601.png"
 ALT="$ A_2$">|; 

$key = q/displaystyle{calC}_V=frac{skew{3}bar{C}_V}{m}=k.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5305.png"
 ALT="$\displaystyle {\cal C}_V = \frac{\skew{3}\bar{C}_V}{m} = k.$">|; 

$key = q/displaystyleK=frac{1}{2},rho,V^{,2},pileft(a^{,2},sin^2alpha+b^{,2},cos^2alpharight),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2514.png"
 ALT="$\displaystyle K = \frac{1}{2} \rho V^{ 2} \pi\left(a^{ 2} \sin^2\alpha+b^{ 2} \cos^2\alpha\right),
$">|; 

$key = q/{rmFr}sim{calO}(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="$ {\rm Fr}\sim{\cal O}(1)$">|; 

$key = q/{bfv}cdot{bfe}_r=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2641.png"
 ALT="$ {\bf v}\cdot{\bf e}_r=0$">|; 

$key = q/{bfV}=V_z(t),{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2659.png"
 ALT="$ {\bf V} = V_z(t) {\bf e}_z$">|; 

$key = q/displaystylefrac{d}{l}simeqfrac{k}{2},costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1216.png"
 ALT="$\displaystyle \frac{d}{l}\simeq \frac{k}{2} \cos\theta.$">|; 

$key = q/displaystylefrac{partial}{partialy}!left(frac{v}{u}right)simeq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3128.png"
 ALT="$\displaystyle \frac{\partial}{\partial y}\!\left(\frac{v}{u}\right)\simeq 0.$">|; 

$key = q/x_pm=(cslash2),costheta_pm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3367.png"
 ALT="$ x_\pm = (c/2) \cos\theta_\pm$">|; 

$key = q/displaystyle=nabla^{,2}(r^{,m+n},{calS}_n)=frac{1}{r^{,2}},frac{d}{dr}left[r^{,2c{d}{dr}(r^{,m+n},{calS}_n)right]-n,(n+1),r^{,m+n-2},{calS}_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="536" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4335.png"
 ALT="$\displaystyle = \nabla^{ 2}(r^{ m+n} {\cal S}_n) = \frac{1}{r^{ 2}} \frac{...
...,\frac{d}{dr}(r^{ m+n} {\cal S}_n)\right] - n (n+1) r^{ m+n-2} {\cal S}_n$">|; 

$key = q/displaystylepsi(x,y)=frac{{mitGamma}}{pi},lnleft[frac{cosh(pi,x,a)-cos(pi,yslasha)}{cosh(pi,xslasha)+cos(pi,yslasha)}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2109.png"
 ALT="$\displaystyle \psi(x,y)= \frac{{\mit\Gamma}}{\pi} \ln\left[\frac{\cosh(\pi x a)-\cos(\pi y/a) }{\cosh(\pi x/a)+\cos(\pi y/a)}\right],$">|; 

$key = q/m(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5217.png"
 ALT="$ m(0)=0$">|; 

$key = q/displaystyleleft|frac{dF}{dz}right|^{,2}dskew{2}bar{z}=frac{dF}{dz},frac{dbar{F}}{dskew{2}bar{z}},dskew{2}bar{z}=frac{dF}{dz},dbar{F},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2402.png"
 ALT="$\displaystyle \left\vert\frac{dF}{dz}\right\vert^{ 2}d\skew{2}\bar{z} = \frac{...
...,\frac{d\bar{F}}{d\skew{2}\bar{z}} d\skew{2}\bar{z} = \frac{dF}{dz} d\bar{F},$">|; 

$key = q/displaystyle=a,(1+zslashd),sin(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3958.png"
 ALT="$\displaystyle = a (1+z/d) \sin(\omega t-k x),$">|; 

$key = q/displaystylefrac{Dv_y}{Dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="$\displaystyle \frac{Dv_y}{Dt}$">|; 

$key = q/dtau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6506.png"
 ALT="$ d\tau$">|; 

$key = q/{mitDelta}{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3384.png"
 ALT="$ {\mit\Delta}{\bf v}$">|; 

$key = q/dFslashdz=V,{rme}^{,{rmi},alpha};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3276.png"
 ALT="$ dF/dz=V {\rm e}^{ {\rm i} \alpha}$">|; 

$key = q/displaystyle{bfv}=nablavarphitimesnablapsi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2552.png"
 ALT="$\displaystyle {\bf v} = \nabla\varphi\times \nabla \psi.$">|; 

$key = q/displaystyle-theta+{rmconst.}=int^{rmMa}sqrt{{rmMa}^{,2}-1},frac{dw}{w}equivnu({rmMa}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5765.png"
 ALT="$\displaystyle -\theta + {\rm const.} =\int^{\rm Ma} \sqrt{{\rm Ma}^{ 2}-1} \frac{dw}{w}\equiv \nu ({\rm Ma}).$">|; 

$key = q/A'B';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1364.png"
 ALT="$ A'B'$">|; 

$key = q/displaystylecdotd{bfS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1552.png"
 ALT="$\displaystyle \cdot d{\bf S}.$">|; 

$key = q/v'slashU;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5939.png"
 ALT="$ v'/U$">|; 

$key = q/OQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6383.png"
 ALT="$ OQ$">|; 

$key = q/L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="$ L$">|; 

$key = q/A_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4412.png"
 ALT="$ A_n$">|; 

$key = q/dx=a,dtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5059.png"
 ALT="$ dx=a d\theta$">|; 

$key = q/displaystylefrac{1}{gamma-1}left(frac{Dp}{Dt}-frac{gamma,p}{rho},frac{Drho}{Dt}right)=chi+frac{kappa,M}R,nabla^{,2}!left(frac{p}{rho}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="$\displaystyle \frac{1}{\gamma-1}\left(\frac{Dp}{Dt} - \frac{\gamma p}{\rho} \...
...}\right) = \chi + \frac{\kappa M}R \nabla^{ 2}\!\left(\frac{p}{\rho}\right),$">|; 

$key = q/bar{F}=skew{3}bar{phi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2301.png"
 ALT="$ \bar{F}=\skew{3}\bar{\phi}$">|; 

$key = q/displaystylep,v={calR},T,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5290.png"
 ALT="$\displaystyle p v = {\cal R} T,$">|; 

$key = q/{rmMa}_{,e};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5422.png"
 ALT="$ {\rm Ma}_{ e}$">|; 

$key = q/ds;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1634.png"
 ALT="$ ds$">|; 

$key = q/omega=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2307.png"
 ALT="$ \omega=\pi$">|; 

$key = q/{calH};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5862.png"
 ALT="$ {\cal H}$">|; 

$key = q/displaystyle=frac{x}{L},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2877.png"
 ALT="$\displaystyle = \frac{x}{L},$">|; 

$key = q/displaystyleH=frac{1}{2},v^{,2}+{calE}+frac{p}{rho}+{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img554.png"
 ALT="$\displaystyle H = \frac{1}{2} v^{ 2} + {\cal E}+ \frac{p}{\rho} + {\mit\Psi}
$">|; 

$key = q/displaystyle=int_0^az,dzint_{-(a-z)slash2}^{(a-z)slash2}dyint_{-(a-z)slash2}^{(a-z)slash2}dx=int_0^az,(a-z)^2,dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6513.png"
 ALT="$\displaystyle = \int_0^a z dz \int_{-(a-z)/2}^{(a-z)/2} dy\int_{-(a-z)/2}^{(a-z)/2} dx = \int_0^a z (a-z)^2 dz$">|; 

$key = q/(0,,+a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2834.png"
 ALT="$ (0, +a)$">|; 

$key = q/P_1P_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1624.png"
 ALT="$ P_1 P_2$">|; 

$key = q/displaystyled{calE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5333.png"
 ALT="$\displaystyle d{\cal E}$">|; 

$key = q/displaystylev_z(x,z,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3943.png"
 ALT="$\displaystyle v_z(x,z,t)$">|; 

$key = q/u_2=w_2,sin(beta-theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5699.png"
 ALT="$ u_2= w_2 \sin(\beta-\theta)$">|; 

$key = q/d^{,2}uslashdz^{,2}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img792.png"
 ALT="$ d^{ 2}u/dz^{ 2}&gt;0$">|; 

$key = q/displaystyle=frac{2}{pi},frac{skew{3}bar{psi}_1}{V}left[tanh^{-1}(costheta)-costheta)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2223.png"
 ALT="$\displaystyle =\frac{2}{\pi} \frac{\skew{3}\bar{\psi}_1}{V}\left[\tanh^{-1}(\cos\theta)-\cos\theta)\right],$">|; 

$key = q/displaystyle=frac{partial}{partialt}+v_x,frac{partial}{partialx}+v_y,frac{partial}{partialy}+v_z,frac{partial}{partialz},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="$\displaystyle = \frac{\partial}{\partial t} + v_x \frac{\partial }{\partial x} + v_y \frac{\partial}{\partial y} + v_z \frac{\partial}{\partial z},$">|; 

$key = q/{bfa}cdot{bfc}=a_x,c_x+a_y,c_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6427.png"
 ALT="$ {\bf a}\cdot {\bf c} = a_x  c_x + a_y  c_y$">|; 

$key = q/nablacdot(nablaAtimesnablaB)equiv0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1638.png"
 ALT="$ \nabla\cdot(\nabla A\times \nabla B)\equiv 0$">|; 

$key = q/1ll{rmRe}ll9times10^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3071.png"
 ALT="$ 1\ll {\rm Re}\ll 9\times 10^4$">|; 

$key = q/dbar{F}=dF;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2407.png"
 ALT="$ d\bar{F}=dF$">|; 

$key = q/rho,S_2,v_2,deltattimes{bfv}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1388.png"
 ALT="$ \rho S_2 v_2 \delta t\times {\bf v}_2$">|; 

$key = q/mu=beta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="$ \mu=\beta$">|; 

$key = q/displaystyle{mitPsi}''simeq-left[frac{1}{2}-frac{x_1}{R}+frac{(1slash2),x_1^{,2}+(1slash2),x_2^{,2}}{R^{,2}}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img982.png"
 ALT="$\displaystyle {\mit\Psi}'' \simeq -\left[\frac{1}{2}-\frac{x_1}{R}+\frac{(1/2) x_1^{ 2}+(1/2) x_2^{ 2}}{R^{ 2}}\right],$">|; 

$key = q/displaystyle=a-BQ'slash3=a,(1-omega,z),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img783.png"
 ALT="$\displaystyle = a-BQ'/3=a (1-\omega z),$">|; 

$key = q/r=(x^{,2}+y^{,2}+z^{,2})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2527.png"
 ALT="$ r=(x^{ 2}+y^{ 2}+z^{ 2})^{1/2}$">|; 

$key = q/displaystyleUleft(frac{dy}{dx}right)_U=-frac{partialphi(x,0_+)}{partialx}=lambda,f'(x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6052.png"
 ALT="$\displaystyle U\left(\frac{dy}{dx}\right)_U = -\frac{\partial\phi(x,0_+)}{\partial x} = \lambda f'(x),$">|; 

$key = q/theta({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="$ \theta({\bf r}, t)$">|; 

$key = q/partialx_islashpartialx_j=delta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="$ \partial x_i/\partial x_j = \delta_{ij}$">|; 

$key = q/0<{rmFr}<{rmFr}_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4007.png"
 ALT="$ 0&lt;{\rm Fr}&lt;{\rm Fr}_c$">|; 

$key = q/psi(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1950.png"
 ALT="$ \psi(x,y)$">|; 

$key = q/displaystylev_x(x,y)=U_0,f'(eta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2982.png"
 ALT="$\displaystyle v_x(x,y) = U_0 f'(\eta),$">|; 

$key = q/displaystylefrac{partialphi}{partialy}=frac{partialskew{3}bar{phi}}{partialeta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6030.png"
 ALT="$\displaystyle \frac{\partial\phi}{\partial y} =\frac{\partial\skew{3}\bar{\phi}}{\partial \eta},$">|; 

$key = q/displaystylephi=frac{Q}{4pi}left(frac{1}{r_2}-frac{1}{r_1}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2618.png"
 ALT="$\displaystyle \phi = \frac{Q}{4\pi}\left(\frac{1}{r_2}-\frac{1}{r_1}\right),$">|; 

$key = q/displaystyle=(1-a_3^{,2}slasha_1^{,2})^{1slash2}=singamma,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img933.png"
 ALT="$\displaystyle = (1-a_3^{ 2}/a_1^{ 2})^{1/2} = \sin\gamma,$">|; 

$key = q/displaystyle=frac{1}{2},V_z(t),frac{a^{,3}}{r^{,3}},sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2663.png"
 ALT="$\displaystyle = \frac{1}{2} V_z(t) \frac{a^{ 3}}{r^{ 3}} \sin\theta.$">|; 

$key = q/partialuslashpartialx=-partialvslashpartialy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3126.png"
 ALT="$ \partial u/\partial x = -\partial v/\partial y$">|; 

$key = q/u=a_1^{,2},lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img876.png"
 ALT="$ u=a_1^{ 2} \lambda$">|; 

$key = q/x=y=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img635.png"
 ALT="$ x=y=0$">|; 

$key = q/displaystylefrac{1}{V},frac{dbar{F}}{dz}=-{rmi}left(frac{k,V}{bar{F}}right)^{1slash2}+left(1-frac{k,V}{bar{F}}right)^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2261.png"
 ALT="$\displaystyle \frac{1}{V} \frac{d\bar{F}}{dz} =-{\rm i}\left(\frac{k V}{\bar{F}}\right)^{1/2}+\left(1-\frac{k V}{\bar{F}}\right)^{1/2},$">|; 

$key = q/displaystyleG(z)=F(z^{,2})=frac{Q}{2pi},ln(z^{,4}+4,a^{,4}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2118.png"
 ALT="$\displaystyle G(z)=F(z^{ 2}) = \frac{Q}{2\pi} \ln(z^{ 4}+4 a^{ 4}),$">|; 

$key = q/beta=(p_2-p_1)slashp_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5532.png"
 ALT="$ \beta=(p_2-p_1)/p_1$">|; 

$key = q/displaystyleM={rmRe}left[2pi,rho,V,B,{rme}^{,{rmi},(alpha-pislash2)}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3292.png"
 ALT="$\displaystyle M = {\rm Re} \left[2\pi \rho V B {\rm e}^{ {\rm i} (\alpha-\pi/2)}\right].$">|; 

$key = q/|n|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6245.png"
 ALT="$ \vert n\vert$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partialP'(theta,phi_pm,t)}{partialphi}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4750.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial P'(\theta,\phi_\pm,t)}{\partial\phi} = 0.$">|; 

$key = q/chi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="$ \chi$">|; 

$key = q/A'B'C'CBA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2194.png"
 ALT="$ A'B'C'CBA$">|; 

$key = q/displaystyleM_0=left[frac{1}{(pi,G)^3}left(frac{k}{m_p}right)^4,frac{3}{alpha}right]^{1slash2}4pi,y_1=3.586times10^{31},{rmkg}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5229.png"
 ALT="$\displaystyle M_0 = \left[\frac{1}{(\pi G)^3}\left(\frac{k}{m_p}\right)^4 \frac{3}{\alpha}\right]^{1/2}4\pi y_1 = 3.586\times 10^{31} {\rm kg}.$">|; 

$key = q/displaystyle{calH}_1+frac{1}{2},u_1^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5466.png"
 ALT="$\displaystyle {\cal H}_1+\frac{1}{2} u_1^{ 2}$">|; 

$key = q/displaystylenablacdotnablatimes{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6687.png"
 ALT="$\displaystyle \nabla\cdot\nabla \times {\bf A}$">|; 

$key = q/pi,rho,a^{,2},V^{,2}slash16;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2850.png"
 ALT="$ \pi \rho a^{ 2} V^{ 2}/16$">|; 

$key = q/displaystyle=xcostheta+ysintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6277.png"
 ALT="$\displaystyle = x\cos\theta + y\sin \theta,$">|; 

$key = q/displaystyleV_y(X,Y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2881.png"
 ALT="$\displaystyle V_y(X,Y)$">|; 

$key = q/displaystylephantom{simeqfrac{G,m',a^{2}}{R^{,3}}}+frac{e}{8},{calR}_2^{,(-2)}le(r,theta,varphi-[(1slash2),omega_1+(1slash2),omega_2],tright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="435" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4308.png"
 ALT="$\displaystyle \phantom{\simeq \frac{G m' a^{2}}{R^{ 3}}} +\frac{e}{8} {\cal R}_2^{ (-2)}\left(r,\theta,\varphi-[(1/2) \omega_1+(1/2) \omega_2] t\right)$">|; 

$key = q/displaystylefrac{{mitDelta}p}{p_1}simeqleft(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmMa[(gamma+1),{rmMa}_1^{,4}-4,({rmMa}_1^{,2}-1)right]theta^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="586" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6124.png"
 ALT="$\displaystyle \frac{{\mit\Delta}p}{p_1} \simeq \left(\frac{\gamma {\rm Ma}_1^{...
...\left[(\gamma+1) {\rm Ma}_1^{ 4}-4 ({\rm Ma}_1^{ 2}-1)\right]\theta^{ 2},
$">|; 

$key = q/|dz''|ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2069.png"
 ALT="$ \vert dz''\vert\ll 1$">|; 

$key = q/displaystyley(x)=h,[cosh(aslash2,h)-cosh(xslashh)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7359.png"
 ALT="$\displaystyle y(x) = h [\cosh(a/2 h) - \cosh(x/h)].$">|; 

$key = q/displaystylefrac{partial}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3014.png"
 ALT="$\displaystyle \frac{\partial}{\partial x}$">|; 

$key = q/displaystyleleft[1+frac{2,c_0}{(gamma+1),u_0}right]frac{1}{k},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5670.png"
 ALT="$\displaystyle \left[1+\frac{2 c_0}{(\gamma+1) u_0}\right]\frac{1}{k},
$">|; 

$key = q/xrightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3953.png"
 ALT="$ x\rightarrow 0$">|; 

$key = q/displaystyleK=K_{rmfluid}+K_{rmsphere}=frac{1}{2}left(m+frac{1}{2},m'right)V^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2681.png"
 ALT="$\displaystyle K = K_{\rm fluid}+K_{\rm sphere} = \frac{1}{2}\left(m+\frac{1}{2}  m'\right)V^{ 2}.$">|; 

$key = q/displaystyley^{,2}=4,k,x=left(frac{8,b}{pi+4}right)x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2274.png"
 ALT="$\displaystyle y^{ 2} = 4 k x = \left(\frac{8 b}{\pi+4}\right)x.$">|; 

$key = q/{bfq};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img519.png"
 ALT="$ {\bf q}$">|; 

$key = q/{bfv}=u(x,t),{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5341.png"
 ALT="$ {\bf v}= u(x,t) {\bf e}_x$">|; 

$key = q/displaystyle=a-BP'slash3=a,(1-omega,z^{,-1}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img782.png"
 ALT="$\displaystyle = a- BP'/3= a (1-\omega z^{ -1}),$">|; 

$key = q/int_{mitOmega}F,d{mitOmega}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4822.png"
 ALT="$ \int_{\mit\Omega} F d{\mit\Omega}=0.$">|; 

$key = q/displaystyle=-left(frac{partial{calE}}{partialv}right)_{calS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5579.png"
 ALT="$\displaystyle =- \left(\frac{\partial {\cal E}}{\partial v}\right)_{\cal S},$">|; 

$key = q/displaystylefrac{partialF}{partialy'}=frac{y'}{sqrt{1+y'^{,2}}}=c,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7327.png"
 ALT="$\displaystyle \frac{\partial F}{\partial y'} = \frac{y'}{\sqrt{1+y'^{ 2}}} = c,$">|; 

$key = q/displaystylep_1=rho,g,xi_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3945.png"
 ALT="$\displaystyle p_1 =\rho g \xi_z,$">|; 

$key = q/4.87times10^{-1},{rmN,m^{-1}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1085.png"
 ALT="$ 4.87\times 10^{-1} {\rm N m^{-1}}$">|; 

$key = q/skew{3}bar{psi}=-psi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2184.png"
 ALT="$ \skew{3}\bar{\psi}=-\psi$">|; 

$key = q/phi(xi,eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2084.png"
 ALT="$ \phi(\xi,\eta)$">|; 

$key = q/displaystylesigma_{rz}=sigma_{zr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="$\displaystyle \sigma_{rz}=\sigma_{zr}$">|; 

$key = q/displaystylebeta_{-r,,-s}=-int_{mitOmega}frac{costheta}{sintheta}left(frac{partilphi},frac{partial{mitPsi}_s}{partialtheta}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4885.png"
 ALT="$\displaystyle \beta_{-r, -s}= -\int_{\mit\Omega}\frac{\cos\theta}{\sin\theta}\...
...\partial\phi} \frac{\partial{\mit\Psi}_s}{\partial\theta}\right)d{\mit\Omega}.$">|; 

$key = q/displaystylefrac{dP}{dr}=rho,r,{mitOmega}^{,2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3635.png"
 ALT="$\displaystyle \frac{dP}{dr} = \rho r {\mit\Omega}^{ 2},$">|; 

$key = q/n=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2488.png"
 ALT="$ n=1$">|; 

$key = q/displaystyleV'^{,2}>frac{(1+s)^2}{s},c_1^{,2}=frac{(1+s)^2}{2,s}left(frac{lambda}{lambda_c}+frac{lambda_c}{lambda}right)c_0^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4196.png"
 ALT="$\displaystyle V'^{ 2} &gt; \frac{(1+s)^2}{s} c_1^{ 2}= \frac{(1+s)^2}{2 s}\left(\frac{\lambda}{\lambda_c}+\frac{\lambda_c}{\lambda}\right)c_0^{ 2}.$">|; 

$key = q/-pi<etaleqpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7224.png"
 ALT="$ -\pi&lt;\eta \leq \pi$">|; 

$key = q/c(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3395.png"
 ALT="$ c(z)$">|; 

$key = q/displaystylek_n';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4442.png"
 ALT="$\displaystyle k_n'$">|; 

$key = q/displaystylenablatimesnablau_i=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7056.png"
 ALT="$\displaystyle \nabla\times \nabla u_i = 0,$">|; 

$key = q/displaystyleint_Vfrac{partial(rho,v_i)}{partialt},dV+oint_Srho,v_i,v_j,dS_j=int_VF_i,dV+oint_Ssigma_{ij},dS_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="441" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="$\displaystyle \int_V \frac{\partial(\rho v_i)}{\partial t} dV +\oint_S\rho v_i v_j dS_j = \int_V F_i dV + \oint_S \sigma_{ij} dS_j,$">|; 

$key = q/{bfc}cdot{bfa}-{bfc}cdot{bfb}=c,|{bfa}-{bfb}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6718.png"
 ALT="$ {\bf c}\cdot{\bf a} - {\bf c}\cdot{\bf b} = c \vert{\bf a}-{\bf b}\vert$">|; 

$key = q/displaystyleB(theta,phi_pm,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4787.png"
 ALT="$\displaystyle B(\theta,\phi_\pm,t)$">|; 

$key = q/{bff};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6328.png"
 ALT="$ {\bf f}$">|; 

$key = q/{bfb}=2,{bfe}_x+{bfe}_y+{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6711.png"
 ALT="$ {\bf b} = 2 {\bf e}_x+ {\bf e}_y+{\bf e}_z$">|; 

$key = q/displaystyledeltasimeqepsilon_1+epsilon_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6179.png"
 ALT="$\displaystyle \delta\simeq \epsilon_1+\epsilon_2,
$">|; 

$key = q/displaystyle=2,x,y.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1956.png"
 ALT="$\displaystyle = 2  x y.$">|; 

$key = q/displaystylefrac{dy}{dx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4041.png"
 ALT="$\displaystyle \frac{dy}{dx}$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{Chapter09slashbs.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="191" BORDER="0"
 SRC="|."$dir".q|img3470.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{Chapter09/bs.eps}}
\end{figure}">|; 

$key = q/{bfv}=V,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2598.png"
 ALT="$ {\bf v} = V {\bf e}_z$">|; 

$key = q/zeta=-{mitGamma}slash(4pi,V_0,a);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1997.png"
 ALT="$ \zeta=-{\mit\Gamma}/(4\pi V_0 a)$">|; 

$key = q/displaystyleF_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1783.png"
 ALT="$\displaystyle F_y$">|; 

$key = q/rho_0=p_0slash(g,H);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5132.png"
 ALT="$ \rho_0= p_0/(g H)$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashvx1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img3031.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/vx1.eps}}
\end{figure}">|; 

$key = q/+alpha_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5794.png"
 ALT="$ +\alpha_0$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter08slashfs.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img2975.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter08/fs.eps}}
\end{figure}">|; 

$key = q/f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6517.png"
 ALT="$ f(x)$">|; 

$key = q/d{bfS}=(dS_x,,dS_y,,dS_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6642.png"
 ALT="$ d{\bf S} = (dS_x,  dS_y,  dS_z)$">|; 

$key = q/2,[(m-m')slash(m+m')],(gslashomega^{,2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1865.png"
 ALT="$ 2 [(m-m')/(m+m')] (g/\omega^{ 2})$">|; 

$key = q/displaystyleL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3508.png"
 ALT="$\displaystyle L$">|; 

$key = q/lambda_r,int_{mitOmega}(cdots),{mitPhi}_r,d{mitOmega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4910.png"
 ALT="$ \lambda_r \int_{\mit\Omega} (\cdots) {\mit\Phi}_r d{\mit\Omega}$">|; 

$key = q/delta(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3396.png"
 ALT="$ \delta(z)$">|; 

$key = q/displaystyle=frac{partial}{partialt}+v_r,frac{partial}{partialr}+frac{v_theta}{reta}+frac{v_varphi}{r,sintheta},frac{partial}{partialvarphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img512.png"
 ALT="$\displaystyle = \frac{\partial}{\partial t} + v_r \frac{\partial }{\partial r}...
...l \theta} + \frac{v_\varphi}{r \sin\theta} \frac{\partial}{\partial \varphi},$">|; 

$key = q/displaystyle=frac{1}{sintheta}left[frac{partial}{partialtheta}left(costheta,fract(costheta,frac{partial{mitPhi}_r}{partialtheta}right)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4841.png"
 ALT="$\displaystyle = \frac{1}{\sin\theta}\left[ \frac{\partial}{\partial\theta}\left...
...hi}\left(\cos\theta \frac{\partial{\mit\Phi}_r}{\partial\theta}\right)\right],$">|; 

$key = q/F_2(z)=phi_2(x,y)+{rmi},psi_2(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2009.png"
 ALT="$ F_2(z)=\phi_2(x,y)+{\rm i} \psi_2(x,y)$">|; 

$key = q/displaystyle=int_0^piint_{-1}^{1}left(frac{partial{mitPhi}_n^{,m}}{partialmu},frfrac{partial{mitPhi}_{n'}^{,m'}}{partialmu}right)mu,dmu,dphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4949.png"
 ALT="$\displaystyle = \int_0^\pi\int_{-1}^{1}\left( \frac{\partial{\mit\Phi}_n^{ m}}...
...}  \frac{\partial{\mit\Phi}_{n'}^{ m'}}{\partial\mu}\right) \mu d\mu d\phi,$">|; 

$key = q/p_0-rho,g,zeta+left.p_1right|_{z=0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4142.png"
 ALT="$ p_0-\rho g \zeta+\left.p_1\right\vert _{z=0}$">|; 

$key = q/displaystylev_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1644.png"
 ALT="$\displaystyle v_r$">|; 

$key = q/displaystyleU=frac{1}{4},(rho,g+gamma,k^{,2}),a^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4213.png"
 ALT="$\displaystyle U= \frac{1}{4} (\rho g+\gamma k^{ 2}) a^{ 2}.
$">|; 

$key = q/displaystyleH(k)=kleft.left[lnleft(frac{1+k}{1-k}right)-frac{3,k}{2}right]rightslashleft[lnleft(frac{1+k}{1-k}right)-2,kright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3721.png"
 ALT="$\displaystyle H(k) = k\left.\left[\ln\left(\frac{1+k}{1-k}\right)-\frac{3 k}{2}\right]\right/\left[\ln\left(\frac{1+k}{1-k}\right)-2 k\right].$">|; 

$key = q/displaystylecosgamma=frac{{bfr}cdot{bfr}'}{r,r'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4236.png"
 ALT="$\displaystyle \cos\gamma =\frac{{\bf r}\cdot{\bf r}'}{r r'}.$">|; 

$key = q/[p(z_2)slashp(z_1)]^{1slashgamma},rho(z_1)<rho(z_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5166.png"
 ALT="$ [p(z_2)/p(z_1)]^{1/\gamma} \rho(z_1) &lt; \rho(z_2)$">|; 

$key = q/skew{5}tilde{zeta}_6=-10.36times10^{-2},{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5039.png"
 ALT="$ \skew{5}\tilde{\zeta}_6=-10.36\times 10^{-2} {\rm m}$">|; 

$key = q/displaystylem_{rmadded}=frac{K}{(1slash2),V^{,2}}=frac{2pi}{3},,a^{,3},rho.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2750.png"
 ALT="$\displaystyle m_{\rm added} = \frac{K}{(1/2) V^{ 2}} = \frac{2\pi}{3}  a^{ 3} \rho.$">|; 

$key = q/displaystylexi=left[frac{(n+1),K}{4pi,G},rho_c^{,(n-1)slashn}right]^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5275.png"
 ALT="$\displaystyle \xi=\left[\frac{(n+1) K}{4\pi G} \rho_c^{ (n-1)/n}\right]^{1/2}.
$">|; 

$key = q/displaystylefrac{dy_L}{dx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5843.png"
 ALT="$\displaystyle \frac{dy_L}{dx}$">|; 

$key = q/u=a_3^{,2},tan^2theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7264.png"
 ALT="$ u=a_3^{ 2} \tan^2\theta$">|; 

$key = q/displaystylenu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="$\displaystyle \nu$">|; 

$key = q/displaystylefrac{partial}{partialtheta},(sintheta,Y')-frac{partialX'}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4747.png"
 ALT="$\displaystyle \frac{\partial}{\partial\theta} (\sin\theta Y')-\frac{\partial X'}{\partial\phi}$">|; 

$key = q/pm1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2102.png"
 ALT="$ \pm 1)$">|; 

$key = q/displaystylea(n),z_{n-2}+b(n),z_n+c(n),z_{n+2}=frac{6}{beta},(1+k_2-h_2),skew{5}tilde{zeta}_j,delta_{n,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4698.png"
 ALT="$\displaystyle a(n) z_{n-2} + b(n) z_n + c(n) z_{n+2} = \frac{6}{\beta} (1+k_2-h_2) \skew{5}\tilde{\zeta}_j \delta_{n 2}$">|; 

$key = q/{mitDelta}Pslashl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3676.png"
 ALT="$ {\mit\Delta} P/l$">|; 

$key = q/mrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2962.png"
 ALT="$ m\rightarrow \infty$">|; 

$key = q/psi_2-psi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1625.png"
 ALT="$ \psi_2-\psi_1$">|; 

$key = q/{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6256.png"
 ALT="$ {\bf e}_x$">|; 

$key = q/displaystyle=frac{2}{m^{,2}-m'^{,2}}int_{-1}^1left(m^{,2},P_n^{,m},frac{dP_{n'}^}{dmu}+m'^{,2},frac{dP_n^{,m}}{dmu},P_{n'}^{,m'}right)mu,dmu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4959.png"
 ALT="$\displaystyle = \frac{2}{m^{ 2}-m'^{ 2}}\int_{-1}^1 \left(m^{ 2} P_n^{ m} ...
...,m'}}{d\mu} + m'^{ 2} \frac{dP_n^{ m}}{d\mu} P_{n'}^{ m'}\right)\mu d\mu,$">|; 

$key = q/etarightarrowtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2745.png"
 ALT="$ \eta\rightarrow\theta$">|; 

$key = q/displaystyle=delta_{ij},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6796.png"
 ALT="$\displaystyle = \delta_{ij},$">|; 

$key = q/displaystyle=frac{2,omega^{,2},(z^{,2}-1),(z^{,2}-omega^{-1},z+1)}{z^{,3}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img787.png"
 ALT="$\displaystyle = \frac{2 \omega^{ 2} (z^{ 2}-1) (z^{ 2}-\omega^{-1} z+1)}{z^{ 3}},$">|; 

$key = q/displaystyled_{ij}=2,muleft(e_{ij}-frac{1}{3},e_{kk},delta_{ij}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="$\displaystyle d_{ij} = 2 \mu\left(e_{ij} - \frac{1}{3} e_{kk} \delta_{ij}\right),$">|; 

$key = q/displaystylefrac{dG}{dx}=2,gamma,(gamma-1),(1-x)^{,2},frac{[2+(gamma-1),x]^{,gamma-1}}{[(gamma+1),x]^{,gamma+1}}geq0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5496.png"
 ALT="$\displaystyle \frac{dG}{dx} = 2 \gamma (\gamma-1) (1-x)^{ 2} \frac{[2+(\gamma-1) x]^{ \gamma-1}}{[(\gamma+1) x]^{ \gamma+1}}\geq 0.$">|; 

$key = q/=nablatimes{bfv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3380.png"
 ALT="$ =\nabla\times {\bf v}$">|; 

$key = q/displaystyle=Rleft{1-e,cos[(omega_1-omega_2),t]+{calO}(e^{,2})right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4296.png"
 ALT="$\displaystyle =R \left\{1-e \cos[(\omega_1-\omega_2) t]+{\cal O}(e^{ 2})\right\}.$">|; 

$key = q/displaystyle=left(frac{n-m+1}{2,n+1}right)P_{n+1}^{,m}(mu)+left(frac{n+m}{2,n+1}right)P_{n-1}^{,m}(mu),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="391" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4690.png"
 ALT="$\displaystyle = \left(\frac{n-m+1}{2 n+1}\right)P_{n+1}^{ m}(\mu)+\left(\frac{n+m}{2 n+1}\right)P_{n-1}^{ m}(\mu),$">|; 

$key = q/displaystylebar{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img748.png"
 ALT="$\displaystyle \bar{y}$">|; 

$key = q/theta=theta_{rmmax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$ \theta =\theta _{\rm max}$">|; 

$key = q/displaystyle=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7120.png"
 ALT="$\displaystyle = 1.$">|; 

$key = q/U(theta)=2,U_0,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3219.png"
 ALT="$ U(\theta)=2 U_0 \sin\theta$">|; 

$key = q/displaystyle=frac{w}{2},sinhleft(pi,frac{x}{w}right)sinleft(pi,frac{y}{w}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2165.png"
 ALT="$\displaystyle = \frac{w}{2} \sinh\left(\pi \frac{x}{w}\right)\sin\left(\pi \frac{y}{w}\right).$">|; 

$key = q/xi=xi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2463.png"
 ALT="$ \xi=\xi_0$">|; 

$key = q/psi={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2405.png"
 ALT="$ \psi={\rm constant}$">|; 

$key = q/p_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1253.png"
 ALT="$ p_0=0$">|; 

$key = q/displaystyle=-left[frac{(m+1),U_0^{,3},x^{,3,m-1}}{2,nu}right]^{1slash2}f'',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2931.png"
 ALT="$\displaystyle =-\left[\frac{(m+1) U_0^{ 3} x^{ 3 m-1}}{2 \nu}\right]^{1/2}f'',$">|; 

$key = q/widehat{omega}ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img903.png"
 ALT="$ \widehat{\omega}\ll 1$">|; 

$key = q/displaystyle=lambda,delta_{il},delta_{jk}+mu,delta_{ij},delta_{kl}+nu,delta_{ik},delta_{jl}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6969.png"
 ALT="$\displaystyle =\lambda \delta_{il} \delta_{jk} + \mu \delta_{ij} \delta_{kl} +\nu \delta_{ik} \delta_{jl}.$">|; 

$key = q/displaystyleD=1.33,frac{rho,U_0^{,2},L}{{rmRe}^{1slash2}}=1.33,rho,U_0,(nu,U_0,L)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2995.png"
 ALT="$\displaystyle D = 1.33 \frac{\rho U_0^{ 2} L}{{\rm Re}^{1/2}}=1.33 \rho U_0 (\nu U_0 L)^{1/2}.$">|; 

$key = q/displaystyle-{rmRe}oint_Cfrac{dF}{dz},dz={mitGamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3278.png"
 ALT="$\displaystyle -{\rm Re}\oint_C \frac{dF}{dz} dz={\mit\Gamma}$">|; 

$key = q/a_2=a_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img871.png"
 ALT="$ a_2=a_1$">|; 

$key = q/displaystyleint_S{bfv}cdotd{bfS}=2pi,r,v_r(r)=Q,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1671.png"
 ALT="$\displaystyle \int_S {\bf v}\cdot d{\bf S} = 2\pi r v_r(r)= Q,$">|; 

$key = q/O_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4556.png"
 ALT="$ O_1$">|; 

$key = q/displaystyle=-V_0,r,costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1972.png"
 ALT="$\displaystyle =-V_0 r \cos\theta,$">|; 

$key = q/{bfp}=p,{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7209.png"
 ALT="$ {\bf p} = p {\bf e}_z$">|; 

$key = q/dz=dx+{rmi},dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2341.png"
 ALT="$ dz=dx+{\rm i} dy$">|; 

$key = q/pislash4+thetaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1203.png"
 ALT="$ \pi/4+\theta/2$">|; 

$key = q/(1-k^{,2})^{1slash2}leqrslashalphaleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1314.png"
 ALT="$ (1-k^{ 2})^{1/2}\leq r/\alpha\leq 1$">|; 

$key = q/2tmspace+thinmuskip{.1667em}alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$ 2\tmspace +\thinmuskip {.1667em}\alpha $">|; 

$key = q/s=mslashm';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1859.png"
 ALT="$ s=m/m'$">|; 

$key = q/theta=25^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$ \theta =25^\circ $">|; 

$key = q/displaystyle=-nablaleft(frac{p}{skew{3}bar{rho}}+{mitPhi}+chiright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4450.png"
 ALT="$\displaystyle = - \nabla\left(\frac{p}{\skew{3}\bar{\rho}} + {\mit\Phi} + \chi\right),$">|; 

$key = q/displaystylev_r(r)=frac{Q}{4pi,r^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2607.png"
 ALT="$\displaystyle v_r(r) = \frac{Q}{4\pi r^{ 2}}.$">|; 

$key = q/displaystyle=-left(frac{cosalpha-cosomega}{sin^4alpha}right)sinomega,domega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2309.png"
 ALT="$\displaystyle =-\left(\frac{\cos\alpha-\cos\omega}{\sin^4\alpha}\right)\sin\omega d\omega,$">|; 

$key = q/dbar{F}slashdz=v_x-{rmi},v_y=-{rmi},partialskew{3}bar{phi}slashpartialy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2282.png"
 ALT="$ d\bar{F}/dz=v_x-{\rm i} v_y=-{\rm i} \partial \skew{3}\bar{\phi}/\partial y$">|; 

$key = q/dr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2544.png"
 ALT="$ dr$">|; 

$key = q/2,{rmh},25,{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1079.png"
 ALT="$ 2 {\rm h} 25 {\rm m}$">|; 

$key = q/displaystylec_{x'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6359.png"
 ALT="$\displaystyle c_{x'}$">|; 

$key = q/displaystyleF(zeta)=frac{Q}{2pi},ln(zeta^{,2}+4,a^{,4}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2117.png"
 ALT="$\displaystyle F(\zeta) = \frac{Q}{2\pi} \ln(\zeta^{ 2}+4 a^{ 4}),$">|; 

$key = q/d=c,tanmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6064.png"
 ALT="$ d=c \tan\mu$">|; 

$key = q/displaystyle(AP'+BQ'),a=4,a^{,2},s.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img739.png"
 ALT="$\displaystyle (AP' + BQ') a = 4 a^{ 2} s.$">|; 

$key = q/displaystylefrac{p-p_1}{p_1}simeqleft(frac{gamma,{rmMa}_1^{,2}}{sqrt{{rmMa}_1^{,2}-1}}right)theta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5786.png"
 ALT="$\displaystyle \frac{p-p_1}{p_1}\simeq \left(\frac{\gamma {\rm Ma}_1^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}\right)\theta,$">|; 

$key = q/displaystyle=frac{partial(zeta_b+zeta)}{partialt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4473.png"
 ALT="$\displaystyle = \frac{\partial(\zeta_b+\zeta)}{\partial t}.$">|; 

$key = q/displaystyle=-frac{1}{sintheta}left[frac{partial}{partialtheta},(sintheta,costheta,xi)+frac{partial}{partialphi},(costheta,eta)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4772.png"
 ALT="$\displaystyle = -\frac{1}{\sin\theta}\left[\frac{\partial}{\partial\theta} (\s...
...a \cos\theta \xi) +\frac{\partial}{\partial \phi} (\cos\theta \eta)\right],$">|; 

$key = q/displaystyleu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img784.png"
 ALT="$\displaystyle u$">|; 

$key = q/bslasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4133.png"
 ALT="$ b/a$">|; 

$key = q/displaystyleG(y)=B_1,{rme}^{-,m,k,y}+B_2,{rme}^{+m,k,y},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5967.png"
 ALT="$\displaystyle G(y) = B_1 {\rm e}^{- m k y}+B_2 {\rm e}^{+m k y},$">|; 

$key = q/displaystyleint_0^picos(m,phi),cos(m',phi),dphi=left{{array}{lll}pi&mbox{hspace{m=m'=0[0.5ex]pislash2&&m=m'neq0[0.5ex]0&&mneqm'{array}right..;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="98" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4936.png"
 ALT="$\displaystyle \int_0^\pi \cos(m \phi) \cos(m' \phi) d\phi =\left\{ \begin{a...
...}&amp;m=m'=0\ [0.5ex] \pi/2&amp;&amp;m=m'\neq 0\ [0.5ex] 0&amp;&amp;m\neq m' \end{array} \right..$">|; 

$key = q/displaystyle(m+1),f'''+(m+1),f,f''-2m,f'^{,2}=-frac{1}{U_0^{,2},x^{,2,m-1}},frac{dU^{,2}}{dx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="464" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2937.png"
 ALT="$\displaystyle (m+1) f'''+ (m+1) f f'' -2m f'^{ 2} = -\frac{1}{U_0^{ 2} x^{ 2 m-1}} \frac{dU^{ 2}}{dx}.$">|; 

$key = q/theta_{n,,n'}^{,m,,m'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4991.png"
 ALT="$ \theta_{n, n'}^{ m, m'}= 0$">|; 

$key = q/displaystyle=-frac{g,d}{a^{,2}},frac{partial}{partialtheta},(zeta-{calF}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4727.png"
 ALT="$\displaystyle =- \frac{g d}{a^{ 2}} \frac{\partial }{\partial\theta} (\zeta-{\cal F}),$">|; 

$key = q/displaystylefrac{p}{p_0}=left(1-frac{z}{c}right)^{cslashH}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5260.png"
 ALT="$\displaystyle \frac{p}{p_0} = \left(1-\frac{z}{c}\right)^{c/H}.
$">|; 

$key = q/displaystylePsimg,zetasima,{mitOmega},vggd,{mitOmega},v.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4497.png"
 ALT="$\displaystyle P\sim g \zeta\sim a {\mit\Omega} v\gg d {\mit\Omega} v.$">|; 

$key = q/displaystyle=p_0-rho,g,z+p_1({bfr},t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4069.png"
 ALT="$\displaystyle =p_0 - \rho g z + p_1({\bf r},t),$">|; 

$key = q/displaystyleC_L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5799.png"
 ALT="$\displaystyle C_L$">|; 

$key = q/displaystylev_z(x,y)=frac{G}{2,mu},frac{a^{,2},b^{,2}-b^{,2},x^{,2}-a^{,2},y^{,2}}{a^{,2}+b^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3881.png"
 ALT="$\displaystyle v_z(x,y) = \frac{G}{2 \mu} \frac{a^{ 2} b^{ 2}-b^{ 2} x^{ 2}-a^{ 2} y^{ 2}}{a^{ 2}+b^{ 2}}
$">|; 

$key = q/kappa_y^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img661.png"
 ALT="$ \kappa_y^{ 2}$">|; 

$key = q/displaystyle=1+({rmMa}_1^{,2}-1),frac{tan^{,2}z}{tan^{,2}z_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6134.png"
 ALT="$\displaystyle = 1+ ({\rm Ma}_1^{ 2}-1) \frac{\tan^{ 2} z}{\tan^{ 2} z_1},$">|; 

$key = q/{figure}{epsfysize{=1.1in{centerline{epsffile{AppendixAslashfigA.01.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="122" BORDER="0"
 SRC="|."$dir".q|img6222.png"
 ALT="\begin{figure}
\epsfysize =1.1in
\centerline{\epsffile{AppendixA/figA.01.eps}}
\end{figure}">|; 

$key = q/D=(0,,+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2838.png"
 ALT="$ D=(0, +1)$">|; 

$key = q/displaystyleh_xi=h_eta=left[left(frac{partialz}{partialxi}right)^2+left(frac{parleft(frac{partialvarpi}{partialeta}right)^2right]^{,1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="516" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2690.png"
 ALT="$\displaystyle h_\xi=h_\eta= \left[\left(\frac{\partial z}{\partial\xi}\right)^2...
...}\right)^2+ \left(\frac{\partial \varpi}{\partial\eta}\right)^2\right]^{ 1/2},$">|; 

$key = q/{calS}_2-{calS}_1geq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5500.png"
 ALT="$ {\cal S}_2-{\cal S}_1\geq 0$">|; 

$key = q/displaystyle2,a_{ijk}+a_{kji}+a_{ikj};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6919.png"
 ALT="$\displaystyle 2 a_{ijk} +a_{kji} +a_{ikj}$">|; 

$key = q/360^circslash{mitPi}_M=8.847,{rmyears};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4589.png"
 ALT="$ 360^\circ/{\mit\Pi}_M=8.847 {\rm years}$">|; 

$key = q/|x|>c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2471.png"
 ALT="$ \vert x\vert&gt;c$">|; 

$key = q/(1-k^{,2})^{1slash4}leqrslashalphaleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1324.png"
 ALT="$ (1-k^{ 2})^{1/4}\leq r/\alpha \leq 1$">|; 

$key = q/displaystyleQ=h(0),v(0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1532.png"
 ALT="$\displaystyle Q = h(0) v(0),$">|; 

$key = q/n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2479.png"
 ALT="$ n$">|; 

$key = q/displaystylephi=frac{Q}{4pi,r}=frac{Q}{4pi,(varpi^{,2}+z^{,2})^{1slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2614.png"
 ALT="$\displaystyle \phi = \frac{Q}{4\pi r} = \frac{Q}{4\pi (\varpi^{ 2}+z^{ 2})^{1/2}}.$">|; 

$key = q/displaystyle=-ointp(a,theta,t),sintheta,a,dtheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1815.png"
 ALT="$\displaystyle =-\oint p(a,\theta,t) \sin\theta a d\theta.$">|; 

$key = q/displaystyleh_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1892.png"
 ALT="$\displaystyle h_2$">|; 

$key = q/displaystylephi(x,z,t)=A,[1+k^{,2},(z+d)^2slash2],cos(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="386" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3956.png"
 ALT="$\displaystyle \phi(x,z,t) = A [1+k^{ 2} (z+d)^2/2] \cos(\omega t-k x).$">|; 

$key = q/x<x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3104.png"
 ALT="$ x&lt;x_0$">|; 

$key = q/displaystyle=tan^{-1}left(sqrt{frac{chi_w}{1-chi_w}},right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6104.png"
 ALT="$\displaystyle = \tan^{-1}\left(\sqrt{\frac{\chi_w}{1-\chi_w}} \right),$">|; 

$key = q/partialslashpartialtsim{calO}({mitOmega});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4495.png"
 ALT="$ \partial/\partial t\sim{\cal O}({\mit\Omega})$">|; 

$key = q/sigma_{ji}=sigma_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7021.png"
 ALT="$ \sigma_{ji}=\sigma_{ij}$">|; 

$key = q/xi_1=6.897;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5222.png"
 ALT="$ \xi_1=6.897$">|; 

$key = q/displaystylestackrel{displaystylerightarrow}{PR},=,stackrel{displaystylerightarrlaystylerightarrow}{PS}+stackrel{displaystylerightarrow}{SR}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6232.png"
 ALT="$\displaystyle \stackrel{\displaystyle \rightarrow}{PR}  =   \stackrel{\displa...
...ackrel{\displaystyle \rightarrow}{PS}+\stackrel{\displaystyle \rightarrow}{SR}.$">|; 

$key = q/displaystyleA_x(x+dx),dy,dz-A_x(x),dy,dz=frac{partialA_x}{partialx},dx,dy,dz=frac{partialA_x}{partialx},dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6579.png"
 ALT="$\displaystyle A_x(x+dx)  dy dz - A_x(x)  dy dz = \frac{\partial A_x}{\partial x} dx dy dz = \frac{\partial A_x}{\partial x} dV,$">|; 

$key = q/displaystylefrac{d}{l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3659.png"
 ALT="$\displaystyle \frac{d}{l}$">|; 

$key = q/displaystylewidehat{P}(X,Y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2883.png"
 ALT="$\displaystyle \widehat{P}(X,Y)$">|; 

$key = q/p_e=p_{rmatm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5424.png"
 ALT="$ p_e=p_{\rm atm}$">|; 

$key = q/dxslashdt=u+c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5662.png"
 ALT="$ dx/dt=u+c$">|; 

$key = q/displaystyle=(1-a_3^{,2}slasha_2^{,2})^{1slash2}=sinbeta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img932.png"
 ALT="$\displaystyle =(1-a_3^{ 2}/a_2^{ 2})^{1/2}= \sin\beta,$">|; 

$key = q/displaystyle{rmRe}left[-frac{1}{2},rhooint_Cleft(frac{dF}{dz}right)^{,2}z,dzright]=M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3289.png"
 ALT="$\displaystyle {\rm Re}\left[-\frac{1}{2} \rho \oint_C \left(\frac{dF}{dz}\right)^{ 2} z dz\right]=M$">|; 

$key = q/displaystyle[{bfa}times({bfb}times{bfc})]_i=epsilon_{kij},epsilon_{klm},a_j,b_l,delta_{il},delta_{jm}-delta_{im},delta_{jl}right)a_j,b_l,c_m,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="486" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6818.png"
 ALT="$\displaystyle [{\bf a}\times ({\bf b}\times {\bf c})]_i = \epsilon_{kij} \epsi...
... = \left(\delta_{il} \delta_{jm}-\delta_{im} \delta_{jl}\right)a_j b_l c_m,$">|; 

$key = q/F=(-h,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2843.png"
 ALT="$ F=(-h, 0)$">|; 

$key = q/alpha=pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2507.png"
 ALT="$ \alpha=\pi/2$">|; 

$key = q/0.28125<s<0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img805.png"
 ALT="$ 0.28125&lt;s&lt;0.5$">|; 

$key = q/delta{mitPhi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4480.png"
 ALT="$ \delta{\mit\Phi}$">|; 

$key = q/displaystylerightarrowbeta_{n,,-n'}^{,m,,-m'}=int_{mitOmega}costhetaleft(frac{parac{partial{mitPsi}_{n'}^{,m'}}{partialphi}right)d{mitOmega},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4946.png"
 ALT="$\displaystyle \rightarrow \beta_{n, -n'}^{ m, -m'}= \int_{\mit\Omega}\cos\th...
...hi}  \frac{\partial{\mit\Psi}_{n'}^{ m'}}{\partial\phi}\right) d{\mit\Omega},$">|; 

$key = q/u=u(theta,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4474.png"
 ALT="$ u=u(\theta,\phi,t)$">|; 

$key = q/psi_2(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2011.png"
 ALT="$ \psi_2(x,y)$">|; 

$key = q/displaystyle{calS}=frac{S}{m}={calC}_Vlnleft(frac{p}{rho^{,gamma}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5328.png"
 ALT="$\displaystyle {\cal S} =\frac{S}{m}={\cal C}_V\ln\left(\frac{p}{\rho^{ \gamma}}\right).$">|; 

$key = q/displaystyletimes{bfr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6651.png"
 ALT="$\displaystyle \times {\bf r}.$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{Chapter03slashfig6.03.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="457" HEIGHT="397" BORDER="0"
 SRC="|."$dir".q|img1138.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{Chapter03/fig6.03.eps}}
\end{figure}">|; 

$key = q/-sindelta,zeta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4531.png"
 ALT="$ -\sin\delta \zeta_2$">|; 

$key = q/displaystylemu,P_n^{,m}(mu);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4689.png"
 ALT="$\displaystyle \mu P_n^{ m}(\mu)$">|; 

$key = q/y=pmaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$ y=\pm a/2$">|; 

$key = q/displaystyle=-frac{nablap}{rho}-nabla{mitPsi}+nu,nabla^{,2}{bfv}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="$\displaystyle = - \frac{\nabla p}{\rho} -\nabla{\mit\Psi} + \nu \nabla^{ 2} {\bf v}.$">|; 

$key = q/beta_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6099.png"
 ALT="$ \beta_s$">|; 

$key = q/bar{u};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6042.png"
 ALT="$ \bar{u}$">|; 

$key = q/displaystyleepsilon_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6180.png"
 ALT="$\displaystyle \epsilon_1$">|; 

$key = q/displaystylerho,frac{D{calE}}{Dt}=frac{partialv_i}{partialx_j},sigma_{ij}+frac{pial}{partialx_j}!left(kappa,frac{partialT}{partialx_j}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="$\displaystyle \rho \frac{D{\cal E}}{Dt} = \frac{\partial v_i}{\partial x_j}  ...
...{\partial}{\partial x_j}\!\left(\kappa \frac{\partial T}{\partial x_j}\right).$">|; 

$key = q/displaystyle={bfn}times{mitDelta}{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3383.png"
 ALT="$\displaystyle = {\bf n} \times{\mit \Delta} {\bf v},$">|; 

$key = q/4pi,a,gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1352.png"
 ALT="$ 4\pi a \gamma$">|; 

$key = q/|theta|ll1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6156.png"
 ALT="$ \vert\theta\vert\ll 1$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter13slashfig4.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img5215.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter13/fig4.02.eps}}
\end{figure}">|; 

$key = q/displaystyle=X_1,sectheta,left(tan^2theta-frac{1}{2}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4221.png"
 ALT="$\displaystyle = X_1 \sec\theta \left(\tan^2\theta-\frac{1}{2}\right),$">|; 

$key = q/X_0=2,j,lambda_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4060.png"
 ALT="$ X_0=2 j \lambda_0$">|; 

$key = q/displaystyle{bfe}_perp={bfe}_paralleltimes{bfe}_z=sinalpha,{bfe}_x+cosalpha,{bfe}_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3322.png"
 ALT="$\displaystyle {\bf e}_\perp= {\bf e}_\parallel \times{\bf e}_z= \sin\alpha {\bf e}_x+\cos\alpha {\bf e}_y$">|; 

$key = q/zeta=(0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2101.png"
 ALT="$ \zeta = (0$">|; 

$key = q/{bfe}_xequiv(1,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6264.png"
 ALT="$ {\bf e}_x\equiv (1, 0, 0)$">|; 

$key = q/displaystyle=frac{2pi,a_1,a_2,a_3}{a_i^{,2}}int_0^inftyfrac{du}{mitDelta}-frac{1la_i}!left(2pi,a_1,a_2,a_3int_0^inftyfrac{du}{mitDelta}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7268.png"
 ALT="$\displaystyle = \frac{2\pi a_1 a_2 a_3}{a_i^{ 2}} \int_0^\infty\frac{du}{\m...
...tial a_i}\!\left( 2\pi a_1 a_2 a_3\int_0^\infty \frac{du}{\mit\Delta}\right)$">|; 

$key = q/V_1=V-V_0=(1-s),V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img603.png"
 ALT="$ V_1= V-V_0 = (1-s) V$">|; 

$key = q/displaystyle=c_{ij}'={calR}_{il},{calR}_{jm},c_{lm}={calR}_{il},{calR}_{jm},a_{lcalR}_{il},{calR}_{jm},a_{lk},{calR}_{pm},{calR}_{qk},b_{pq}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6861.png"
 ALT="$\displaystyle =c_{ij}' = {\cal R}_{il} {\cal R}_{jm} c_{lm}= {\cal R}_{il} {...
...mk}={\cal R}_{il} {\cal R}_{jm} a_{lk} {\cal R}_{pm} {\cal R}_{qk} b_{pq}'$">|; 

$key = q/displaystylegamma,oint_C{bft}times{bfn},dr=-gammaint_S(nablacdot{bfn}),{bfn},dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1105.png"
 ALT="$\displaystyle \gamma \oint_C {\bf t}\times{\bf n} dr = - \gamma\int_S (\nabla \cdot{\bf n}) {\bf n} dS,$">|; 

$key = q/displaystyle{bfa}+{bfb}equiv(a_x+b_x,,a_y+b_y,,a_z+b_z).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6272.png"
 ALT="$\displaystyle {\bf a} + {\bf b} \equiv (a_x+b_x, a_y+b_y, a_z+b_z).$">|; 

$key = q/{figure}{epsfysize{=3.25in{centerline{epsffile{Chapter06slashplate.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="521" HEIGHT="373" BORDER="0"
 SRC="|."$dir".q|img2253.png"
 ALT="\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter06/plate.eps}}
\end{figure}">|; 

$key = q/displaystyle=2,l,sinh[ln(cslashl)]=c-l^{,2}slashc.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2130.png"
 ALT="$\displaystyle = 2 l \sinh[\ln(c/l)] = c - l^{ 2}/c.$">|; 

$key = q/0.47-6tmspace+thinmuskip{.1667em}lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$ 0.47-6\tmspace +\thinmuskip {.1667em}\lambda $">|; 

$key = q/f(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4798.png"
 ALT="$ f(\theta,\phi)$">|; 

$key = q/x=pml;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$ x=\pm l$">|; 

$key = q/gamma<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1086.png"
 ALT="$ \gamma&lt;0$">|; 

$key = q/y={rmIm}(z_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2176.png"
 ALT="$ y={\rm Im}(z_0)$">|; 

$key = q/displaystyle=-p+2,mu,frac{partialv_x}{partialx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="$\displaystyle = -p + 2 \mu \frac{\partial v_x}{\partial x},$">|; 

$key = q/v_r(a,theta)equiv-(1slasha),partialpsislashpartialtheta|_{r=a}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1766.png"
 ALT="$ v_r(a,\theta) \equiv -(1/a) \partial\psi/\partial \theta\vert _{r=a} = 0$">|; 

$key = q/displaystyle=F(xmpc_0,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5359.png"
 ALT="$\displaystyle =F(x\mp c_0 t),$">|; 

$key = q/displaystyle=frac{partial^{,2}}{partialx^{,2}}+frac{partial^{,2}}{partialy^{,2}}+frac{partial^{,2}}{partialz^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="$\displaystyle =\frac{\partial^{ 2}}{\partial x^{ 2}} + \frac{\partial^{ 2}}{\partial y^{ 2}}+\frac{\partial^{ 2}}{\partial z^{ 2}},$">|; 

$key = q/r=(x^{,2}+y^{,2})^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1670.png"
 ALT="$ r=(x^{ 2}+y^{ 2})^{1/2}$">|; 

$key = q/displaystylebeta_{n,,n'}^{,m,,m'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4948.png"
 ALT="$\displaystyle \beta_{n, n'}^{ m, m'}$">|; 

$key = q/displaystylefrac{d}{dx}!left(y',frac{partialF}{partialy'}right)=y',frac{d}{dx}!lrac{partialF}{partialy'}right)+y'',frac{partialF}{partialy'}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7321.png"
 ALT="$\displaystyle \frac{d}{dx}\!\left(y' \frac{\partial F}{\partial y'}\right) = y...
...eft(\frac{\partial F}{\partial y'}\right)+ y'' \frac{\partial F}{\partial y'}.$">|; 

$key = q/displaystylek=frac{2,b}{pi+4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2269.png"
 ALT="$\displaystyle k=\frac{2 b}{\pi+4}.$">|; 

$key = q/displaystylesigma_{ii}=sigma_{11}'+sigma_{22}'+sigma_{33}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="$\displaystyle \sigma_{ii} = \sigma_{11}'+\sigma_{22}'+\sigma_{33}'.$">|; 

$key = q/sigma_{ij}({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$ \sigma_{ij}({\bf r}, t)$">|; 

$key = q/dT=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6545.png"
 ALT="$ dT=0$">|; 

$key = q/|z|>bslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3501.png"
 ALT="$ \vert z\vert&gt;b/2$">|; 

$key = q/displaystylec^{,2}=frac{g,(rho-rho')}{k,(rho+rho')}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4113.png"
 ALT="$\displaystyle c^{ 2} = \frac{g (\rho-\rho')}{k (\rho+\rho')}.$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter11slashgroup.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="281" BORDER="0"
 SRC="|."$dir".q|img4030.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter11/group.eps}}
\end{figure}">|; 

$key = q/displaystylepsi(a,theta)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1767.png"
 ALT="$\displaystyle \psi(a,\theta)= 0,$">|; 

$key = q/e_{13}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img874.png"
 ALT="$ e_{13}=0$">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{Chapter05slashwedge2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="308" BORDER="0"
 SRC="|."$dir".q|img1877.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{Chapter05/wedge2.eps}}
\end{figure}">|; 

$key = q/displaystyleint_Sleft[(p_1-p_2)-gamma,(nablacdot{bfn})right]{bfn},dS=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1106.png"
 ALT="$\displaystyle \int_S\left[(p_1-p_2)-\gamma (\nabla\cdot{\bf n})\right]{\bf n} dS = 0.$">|; 

$key = q/{mitLambda}>12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3173.png"
 ALT="$ {\mit\Lambda}&gt; 12$">|; 

$key = q/{rmMa}_2rightarrow{rmMa}_2,sin(beta-theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5700.png"
 ALT="$ {\rm Ma}_2\rightarrow {\rm Ma}_2 \sin(\beta-\theta)$">|; 

$key = q/rslasha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3826.png"
 ALT="$ r/a$">|; 

$key = q/z=-dslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2616.png"
 ALT="$ z=-d/2$">|; 

$key = q/0.8<{rmMa}_infty<1.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5922.png"
 ALT="$ 0.8&lt;{\rm Ma}_\infty &lt; 1.2$">|; 

$key = q/displaystylehsimeqfrac{d}{1-1slash{rmFr}^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1463.png"
 ALT="$\displaystyle h\simeq \frac{d}{1-1/{\rm Fr}^{ 2}}.$">|; 

$key = q/xi={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7227.png"
 ALT="$ \xi={\rm constant}$">|; 

$key = q/{bfv}cdotd{bfv}=d(v^{,2}slash2)=nabla(v^{,2}slash2)cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1559.png"
 ALT="$ {\bf v}\cdot d{\bf v} = d(v^{ 2}/2)=\nabla(v^{ 2}/2)\cdot d{\bf r}$">|; 

$key = q/displaystyle{calS}_2-{calS}_1={calC}_Vlnleft[frac{p_2}{p_1}left(frac{rho_1}{rho_2}right)^gammaright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5493.png"
 ALT="$\displaystyle {\cal S}_2-{\cal S}_1={\cal C}_V\ln\left[\frac{p_2}{p_1}\left(\frac{\rho_1}{\rho_2}\right)^\gamma\right].$">|; 

$key = q/displaystyle=int_0^phi(1-k^{,2},sin^2phi)^{-1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1183.png"
 ALT="$\displaystyle = \int_0^\phi (1-k^{ 2} \sin^2\phi)^{-1/2},$">|; 

$key = q/displaystyleC_D=frac{D}{q_1,c}=frac{4,epsilon^{,2}}{sqrt{{rmMa}_1^{,2}-1}}=frac{4}{sqrt{{rmMa}_1^{,2}-1}}left(frac{t}{c}right)^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5814.png"
 ALT="$\displaystyle C_D = \frac{D}{q_1 c}= \frac{4 \epsilon^{ 2}}{\sqrt{{\rm Ma}_1^{ 2}-1}}=\frac{4}{\sqrt{{\rm Ma}_1^{ 2}-1}}\left(\frac{t}{c}\right)^2.$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{AppendixAslashfigA.02.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="253" BORDER="0"
 SRC="|."$dir".q|img6226.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{AppendixA/figA.02.eps}}
\end{figure}">|; 

$key = q/dQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5319.png"
 ALT="$ dQ$">|; 

$key = q/displaystyle{bfgrad},f=nablafequivleft(frac{partialf}{partialx},,frac{partialf}{partialy},,frac{partialf}{partialz}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6563.png"
 ALT="$\displaystyle {\bf grad} f = \nabla f \equiv \left(\frac{\partial f}{\partial x},  \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right).$">|; 

$key = q/displaystyleskew{5}bar{zeta}_2(theta,phi)=-g^{,-1}left.{mitPhi}_{rmext}right|_{r=a}=-g^{,-1},({mitPhi}_{rmtide}+chi_2)_{r=a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4407.png"
 ALT="$\displaystyle \skew{5}\bar{\zeta}_2(\theta,\phi) =-g^{ -1}\left.{\mit\Phi}_{\rm ext}\right\vert _{r=a}= - g^{ -1} ({\mit\Phi}_{\rm tide}+ \chi_2)_{r=a}$">|; 

$key = q/displaystyle{mitPhi}(mu,phi,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4675.png"
 ALT="$\displaystyle {\mit\Phi}(\mu,\phi,t)$">|; 

$key = q/theta({bfr},t),dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="$ \theta({\bf r},t) dV$">|; 

$key = q/{mitOmega}ggomega_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4337.png"
 ALT="$ {\mit\Omega}\gg \omega_1$">|; 

$key = q/i=1,{calF};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7364.png"
 ALT="$ i=1,{\cal F}$">|; 

$key = q/displaystylev;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4661.png"
 ALT="$\displaystyle v$">|; 

$key = q/{mitDelta}=(a_1^{,2}+u)^{1slash2},(a_2^{,2}+u)^{1slash2},(a_3^{,2}+u)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7296.png"
 ALT="$ {\mit\Delta} = (a_1^{ 2}+u)^{1/2} (a_2^{ 2}+u)^{1/2} (a_3^{ 2}+u)^{1/2}$">|; 

$key = q/e_{13}rightarrow1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img875.png"
 ALT="$ e_{13}\rightarrow 1$">|; 

$key = q/displaystylefrac{1}{sintheta},frac{partial{mitPhi}(theta,phi_pm,t)}{partialphi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4763.png"
 ALT="$\displaystyle \frac{1}{\sin\theta} \frac{\partial{\mit\Phi}(\theta,\phi_\pm,t)}{\partial \phi}$">|; 

$key = q/displaystyle=frac{2}{sqrt{{rmMa}_1^{,2}-1}}left(-frac{dy_L}{dx}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5840.png"
 ALT="$\displaystyle =\frac{2}{\sqrt{{\rm Ma}_1^{ 2}-1}}\left(-\frac{dy_L}{dx}\right),$">|; 

$key = q/displaystylep=p_0-frac{1}{2},rholeft[left(frac{3}{2},G,M,alpha_1-omega^{,2}rightega^{,2}right)x_2^{,2}+frac{3}{2},G,M,alpha_3,x_3^{,2}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="630" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img863.png"
 ALT="$\displaystyle p = p_0 - \frac{1}{2} \rho\left[\left(\frac{3}{2} G M \alpha_...
..._2-\omega^{ 2}\right)x_2^{ 2}+ \frac{3}{2} G M \alpha_3 x_3^{ 2}\right],$">|; 

$key = q/displaystylepsi(x,y)=-frac{Q}{2pi},tan^{-1}left(frac{2,x,y}{x^{,2}-y^{,2}+a^{,2}}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2022.png"
 ALT="$\displaystyle \psi(x,y) =-\frac{Q}{2\pi} \tan^{-1}\left(\frac{2 x y}{x^{ 2}-y^{ 2}+a^{ 2}}\right).$">|; 

$key = q/displaystyleoint_C{bfA}cdotd{bfr}=int_Snablatimes{bfA}cdotd{bfS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6662.png"
 ALT="$\displaystyle \oint_C {\bf A} \cdot d{\bf r} = \int_S \nabla\times{\bf A}\cdot d{\bf S},$">|; 

$key = q/omega_1=omega_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4600.png"
 ALT="$ \omega_1=\omega_E$">|; 

$key = q/displaystyle=-sum_{r=1,infty}left(p_r,frac{partial{mitPhi}_r}{partialtheta}+p_{-,frac{1}{sintheta},frac{partial{mitPsi}_r}{partialphi}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4892.png"
 ALT="$\displaystyle = -\sum_{r=1,\infty}\left(p_r \frac{\partial{\mit\Phi}_r}{\parti...
...p_{-r} \frac{1}{\sin\theta} \frac{\partial{\mit\Psi}_r}{\partial\phi}\right),$">|; 

$key = q/displaystyle=r^{,2},sintheta,dtheta,dphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7174.png"
 ALT="$\displaystyle = r^{ 2} \sin\theta d\theta d\phi,$">|; 

$key = q/displaystyle=coslambda,sindelta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4291.png"
 ALT="$\displaystyle =\cos\lambda \sin\delta,$">|; 

$key = q/displaystyleleft(frac{partialX}{partialY}right)_Z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5573.png"
 ALT="$\displaystyle \left(\frac{\partial X}{\partial Y}\right)_Z$">|; 

$key = q/displaystylelambda'+frac{y}{h}=-coshz.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7350.png"
 ALT="$\displaystyle \lambda' + \frac{y}{h} = -\cosh z.$">|; 

$key = q/x';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2322.png"
 ALT="$ x'$">|; 

$key = q/bfa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6320.png"
 ALT="$ \bf a$">|; 

$key = q/displaystyle=-costheta,frac{partial{mitPhi}_r}{partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4844.png"
 ALT="$\displaystyle = -\cos\theta \frac{\partial{\mit\Phi}_r}{\partial\theta},$">|; 

$key = q/displaystylef(x,y)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5934.png"
 ALT="$\displaystyle f(x,y)=0$">|; 

$key = q/lambdarightarrowcosalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2291.png"
 ALT="$ \lambda\rightarrow \cos\alpha$">|; 

$key = q/z_{GH}=z_G-z_H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img653.png"
 ALT="$ z_{GH} = z_G-z_H$">|; 

$key = q/displaystyle2,a_{ijkl}+a_{jikl}+a_{ikjl}+a_{iljk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6940.png"
 ALT="$\displaystyle 2 a_{ijkl} + a_{jikl} + a_{ikjl} +a_{iljk}$">|; 

$key = q/m_ast=-0.0905;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3232.png"
 ALT="$ m_\ast=-0.0905$">|; 

$key = q/displaystyledzeta''&=f'(z_0),dz'',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2072.png"
 ALT="$\displaystyle d\zeta''&amp;= f'(z_0) dz'',$">|; 

$key = q/{rmRe},{rmFr}gg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="$ {\rm Re}, {\rm Fr} \gg 1$">|; 

$key = q/displaystylexi_x(x,z,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3937.png"
 ALT="$\displaystyle \xi_x(x,z,t)$">|; 

$key = q/displaystyle{bfa}times{bfb}equiv(a_y,b_z-a_z,b_y,,a_z,b_x-a_x,b_z,,a_x,b_y-a_y,b_x)={bfc}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="445" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6358.png"
 ALT="$\displaystyle {\bf a}\times{\bf b} \equiv (a_y   b_z-a_z  b_y,  a_z  b_x - a_x  b_z,  a_x  b_y - a_y  b_x) ={\bf c}.$">|; 

$key = q/h_1=h_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1477.png"
 ALT="$ h_1=h_2$">|; 

$key = q/displaystyle-frac{pi}{2},frac{dz}{b}=frac{dlambda}{lambda}left[-{rmi},lambda+(1-lambda^{,2})^{1slash2}right]^{,2,alphaslashpi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2245.png"
 ALT="$\displaystyle -\frac{\pi}{2} \frac{dz}{b}=\frac{d\lambda}{\lambda} \left[-{\rm i} \lambda + (1-\lambda^{ 2})^{1/2}\right]^{ 2 \alpha/\pi}.$">|; 

$key = q/rho_0,V_0,g=W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img608.png"
 ALT="$ \rho_0  V_0 g=W$">|; 

$key = q/displaystyle=-frac{1}{2},V,c^{,2},{rme}^{,3,xi_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2739.png"
 ALT="$\displaystyle =-\frac{1}{2} V c^{ 2} {\rm e}^{ 3 \xi_0},$">|; 

$key = q/displaystyle=frac{1}{r,sintheta},frac{partialpsi}{partialr}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2551.png"
 ALT="$\displaystyle = \frac{1}{r \sin\theta} \frac{\partial\psi}{\partial r}.$">|; 

$key = q/{mitPhi}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4795.png"
 ALT="$ {\mit\Phi}'$">|; 

$key = q/oint_S{bfh}cdotd{bfS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6606.png"
 ALT="$ \oint_S {\bf h}\cdot d{\bf S}$">|; 

$key = q/displaystylef'(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3163.png"
 ALT="$\displaystyle f'(1)$">|; 

$key = q/int_{-infty}^inftyexp(-t^{,2}),dt=sqrt{pi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3029.png"
 ALT="$ \int_{-\infty}^\infty \exp(-t^{ 2}) dt = \sqrt{\pi}$">|; 

$key = q/displaystylesimeq1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4139.png"
 ALT="$\displaystyle \simeq 1.$">|; 

$key = q/displaystyle=P_n(costheta),P_n(costheta');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4258.png"
 ALT="$\displaystyle = P_n(\cos\theta) P_n(\cos\theta')$">|; 

$key = q/displaystyle{rmMa}_1simeq1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5522.png"
 ALT="$\displaystyle {\rm Ma}_1\simeq 1.$">|; 

$key = q/{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6223.png"
 ALT="$ {\bf a}$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter15slashpran.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="255" BORDER="0"
 SRC="|."$dir".q|img6125.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter15/pran.eps}}
\end{figure}">|; 

$key = q/displaystylef(r)=frac{A}{r}+B,r+C,r^{,2}+D,r^{,4},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3776.png"
 ALT="$\displaystyle f(r) = \frac{A}{r} + B r + C r^{ 2} + D r^{ 4},$">|; 

$key = q/z=f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1154.png"
 ALT="$ z=f(x)$">|; 

$key = q/displaystyleA_i=frac{J}{a_i^{,2}}-frac{1}{a_i},frac{partialJ}{partiala_i}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7256.png"
 ALT="$\displaystyle A_i = \frac{J}{a_i^{ 2}}-\frac{1}{a_i} \frac{\partial J}{\partial a_i}.$">|; 

$key = q/displaystylec_{ijk}=a_i,b_{jk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6847.png"
 ALT="$\displaystyle c_{ijk} = a_i b_{jk},$">|; 

$key = q/0.23,{rmmslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4152.png"
 ALT="$ 0.23 {\rm m/s}$">|; 

$key = q/displaystylecostheta,frac{partial{mitPsi}_r}{partialtheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4860.png"
 ALT="$\displaystyle \cos\theta \frac{\partial{\mit\Psi}_r}{\partial\theta}$">|; 

$key = q/lambda=+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2257.png"
 ALT="$ \lambda=+1$">|; 

$key = q/displaystyle{bff}=(-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$\displaystyle {\bf f} = (-$">|; 

$key = q/DslashL^{,2}=({rmMa}_1^{,2}-1)^{1slash2}slash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5807.png"
 ALT="$ D/L^{ 2}=({\rm Ma}_1^{ 2}-1)^{1/2}/4$">|; 

$key = q/z_{GH};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img709.png"
 ALT="$ z_{GH}$">|; 

$key = q/displaystylechi(r,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4377.png"
 ALT="$\displaystyle \chi(r,\theta)$">|; 

$key = q/displaystylenablatimes;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3739.png"
 ALT="$\displaystyle \nabla\times$">|; 

$key = q/displaystyle=frac{(p_L-p_U),c,sinalpha_0}{q_1,c}=(C_{pL}-C_{pU}),sinalpha_0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5802.png"
 ALT="$\displaystyle = \frac{(p_L-p_U) c \sin \alpha_0}{q_1 c}= (C_{pL}-C_{pU}) \sin\alpha_0.$">|; 

$key = q/displaystyleU,frac{df}{dx}simeq-frac{partialphi(x,0)}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6028.png"
 ALT="$\displaystyle U \frac{df}{dx} \simeq- \frac{\partial\phi(x,0)}{\partial y}$">|; 

$key = q/displaystyleint_Vleft{frac{partial}{partialt}!left[rholeft({calE}+frac{1}{2},v_ileft[rholeft({calE}+frac{1}{2},v_i,v_iright)v_jright]right}dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="472" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="$\displaystyle \int_V\left\{\frac{\partial}{\partial t}\!\left[\rho\left({\cal E...
...x_j}\!\left[\rho\left({\cal E}+\frac{1}{2} v_i v_i\right)v_j\right]\right\}dV$">|; 

$key = q/(pmaslash2,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7339.png"
 ALT="$ (\pm a/2, 0)$">|; 

$key = q/{bfe}_phi=nablaphislash|nablaphi|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7170.png"
 ALT="$ {\bf e}_\phi=\nabla \phi/\vert\nabla \phi\vert$">|; 

$key = q/displaystyle{rmFr}_2=frac{v_2}{sqrt{g,h_2}}=frac{Q}{g^{,1slash2},h_2^{,3slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1487.png"
 ALT="$\displaystyle {\rm Fr}_2 = \frac{v_2}{\sqrt{g h_2}} = \frac{Q}{g^{ 1/2} h_2^{ 3/2}}.$">|; 

$key = q/={mitOmega}_x,{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3377.png"
 ALT="$ = {\mit\Omega}_x {\bf e}_x$">|; 

$key = q/p_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1322.png"
 ALT="$ p_0&gt;0$">|; 

$key = q/displaystyle|{bfcurl},{bfA}|=lim_{dSrightarrow0}frac{oint{bfA}cdotd{bfr}}{dS};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6629.png"
 ALT="$\displaystyle \vert{\bf curl} {\bf A}\vert = \lim_{dS\rightarrow 0}\frac{\oint {\bf A}\cdot d{\bf r}}{dS}$">|; 

$key = q/omega_1=omega_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4584.png"
 ALT="$ \omega_1=\omega_M$">|; 

$key = q/omega>1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img797.png"
 ALT="$ \omega&gt;1/2$">|; 

$key = q/{bfa}={bfb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6310.png"
 ALT="$ {\bf a} = {\bf b}$">|; 

$key = q/displaystyleleft.frac{partialphi}{partialr}right|_{r=a}=-V_z,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2660.png"
 ALT="$\displaystyle \left.\frac{\partial\phi}{\partial r}\right\vert _{r=a}=-V_z \cos\theta.$">|; 

$key = q/OAPBO;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1574.png"
 ALT="$ OAPBO$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{AppendixAslashfigA.25.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="226" BORDER="0"
 SRC="|."$dir".q|img6635.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{AppendixA/figA.25.eps}}
\end{figure}">|; 

$key = q/displaystyle=sum_{i=1,3}frac{x_i^{,2}}{a_i^{,2}}-1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7245.png"
 ALT="$\displaystyle =\sum_{i=1,3}\frac{x_i^{ 2}}{a_i^{ 2}}-1.$">|; 

$key = q/displaystyle={bfV}+{bfv}_{mitSigma}(P_+)+{bfv}_S(P_+),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3421.png"
 ALT="$\displaystyle = {\bf V} + {\bf v}_{\mit\Sigma}(P_+)+ {\bf v}_S(P_+),$">|; 

$key = q/displaystyle=muleft(frac{partialv_x}{partialy}+frac{partialv_y}{partialx}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="$\displaystyle = \mu\left(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right),$">|; 

$key = q/displaystyle=-frac{1}{sintheta},frac{partial{mitPhi}_r''}{partialphi}+frac{partial{mitPsi}_r'}{partialtheta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4863.png"
 ALT="$\displaystyle = -\frac{1}{\sin\theta} \frac{\partial{\mit\Phi}_r''}{\partial\phi} + \frac{\partial{\mit\Psi}_r'}{\partial\theta}.$">|; 

$key = q/epsilon_{ijk},a_j,b_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6869.png"
 ALT="$ \epsilon_{ijk} a_j b_k$">|; 

$key = q/displaystylefrac{d{mitGamma}}{dt}=-V,{mitOmega}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3329.png"
 ALT="$\displaystyle \frac{d{\mit\Gamma}}{dt} = - V {\mit\Omega}_z,$">|; 

$key = q/hllH;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1453.png"
 ALT="$ h\ll H$">|; 

$key = q/kappa_{n,,n'}^{,m,,m'}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4983.png"
 ALT="$ \kappa_{n, n'}^{ m, m'}=0$">|; 

$key = q/bfF;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6324.png"
 ALT="$ \bf F$">|; 

$key = q/displaystylenablacdot(nablatimes{bfa})=frac{partial}{partialx_i}!left(epsilon_{ight)=epsilon_{ijk},frac{partiala_k}{partialx_i,partialx_j}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6888.png"
 ALT="$\displaystyle \nabla\cdot( \nabla\times {\bf a}) = \frac{\partial}{\partial x_i...
...j}\right)= \epsilon_{ijk} \frac{\partial a_k}{\partial x_i \partial x_j} = 0,$">|; 

$key = q/displaystyleX-{rmi},Y=frac{{rmi},rho,Q^{,2}}{4pi,a}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2420.png"
 ALT="$\displaystyle X - {\rm i} Y = \frac{{\rm i} \rho Q^{ 2}}{4\pi a}.$">|; 

$key = q/displaystylef_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4686.png"
 ALT="$\displaystyle f_j$">|; 

$key = q/displaystyleskew{3}bar{C}_p=m,({calC}_V+{calR}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5312.png"
 ALT="$\displaystyle \skew{3}\bar{C}_p = m ({\cal C}_V+{\cal R}).$">|; 

$key = q/betall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5540.png"
 ALT="$ \beta\ll 1$">|; 

$key = q/a_2neqa_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img923.png"
 ALT="$ a_2\neq a_1$">|; 

$key = q/varthetaequivpislash2-theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5036.png"
 ALT="$ \vartheta\equiv\pi/2-\theta$">|; 

$key = q/displaystylen=frac{tan{mitLambda}}{sqrt{{rmMa}_1^{,2}-1}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6216.png"
 ALT="$\displaystyle n=\frac{\tan{\mit\Lambda}}{\sqrt{{\rm Ma}_1^{ 2}-1}}.
$">|; 

$key = q/{bfV}=V,{bfe}_parallel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3394.png"
 ALT="$ {\bf V}= V {\bf e}_\parallel$">|; 

$key = q/displaystyleleft(frac{varpi}{b}right)^2+left(frac{z}{a}right)^2=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2798.png"
 ALT="$\displaystyle \left(\frac{\varpi}{b}\right)^2+\left(\frac{z}{a}\right)^2=1.$">|; 

$key = q/displaystyle={rmi}sum_{n=1,infty}{mitPsi}_n,P_n^{,m_j}(mu),{rme}^{,{rmi},(m_j,phi+sigma_j,t)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4678.png"
 ALT="$\displaystyle = {\rm i}\sum_{n=1,\infty}{\mit\Psi}_n P_n^{ m_j}(\mu) {\rm e}^{ {\rm i} (m_j \phi+\sigma_j t)},$">|; 

$key = q/z={rmi},a+epsilon,{rme}^{,{rmi},theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2417.png"
 ALT="$ z={\rm i} a+\epsilon {\rm e}^{ {\rm i} \theta}$">|; 

$key = q/displaystylezeta_{b,n}=h_2,skew{5}bar{zeta}_2,delta_{n,2}+h_n',alpha_n,zeta_n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4430.png"
 ALT="$\displaystyle \zeta_{b n} = h_2 \skew{5}\bar{\zeta}_2 \delta_{n 2} + h_n' \alpha_n \zeta_n,$">|; 

$key = q/displaystyle{bfa}cdot({bfb}+{bfc});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6307.png"
 ALT="$\displaystyle {\bf a}\cdot({\bf b}+{\bf c})$">|; 

$key = q/e_{13}<0.93858;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img964.png"
 ALT="$ e_{13}&lt;0.93858$">|; 

$key = q/t=j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6913.png"
 ALT="$ t=j$">|; 

$key = q/AA'E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4010.png"
 ALT="$ AA'E$">|; 

$key = q/r^{,0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1744.png"
 ALT="$ r^{ 0}$">|; 

$key = q/dW={bff}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6330.png"
 ALT="$ dW={\bf f}\cdot d{\bf r}$">|; 

$key = q/v_x(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2913.png"
 ALT="$ v_x(x,y)$">|; 

$key = q/nabla({bfa}cdot{bfa})=2,{bfa}times(nablatimes{bfa})+2,({bfa}cdotnabla),{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6990.png"
 ALT="$ \nabla ({\bf a}\cdot{\bf a}) = 2 {\bf a}\times (\nabla\times{\bf a}) + 2 ({\bf a}\cdot\nabla) {\bf a}$">|; 

$key = q/displaystylealpha_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1000.png"
 ALT="$\displaystyle \alpha_1$">|; 

$key = q/displaystyle=alpha,beta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1292.png"
 ALT="$\displaystyle =\alpha \beta,$">|; 

$key = q/zeta_{t,n};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4346.png"
 ALT="$ \zeta_{t n}$">|; 

$key = q/Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1426.png"
 ALT="$ Q$">|; 

$key = q/betaslash(1-beta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5231.png"
 ALT="$ \beta/(1-\beta)$">|; 

$key = q/lambda=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2213.png"
 ALT="$ \lambda=1$">|; 

$key = q/displaystylesigma_{thetaz}=sigma_{ztheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="$\displaystyle \sigma_{\theta z} = \sigma_{z\theta}$">|; 

$key = q/displaystyleint_{mitOmega}({mitPhi}_r^{,ast},D,{mitPhi}_r-{mitPhi}_r,D,{mitPhi}_ambda_r^{,ast})int_{mitOmega}|{mitPhi}_r|^{,2},d{mitOmega}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="476" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4807.png"
 ALT="$\displaystyle \int_{\mit\Omega} ({\mit\Phi}_r^{ \ast} D {\mit\Phi}_r - {\mit...
...a_r^{ \ast})\int_{\mit\Omega} \vert{\mit\Phi}_r\vert^{ 2} d{\mit\Omega} = 0.$">|; 

$key = q/c(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5621.png"
 ALT="$ c(x)$">|; 

$key = q/displaystylefrac{x_1^{,2}}{a_1^{,2}}+frac{x_2^{,2}}{a_2^{,2}}+frac{x_3^{,2}}{a_3^{,2}}=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img844.png"
 ALT="$\displaystyle \frac{x_1^{ 2}}{a_1^{ 2}}+ \frac{x_2^{ 2}}{a_2^{ 2}}+\frac{x_3^{ 2}}{a_3^{ 2}}=1,$">|; 

$key = q/displaystyle=-rho,nablacdot{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="$\displaystyle = -\rho \nabla\cdot{\bf v},$">|; 

$key = q/{rmMa}_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5952.png"
 ALT="$ {\rm Ma}_\infty$">|; 

$key = q/oint{bfv}cdotd{bfr}=2pi,r,omega,r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6654.png"
 ALT="$ \oint {\bf v}\cdot d{\bf r} = 2\pi  r  \omega  r$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter06slashschwarz1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="410" HEIGHT="256" BORDER="0"
 SRC="|."$dir".q|img2150.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter06/schwarz1.eps}}
\end{figure}">|; 

$key = q/displaystyle=P_2^{,0}(costheta),P_2^{,0}(costheta')+frac{1}{3},P_2^{,1}(costheta),P_2^{,1}(costheta'),cos(varphi-varphi');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="483" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4261.png"
 ALT="$\displaystyle = P_2^{ 0}(\cos\theta) P_2^{ 0}(\cos\theta')+ \frac{1}{3} P_2^{ 1}(\cos\theta) P_2^{ 1}(\cos\theta') \cos(\varphi-\varphi')$">|; 

$key = q/displaystyle{bfL};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3299.png"
 ALT="$\displaystyle {\bf L}$">|; 

$key = q/A_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="$ A_{ij}$">|; 

$key = q/deltay'=d(deltay)slashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7310.png"
 ALT="$ \delta y' = d(\delta y)/dx$">|; 

$key = q/displaystylephi(x,z,t)=left[A,cosh(k,z)+B,sinh(k,z)right]cos(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4079.png"
 ALT="$\displaystyle \phi(x,z,t)= \left[A \cosh(k z)+B \sinh(k z)\right]\cos(\omega t-k x).$">|; 

$key = q/U_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2885.png"
 ALT="$ U_1$">|; 

$key = q/gamma=7slash5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$ \gamma =7/5$">|; 

$key = q/displaystylecostheta';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4290.png"
 ALT="$\displaystyle \cos\theta'$">|; 

$key = q/x=+l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3036.png"
 ALT="$ x=+l$">|; 

$key = q/p({bfr},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="$ p({\bf r},t)$">|; 

$key = q/displaystyle=frac{int_{-a}^aint_{a-h(x)}^ay,dx,dy}{int_{-a}^aint_{a-h(x)}^adx,dy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img749.png"
 ALT="$\displaystyle =\frac{\int_{-a}^a\int_{a-h(x)}^a y dx dy}{\int_{-a}^a\int_{a-h(x)}^a dx dy}$">|; 

$key = q/{bfa}equiv(1slash!sqrt{2},,1slash!sqrt{2},,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6297.png"
 ALT="$ {\bf a} \equiv (1/\!\sqrt{2},  1/\!\sqrt{2},  0)$">|; 

$key = q/displaystyle=r,sintheta,dr,dphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7175.png"
 ALT="$\displaystyle = r \sin\theta dr d\phi,$">|; 

$key = q/2<n<3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2485.png"
 ALT="$ 2&lt;n&lt;3$">|; 

$key = q/displaystylez=zeta+frac{l^{,2}}{zeta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3303.png"
 ALT="$\displaystyle z = \zeta+ \frac{l^{ 2}}{\zeta}$">|; 

$key = q/alpha'=-mslash(1+m);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2958.png"
 ALT="$ \alpha'=-m/(1+m)$">|; 

$key = q/displaystylea_{z'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6289.png"
 ALT="$\displaystyle a_{z'}$">|; 

$key = q/displaystyle{rmFr}_1=frac{v_1}{sqrt{g,h_1}}=frac{Q}{g^{,1slash2},h_1^{,3slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1481.png"
 ALT="$\displaystyle {\rm Fr}_1 = \frac{v_1}{\sqrt{g h_1}} = \frac{Q}{g^{ 1/2} h_1^{ 3/2}}.$">|; 

$key = q/langlecdotsrangle=int_{x_1}^{x_2}(cdots),dxslashl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3673.png"
 ALT="$ \langle \cdots\rangle = \int_{x_1}^{x_2} (\cdots) dx/l$">|; 

$key = q/theta=cos^{-1}(zslashr);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2528.png"
 ALT="$ \theta=\cos^{-1}(z/r)$">|; 

$key = q/h(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1533.png"
 ALT="$ h(0)$">|; 

$key = q/displaystyleD,{mitPsi}_r''&=frac{1}{sintheta}left[frac{partial}{partialtheta}lefeft(costheta,frac{partial{mitPhi}_r}{partialphi}right)right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="511" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4842.png"
 ALT="$\displaystyle D {\mit\Psi}_r''&amp;= \frac{1}{\sin\theta}\left[ \frac{\partial}{\p...
...hi}\left( \cos\theta \frac{\partial{\mit\Phi}_r}{\partial \phi}\right)\right],$">|; 

$key = q/displaystylen,({bfa}+{bfb});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6250.png"
 ALT="$\displaystyle n ({\bf a} + {\bf b})$">|; 

$key = q/y=pmh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3045.png"
 ALT="$ y=\pm h$">|; 

$key = q/kappa>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="$ \kappa&gt;0$">|; 

$key = q/C'=(R,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img976.png"
 ALT="$ C'=(R, 0, 0)$">|; 

$key = q/{mitGamma}rightarrow-{mitGamma};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2035.png"
 ALT="$ {\mit\Gamma}\rightarrow -{\mit\Gamma}$">|; 

$key = q/displaystylepsi(rrightarrowinfty,theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3763.png"
 ALT="$\displaystyle \psi(r\rightarrow\infty,\theta)$">|; 

$key = q/displaystyle=frac{1}{a_1}int_0^inftyfrac{dlambda}{(1+lambda),(1+lambda-e_{13}^{,ac{2}{a_1,e_{13}}int_0^{e_{13}}frac{dz}{(1-z^{,2})^{1slash2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="485" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img882.png"
 ALT="$\displaystyle = \frac{1}{a_1}\int_0^\infty\frac{d\lambda}{(1+\lambda) (1+\lamb...
...{ 2})^{1/2}} =\frac{2}{a_1 e_{13}}\int_0^{e_{13}}\frac{dz}{(1-z^{ 2})^{1/2}}$">|; 

$key = q/displaystyle=-frac{1}{v}left(frac{partialv}{partialp}right)_{calS},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5601.png"
 ALT="$\displaystyle = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_{\cal S},$">|; 

$key = q/d_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="$ d_{ij}$">|; 

$key = q/delta_{i,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4409.png"
 ALT="$ \delta_{i j}$">|; 

$key = q/displaystyle=-p,dV+T,dS,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5330.png"
 ALT="$\displaystyle =-p dV+T dS,$">|; 

$key = q/FA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2717.png"
 ALT="$ FA$">|; 

$key = q/displaystyle=nabla^{,2}A_i+sum_{j=1,3}left{frac{2}{h_i,h_j}left(frac{1}{h_i},fratialh_j}{partialu_i},frac{partial}{partialu_j}right)A_jright.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="448" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7099.png"
 ALT="$\displaystyle = \nabla^{ 2} A_i +\sum_{j=1,3}\left\{\frac{2}{h_i h_j}\left(\f...
...ac{\partial h_j}{\partial u_i} \frac{\partial}{\partial u_j}\right) A_j\right.$">|; 

$key = q/displaystyle2,a_{ijkl}+a_{ljki}+a_{ilkj}+a_{ijlk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6944.png"
 ALT="$\displaystyle 2 a_{ijkl}+a_{ljki} +a_{ilkj}+a_{ijlk}$">|; 

$key = q/=-{mitGamma},{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3471.png"
 ALT="$ = -{\mit\Gamma} {\bf e}_x$">|; 

$key = q/C'D';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1365.png"
 ALT="$ C'D'$">|; 

$key = q/overline{C};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3854.png"
 ALT="$ \overline{C}$">|; 

$key = q/displaystyleint_0^pihat{P}_n^{,m}(costheta),hat{P}_{n'}^{,m}(costheta),sintheta,dtheta=frac{delta_{n,n'}}{2pi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4922.png"
 ALT="$\displaystyle \int_0^\pi \hat{P}_n^{ m}(\cos\theta) \hat{P}_{n'}^{ m}(\cos\theta) \sin\theta d\theta = \frac{\delta_{n n'}}{2\pi}.$">|; 

$key = q/T(r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5182.png"
 ALT="$ T(r)$">|; 

$key = q/displaystyle=int_0^a(z,a^{,2}-2,a,z^{,2}+z^{,3}),dz=left[a^{,2},z^{,2}slash2-2,a,z^{,3}slash3+z^4slash4right]_0^a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="487" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6514.png"
 ALT="$\displaystyle =\int_0^a (z a^{ 2}-2 a z^{ 2}+z^{ 3}) dz= \left[a^{ 2} z^{ 2}/2-2 a z^{ 3}/3+z^4/4\right]_0^a$">|; 

$key = q/displaystyle{mitGamma}(z)=pi,V,c(z),sinalpha={mitGamma}_0,sinphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3490.png"
 ALT="$\displaystyle {\mit\Gamma}(z)= \pi V c(z) \sin\alpha = {\mit\Gamma}_0 \sin\phi,$">|; 

$key = q/displaystyle=Z_2(x-c,t,0),{rme}^{-s,yslashl}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5099.png"
 ALT="$\displaystyle = Z_2(x-c t,0) {\rm e}^{-s y/l}.$">|; 

$key = q/thetall1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5723.png"
 ALT="$ \theta\ll 1$">|; 

$key = q/p_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5629.png"
 ALT="$ p_4$">|; 

$key = q/costhetasimeq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6396.png"
 ALT="$ \cos\theta\simeq 1$">|; 

$key = q/{bfA};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6476.png"
 ALT="$ {\bf A}$">|; 

$key = q/displaystylef(0)=f'(0)=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2942.png"
 ALT="$\displaystyle f(0) = f'(0) = 0,$">|; 

$key = q/displaystyle=-frac{1}{r,sintheta},frac{partialpsi}{r,partialtheta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2550.png"
 ALT="$\displaystyle = -\frac{1}{r \sin\theta} \frac{\partial\psi}{r \partial\theta},$">|; 

$key = q/displaystyleu_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5087.png"
 ALT="$\displaystyle u_0$">|; 

$key = q/lambda,delta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6896.png"
 ALT="$ \lambda \delta_{ij}$">|; 

$key = q/p_1,S_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1393.png"
 ALT="$ p_1 S_1$">|; 

$key = q/displaystyle2,frac{d}{du}!left(frac{u}{mitDelta}right)=-frac{1}{mitDelta}+sum_{i=1,3}frac{a_i^{,2}}{(a_i^{,2}+u),{mitDelta}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7276.png"
 ALT="$\displaystyle 2 \frac{d}{du}\!\left(\frac{u}{\mit\Delta}\right) = -\frac{1}{\mit\Delta}+ \sum_{i=1,3}\frac{a_i^{ 2}}{(a_i^{ 2}+u) {\mit\Delta}},$">|; 

$key = q/-2.95times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4613.png"
 ALT="$ -2.95\times 10^{-2}$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{Chapter15slashmach.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="225" BORDER="0"
 SRC="|."$dir".q|img5722.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{Chapter15/mach.eps}}
\end{figure}">|; 

$key = q/z=[gammaslash(gamma-1)],Hsimeq28,{rmkm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5156.png"
 ALT="$ z=[\gamma/(\gamma-1)] H\simeq 28 {\rm km}$">|; 

$key = q/coshxi_0=1slashe;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2769.png"
 ALT="$ \cosh\xi_0=1/e$">|; 

$key = q/xi_1>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5277.png"
 ALT="$ \xi_1&gt;0$">|; 

$key = q/displaystylefrac{partial^{,2}v_x}{partialy^{,2}}left[1+{calO}left(frac{rho,v_x,dmu,l}right)+{calO}left(frac{d}{l}right)^2right]=-frac{G}{mu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3658.png"
 ALT="$\displaystyle \frac{\partial^{ 2} v_x}{\partial y^{ 2}}\left[1+{\cal O}\left(...
...2}}{\mu l}\right)+{\cal O}\left(\frac{d}{l}\right)^2\right] = - \frac{G}{\mu}.$">|; 

$key = q/Rgeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6111.png"
 ALT="$ R\geq 0$">|; 

$key = q/epsilon_{ijk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$ \epsilon_{ijk}$">|; 

$key = q/displaystyle{calR}_{ik},{calR}_{jk};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6829.png"
 ALT="$\displaystyle {\cal R}_{ik} {\cal R}_{jk}$">|; 

$key = q/phi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1311.png"
 ALT="$ \phi=0$">|; 

$key = q/z_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2212.png"
 ALT="$ z_1$">|; 

$key = q/displaystyle=rho,frac{partial^{,2}u_i}{partialt^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7018.png"
 ALT="$\displaystyle =\rho \frac{\partial^{ 2} u_i}{\partial t^{ 2}},$">|; 

$key = q/{mitOmega}={rmconstant};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3639.png"
 ALT="$ {\mit\Omega}= {\rm constant}$">|; 

$key = q/dvarpi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2542.png"
 ALT="$ d\varpi$">|; 

$key = q/C=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1222.png"
 ALT="$ C=1$">|; 

$key = q/displaystyle=c_{ijkl},frac{1}{2}!left(frac{partialu_k}{partialx_l}+frac{partialu_l}{partialx_k}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7020.png"
 ALT="$\displaystyle =c_{ijkl} \frac{1}{2}\!\left(\frac{\partial u_k}{\partial x_l}+ \frac{\partial u_l}{\partial x_k}\right),$">|; 

$key = q/(0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$ (0$">|; 

$key = q/displaystyleV^{,-1}left[-A,sinh(k,d)+B,cosh(k,d)right]=a.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4122.png"
 ALT="$\displaystyle V^{ -1}\left[-A \sinh(k d)+ B \cosh(k d)\right] = a.$">|; 

$key = q/y=-aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2095.png"
 ALT="$ y=-a/2$">|; 

$key = q/delta(x)=[2,nuslash(m+1),U_0,x^{,m-1}]^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2920.png"
 ALT="$ \delta(x)=[2 \nu/(m+1) U_0 x^{ m-1}]^{1/2}$">|; 

$key = q/displaystyleL=rho,V,{mitGamma}_infty=pi,rho,V^{,2},(c+delta),sinalpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3350.png"
 ALT="$\displaystyle L = \rho V {\mit\Gamma}_\infty = \pi \rho V^{ 2} (c+\delta) \sin\alpha.$">|; 

$key = q/y=aslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2097.png"
 ALT="$ y=a/2$">|; 

$key = q/displaystylefrac{dE}{dt}+{mitPhi}_E=dot{W}-dot{Q},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="$\displaystyle \frac{dE}{dt} + {\mit\Phi}_E = \dot{W} -\dot{Q},$">|; 

$key = q/V=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4109.png"
 ALT="$ V=0$">|; 

$key = q/displaystyleq=mleft[({calC}_V+{calR}),T-v,dpright].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5310.png"
 ALT="$\displaystyle q = m\left[({\cal C}_V+{\cal R}) T-v dp\right].$">|; 

$key = q/displaystylefrac{d}{dx}left(rho,u,{calE}+p,uright)=-rho,u^{,2},frac{du}{dx}=-fraac{du^{,2}}{dx}=-frac{d}{dx}left(frac{1}{2},rho,u^{,3}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="502" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5457.png"
 ALT="$\displaystyle \frac{d}{dx}\left(\rho u {\cal E} +p u\right)= -\rho u^{ 2}\...
...o u \frac{du^{ 2}}{dx}=-\frac{d}{dx}\left(\frac{1}{2} \rho u^{ 3}\right),$">|; 

$key = q/sigma_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$ \sigma_{12}$">|; 

$key = q/displaystylenablacdot;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1544.png"
 ALT="$\displaystyle \nabla\cdot$">|; 

$key = q/displaystyle=-3,x,(1-x^{,2})^{1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4268.png"
 ALT="$\displaystyle = -3 x (1-x^{ 2})^{1/2},$">|; 

$key = q/displaystylemuleqbetaleqfrac{pi}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5695.png"
 ALT="$\displaystyle \mu\leq \beta\leq \frac{\pi}{2},$">|; 

$key = q/displaystyleP_2^{,2}(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4269.png"
 ALT="$\displaystyle P_2^{ 2}(x)$">|; 

$key = q/A_r={bfA}cdot{bfe}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7114.png"
 ALT="$ A_r = {\bf A}\cdot {\bf e}_r$">|; 

$key = q/displaystyleU_0=left[frac{{mitGamma}}{pi,(c+delta)}-V,sinalpharight]left(frac{c+delta}{delta}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3336.png"
 ALT="$\displaystyle U_0 = \left[\frac{{\mit\Gamma}}{\pi (c+\delta)}-V \sin\alpha\right]\left(\frac{c+\delta}{\delta}\right).$">|; 

$key = q/J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7334.png"
 ALT="$ J$">|; 

$key = q/displaystylev_z(r)=-frac{G}{4,mu},r^{,2}+A,lnr+B,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3613.png"
 ALT="$\displaystyle v_z(r) = -\frac{G}{4 \mu} r^{ 2} + A \ln r +B,$">|; 

$key = q/v_1=V+u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1513.png"
 ALT="$ v_1=V+u_1$">|; 

$key = q/h,v=h_1,v+h_2,v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1886.png"
 ALT="$ h v = h_1 v+h_2 v$">|; 

$key = q/D,{rme}^{,{rmi},alpha};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2452.png"
 ALT="$ D {\rm e}^{ {\rm i} \alpha}$">|; 

$key = q/displaystylerho,g,b,sin(k,x)=p_1(x,0)=rho,k,V,A,sin(k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4125.png"
 ALT="$\displaystyle \rho g b \sin(k x) = p_1(x,0) = \rho k V A \sin(k x),$">|; 

$key = q/7;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4560.png"
 ALT="$ 7$">|; 

$key = q/displaystyleskew{3}bar{psi}=skew{3}bar{psi}_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2219.png"
 ALT="$\displaystyle \skew{3}\bar{\psi}=\skew{3}\bar{\psi}_1,$">|; 

$key = q/displaystylea_{ijss};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6955.png"
 ALT="$\displaystyle a_{ijss}$">|; 

$key = q/CD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1361.png"
 ALT="$ CD$">|; 

$key = q/{figure}{epsfysize{=3.in{centerline{epsffile{Chapter10slashdrop2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img3858.png"
 ALT="\begin{figure}
\epsfysize =3.in
\centerline{\epsffile{Chapter10/drop2.eps}}
\end{figure}">|; 

$key = q/z=d';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4086.png"
 ALT="$ z=d'$">|; 

$key = q/displaystyley',frac{d}{dx}!left(frac{partialF}{partialy'}right)-y',frac{partialF}{partialy}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7320.png"
 ALT="$\displaystyle y' \frac{d}{dx}\!\left(\frac{\partial F}{\partial y'}\right)-y' \frac{\partial F}{\partial y} = 0.$">|; 

$key = q/displaystyleleft[alpha_1-frac{omega^{,2}}{(3slash2),G,M}right]a_1^{,2}=left[alphac{omega^{,2}}{(3slash2),G,M}right]a_2^{,2}=alpha_3,a_3^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img866.png"
 ALT="$\displaystyle \left[\alpha_1-\frac{\omega^{ 2}}{(3/2) G M}\right]a_1^{ 2} =...
...alpha_2-\frac{\omega^{ 2}}{(3/2) G M}\right]a_2^{ 2} = \alpha_3 a_3^{ 2}.$">|; 

$key = q/lambda_c=2pi,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4185.png"
 ALT="$ \lambda_c=2\pi l$">|; 

$key = q/{rmMa}_2simeq{rmMa}_1,(1-delta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6174.png"
 ALT="$ {\rm Ma}_2\simeq {\rm Ma}_1 (1-\delta)$">|; 

$key = q/sigma_{33};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="$ \sigma_{33}$">|; 

$key = q/displaystyle=r,dtheta,dz,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7122.png"
 ALT="$\displaystyle = r d\theta dz,$">|; 

$key = q/displaystyle=rhoint_S{bfv}_{mitSigma}times;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3441.png"
 ALT="$\displaystyle = \rho\int_S {\bf v}_{\mit\Sigma}\times$">|; 

$key = q/displaystyle={mitOmega}_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3631.png"
 ALT="$\displaystyle ={\mit\Omega}_2.$">|; 

$key = q/s=rho'slashrho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4173.png"
 ALT="$ s=\rho'/\rho$">|; 

$key = q/displaystyleleft(frac{partialT}{partialp}right)_{calS}=left(frac{partialv}{partial{calS}}right)_p.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5584.png"
 ALT="$\displaystyle \left(\frac{\partial T}{\partial p}\right)_{\cal S} = \left(\frac{\partial v}{\partial {\cal S}}\right)_p.
$">|; 

$key = q/displaystylefrac{{mitDelta}{calS}}{{calR}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5514.png"
 ALT="$\displaystyle \frac{{\mit\Delta}{\cal S}}{{\cal R}}$">|; 

$key = q/displaystylewidehat{mitGamma}_infty=sinalpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3345.png"
 ALT="$\displaystyle \widehat{\mit\Gamma}_\infty=\sin\alpha.$">|; 

$key = q/displaystyleexpleft(frac{g,M,a}{R,T}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5265.png"
 ALT="$\displaystyle \exp\left(\frac{g M a}{R T}\right),
$">|; 

$key = q/displaystylefrac{partialphi(x,0)}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5948.png"
 ALT="$\displaystyle \frac{\partial\phi(x,0)}{\partial y}$">|; 

$key = q/d{mitOmega}'equivsintheta',dtheta',dphi';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4518.png"
 ALT="$ d{\mit\Omega}'\equiv \sin\theta' d\theta' d\phi'$">|; 

$key = q/displaystyler_1-r_2rightarrowd,costheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2629.png"
 ALT="$\displaystyle r_1-r_2\rightarrow d \cos\theta.$">|; 

$key = q/lambda_1gtreqqless0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img699.png"
 ALT="$ \lambda_1\gtreqqless 0$">|; 

$key = q/displaystyleleft(frac{varpi}{a}right)^2+left(frac{z}{b}right)^2=1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2757.png"
 ALT="$\displaystyle \left(\frac{\varpi}{a}\right)^2+\left(\frac{z}{b}\right)^2=1.$">|; 

$key = q/displaystyleV_x(X,Y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2879.png"
 ALT="$\displaystyle V_x(X,Y)$">|; 

$key = q/2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2512.png"
 ALT="$ 2\pi$">|; 

$key = q/theta=+alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2241.png"
 ALT="$ \theta=+\alpha$">|; 

$key = q/{bf0}equiv(0,,0,,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6267.png"
 ALT="$ {\bf0} \equiv (0, 0, 0)$">|; 

$key = q/displaystylesimeqtheta_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6203.png"
 ALT="$\displaystyle \simeq \theta_1,$">|; 

$key = q/A=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3140.png"
 ALT="$ A=0$">|; 

$key = q/displaystyle=left(frac{rho}{rho_0}right)^gamma.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="$\displaystyle = \left(\frac{\rho}{\rho_0}\right)^\gamma.$">|; 

$key = q/displaystylefrac{p}{rho}+{calT};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1358.png"
 ALT="$\displaystyle \frac{p}{\rho} + {\cal T}$">|; 

$key = q/overline{mu}llmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3866.png"
 ALT="$ \overline{\mu}\ll \mu$">|; 

$key = q/displaystylefrac{1}{r},frac{partial}{partialr}!left(r,frac{partialpsi}{partialr})+frac{1}{r^{,2}},frac{partial^{,2}psi}{partialtheta^{,2}}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1729.png"
 ALT="$\displaystyle \frac{1}{r} \frac{\partial}{\partial r}\!\left(r \frac{\partial...
...right) + \frac{1}{r^{ 2}} \frac{\partial^{ 2}\psi}{\partial\theta^{ 2}} =0.$">|; 

$key = q/displaystylefrac{1}{R},frac{d}{dr}left(r^{,2},frac{dR}{dr}right)=-frac{1}{{mitThfrac{d}{dtheta}left(sintheta,frac{d{mitTheta}}{dtheta}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2574.png"
 ALT="$\displaystyle \frac{1}{R} \frac{d}{dr}\left(r^{ 2} \frac{dR}{dr}\right) = -\...
...heta} \frac{d}{d\theta}\left(\sin\theta \frac{d{\mit\Theta}}{d\theta}\right),$">|; 

$key = q/displaystyler_1-r_2=frac{d,(sintheta_2-sintheta_1)}{2,sin[(theta_2-theta_1)slash2],cos[(theta_2-theta_1)slash2]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2624.png"
 ALT="$\displaystyle r_1-r_2 = \frac{d (\sin\theta_2-\sin\theta_1)}{2 \sin[(\theta_2-\theta_1)/2] \cos[(\theta_2-\theta_1)/2]}.$">|; 

$key = q/h_2=5slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4645.png"
 ALT="$ h_2=5/2$">|; 

$key = q/p=p(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5343.png"
 ALT="$ p=p(x,t)$">|; 

$key = q/y_i(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7363.png"
 ALT="$ y_i(x)$">|; 

$key = q/theta=tan^{-1}(yslashx);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1937.png"
 ALT="$ \theta=\tan^{-1}(y/x)$">|; 

$key = q/displaystyle=deltaint_0^1f,(1-f),deta=deltaleft(frac{37}{315}-frac{{mitLambda}}{945}-frac{{mitLambda}^{,2}}{9072}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3184.png"
 ALT="$\displaystyle =\delta\int_0^1 f (1-f) d\eta=\delta\left(\frac{37}{315}-\frac{{\mit\Lambda}}{945}-\frac{{\mit\Lambda}^{ 2}}{9072}\right).$">|; 

$key = q/2,{mitOmega}-2,omega_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4566.png"
 ALT="$ 2 {\mit\Omega}-2 \omega_M$">|; 

$key = q/displaystylel;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3307.png"
 ALT="$\displaystyle l$">|; 

$key = q/displaystyleoint_S{bfA}cdotd{bfS}=int_Vnablacdot{bfA},,dV.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6598.png"
 ALT="$\displaystyle \oint_S {\bf A}\cdot d{\bf S} = \int_V \nabla\cdot{\bf A}  dV.$">|; 

$key = q/displaystylen,{bfa}equiv(n,a_x,,n,a_y,,n,a_z).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6273.png"
 ALT="$\displaystyle n {\bf a} \equiv (n a_x, n a_y, n a_z).$">|; 

$key = q/(d_1-d_2)slashd_1sim{calO}(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3726.png"
 ALT="$ (d_1-d_2)/d_1\sim{\cal O}(1)$">|; 

$key = q/displaystylepsi(xi_0,eta)=-frac{1}{2},V,varpi^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2712.png"
 ALT="$\displaystyle \psi(\xi_0,\eta)= -\frac{1}{2} V \varpi^{ 2}.$">|; 

$key = q/delta(x)proptox^{,-(m-1)slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2952.png"
 ALT="$ \delta(x)\propto x^{ -(m-1)/2}$">|; 

$key = q/displaystylefrac{{mitDelta}{rmMa}}{{rmMa}_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6163.png"
 ALT="$\displaystyle \frac{{\mit\Delta}{\rm Ma}}{{\rm Ma}_1}$">|; 

$key = q/displaystylefrac{du}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img756.png"
 ALT="$\displaystyle \frac{du}{dt}$">|; 

$key = q/displaystylebeta_{-r,,s}=-int_{mitOmega}costhetaleft(frac{partial{mitPsi}_r}{parialphi},frac{partial{mitPhi}_s}{partialphi}right)d{mitOmega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="430" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4874.png"
 ALT="$\displaystyle \beta_{-r, s}= -\int_{\mit\Omega}\cos\theta\left(\frac{\partial{...
...}{\partial\phi} \frac{\partial{\mit\Phi}_s}{\partial\phi}\right)d{\mit\Omega}.$">|; 

$key = q/nabla{calH}_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5871.png"
 ALT="$ \nabla {\cal H}_0=0$">|; 

$key = q/displaystyleC_p=frac{p-p_infty}{(1slash2),rho_infty,U^{,2}}=frac{2}{gamma,{rmMa}_infty^{,2}},frac{p-p_infty}{p_infty}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5924.png"
 ALT="$\displaystyle C_p = \frac{p-p_\infty}{(1/2) \rho_\infty U^{ 2}}=\frac{2}{\gamma {\rm Ma}_\infty^{ 2}} \frac{p-p_\infty}{p_\infty}.$">|; 

$key = q/beta=tan^{-1}(u_1slashv);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5681.png"
 ALT="$ \beta=\tan^{-1}(u_1/v)$">|; 

$key = q/zeta=b+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2175.png"
 ALT="$ \zeta=b+1$">|; 

$key = q/displaystyle=frac{xi_1}{4,y_1},mu,(1-beta),frac{G,M_ast,m_p}{R,k}=1.34times10^7,{rmK},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5241.png"
 ALT="$\displaystyle = \frac{\xi_1}{4 y_1} \mu (1-\beta) \frac{G M_\ast m_p}{R k} = 1.34\times 10^7 {\rm K},$">|; 

$key = q/displaystyle(widetilde{nabla{bfA}})_{zz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7151.png"
 ALT="$\displaystyle (\widetilde{\nabla {\bf A}})_{zz}$">|; 

$key = q/displaystyle=f(x-lambda,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6047.png"
 ALT="$\displaystyle = f(x-\lambda y)$">|; 

$key = q/c=e,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2771.png"
 ALT="$ c=e a$">|; 

$key = q/deltar;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1126.png"
 ALT="$ \delta r$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{Chapter06slashvortexb.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="221" BORDER="0"
 SRC="|."$dir".q|img2112.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{Chapter06/vortexb.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{rho_2}{rho_1}=frac{u_1}{u_2}=sqrt{frac{T_1}{T_2}},frac{{rmMa}_1}{{rmMa}_2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5475.png"
 ALT="$\displaystyle \frac{\rho_2}{\rho_1}=\frac{u_1}{u_2}=\sqrt{\frac{T_1}{T_2}} \frac{{\rm Ma}_1}{{\rm Ma}_2},$">|; 

$key = q/displaystyle{calR}_{ki},{calR}_{kj}=delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6832.png"
 ALT="$\displaystyle {\cal R}_{ki} {\cal R}_{kj } = \delta_{ij}.$">|; 

$key = q/xi=xi_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5212.png"
 ALT="$ \xi=\xi_1$">|; 

$key = q/(partialslashpartialr)(r,partialpsislashpartialr)=partialpsislashpartialtheta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1709.png"
 ALT="$ (\partial/\partial r)(r \partial\psi/\partial r)= \partial\psi/\partial\theta=0$">|; 

$key = q/1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1419.png"
 ALT="$ 1/2$">|; 

$key = q/10^{-9},{rmm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1081.png"
 ALT="$ 10^{-9} {\rm m}$">|; 

$key = q/C=(1slash2),V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3777.png"
 ALT="$ C= (1/2) V$">|; 

$key = q/{multline}left(1-[2,n,A_n+(2+n),B_n],muright)left.p_nright|_{r=a}=[0.5ex]rho,gle2,n-2}{2,n+1}right)frac{skew{3}bar{rho}}{rho},zeta_nright],{multline};LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="534" HEIGHT="96" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4422.png"
 ALT="\begin{multline}
\left(1-[2 n A_n+(2+n) B_n] \mu\right)\left.p_n\right\vert ...
...}{2 n+1}\right)
\frac{\skew{3}\bar{\rho}}{\rho} \zeta_n\right],
\end{multline}">|; 

$key = q/displaystylecostheta=1-frac{2,Q'}{Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2831.png"
 ALT="$\displaystyle \cos\theta = 1-\frac{2 Q'}{Q}
$">|; 

$key = q/displaystyle=frac{kappa^{,2},A}{V_1}-H_1G=frac{1}{V_1}left[A,kappa^{,2}-left(frac{V_1,V_2}{V_1+V_2}right)H_1H_2right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img695.png"
 ALT="$\displaystyle = \frac{\kappa^{ 2} A}{V_1} - H_1G =\frac{1}{V_1}\left[A \kappa^{ 2}- \left(\frac{V_1 V_2}{V_1+V_2}\right) H_1H_2\right],$">|; 

$key = q/displaystylefrac{p_3'-p_1}{p_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6198.png"
 ALT="$\displaystyle \frac{p_3'-p_1}{p_1}$">|; 

$key = q/partialslashpartialvarphi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3749.png"
 ALT="$ \partial/\partial \varphi = 0$">|; 

$key = q/displaystyleomega_varphi(r,theta)=frac{{calL}(psi)}{r,sintheta}=frac{sintheta}{r^{,2}}-frac{2}{r^{,2}}right)f=frac{3,V,a,sintheta}{2,r^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3783.png"
 ALT="$\displaystyle \omega_\varphi(r,\theta) = \frac{{\cal L}(\psi)}{r \sin\theta} =...
...{dr^{ 2}}-\frac{2}{r^{ 2}}\right) f = \frac{3 V a \sin\theta}{2 r^{ 2}}.$">|; 

$key = q/|z|lla;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4459.png"
 ALT="$ \vert z\vert\ll a$">|; 

$key = q/+(7,eslash4),zeta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4533.png"
 ALT="$ +(7 e/4) \zeta_2$">|; 

$key = q/displaystyle{bfa}cdot{bfb}times{bfc}={bfb}cdot{bfc}times{bfa}={bfc}cdot{bfa}times{bfb},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6412.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b} \times{\bf c} = {\bf b} \cdot {\bf c} \times{\bf a} = {\bf c} \cdot {\bf a} \times{\bf b},$">|; 

$key = q/dh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6522.png"
 ALT="$ dh$">|; 

$key = q/displaystyle=-frac{2,m}{m^{,2}-m'^{,2}}int_{-1}^1left[(1-mu^{,2}),frac{dP_n^{,m}}+frac{m'^{,2}}{1-mu^{,2}},P_n^{,m},P_{n'}^{,m'}right]mu,dmu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="537" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4961.png"
 ALT="$\displaystyle =- \frac{2 m}{m^{ 2}-m'^{ 2}}\int_{-1}^1 \left[(1-\mu^{ 2}) ...
...d\mu} + \frac{m'^{ 2}}{1-\mu^{ 2}} P_n^{ m} P_{n'}^{ m'}\right]\mu d\mu,$">|; 

$key = q/displaystylefrac{delta_1(x)}{x}=frac{1.75}{{rmRe}^{1slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3216.png"
 ALT="$\displaystyle \frac{\delta_1(x)}{x} = \frac{1.75}{{\rm Re}^{1/2}}.$">|; 

$key = q/displaystylefrac{h^{,2}}{b^{,2}}=frac{2,a}{(h^{,2}+a^{,2})^{1slash2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2844.png"
 ALT="$\displaystyle \frac{h^{ 2}}{b^{ 2}} = \frac{2 a}{(h^{ 2}+a^{ 2})^{1/2}}.
$">|; 

$key = q/deltaproptox^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2999.png"
 ALT="$ \delta\propto x^{1/2}$">|; 

$key = q/sigma_{33}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="$ \sigma_{33}'$">|; 

$key = q/partialTslashpartialt<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6622.png"
 ALT="$ \partial T/\partial t&lt;0$">|; 

$key = q/{bfV}=V_x(t),{bfe}_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1796.png"
 ALT="$ {\bf V} = V_x(t) {\bf e}_x$">|; 

$key = q/displaystyle=(nablacdot{bfB}),{bfA}-(nablacdot{bfA}),{bfB}+({bfB}cdotnabla),{bfA}-({bfA}cdotnabla),{bfB}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="415" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6698.png"
 ALT="$\displaystyle = (\nabla\cdot {\bf B}) {\bf A}- (\nabla \cdot{\bf A})  {\bf B}+({\bf B}\cdot\nabla) {\bf A}- ({\bf A}\cdot\nabla) {\bf B}.$">|; 

$key = q/displaystyleBQ';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img742.png"
 ALT="$\displaystyle BQ'$">|; 

$key = q/displaystyle=({bfa}cdot{bfc}),{bfb}-({bfa}cdot{bfb}),{bfc},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6674.png"
 ALT="$\displaystyle = ({\bf a}\cdot{\bf c}) {\bf b} - ({\bf a}\cdot{\bf b}) {\bf c},$">|; 

$key = q/beta=beta_ast=-0.1989;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2971.png"
 ALT="$ \beta=\beta_\ast=-0.1989$">|; 

$key = q/f_3equivf_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img577.png"
 ALT="$ f_3\equiv f_z$">|; 

$key = q/displaystylepsi(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1955.png"
 ALT="$\displaystyle \psi(x, y)$">|; 

$key = q/displaystylec_{ij}=a_{ik},b_{jk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6856.png"
 ALT="$\displaystyle c_{ij} = a_{ik} b_{jk},$">|; 

$key = q/displaystyle=skew{5}ddot{mitPhi}-2,{mitOmega},skew{5}dot{mitPhi}'-frac{g,d}{a^{,2}},(zeta-F),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4782.png"
 ALT="$\displaystyle = \skew{5}\ddot{\mit\Phi} -2 {\mit\Omega} \skew{5}\dot{\mit\Phi}' - \frac{g d}{a^{ 2}} (\zeta-F),$">|; 

$key = q/Qslasha^{,4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3617.png"
 ALT="$ Q/a^{ 4}$">|; 

$key = q/f'(0)<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3177.png"
 ALT="$ f'(0)&lt;0$">|; 

$key = q/displaystyleG({rmFr})=left(frac{Q^{,2}}{g}right)^{1slash2}left(frac{1}{{rmFr}^{,2slash3}}+frac{{rmFr}^{,4slash3}}{2}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1446.png"
 ALT="$\displaystyle G({\rm Fr}) = \left(\frac{Q^{ 2}}{g}\right)^{1/2}\left(\frac{1}{{\rm Fr}^{ 2/3}}+\frac{{\rm Fr}^{ 4/3}}{2}\right).$">|; 

$key = q/1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$ 1$">|; 

$key = q/displaystyle={mitPhi}_r'',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4852.png"
 ALT="$\displaystyle ={\mit\Phi}_r'',$">|; 

$key = q/displaystyleQ={calC},S,v={calC},S,(2,g,h)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1411.png"
 ALT="$\displaystyle Q = {\cal C} S v = {\cal C} S (2 g h)^{1/2}.$">|; 

$key = q/-Aslasha^{,3}+Bslasha-V=2,overline{C}+4,overline{D},a^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3851.png"
 ALT="$ -A/a^{ 3}+B/a-V= 2 \overline{C}+4 \overline{D} a^{ 2}$">|; 

$key = q/deltaslashc=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3371.png"
 ALT="$ \delta/c=1$">|; 

$key = q/displaystyle=~_{lim|deltax|rightarrow0}frac{phi(x+deltax,y)+{rmi},psi(x+deltax,y)-phi(x,y)-{rmi},psi(x,y)}{deltax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="491" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1962.png"
 ALT="$\displaystyle =&nbsp;_{\lim \vert\delta x\vert\rightarrow 0} \frac{\phi(x+\delta x, y) + {\rm i}  \psi(x+\delta x, y) - \phi(x, y) - {\rm i}  \psi(x,y)}{\delta x}$">|; 

$key = q/displaystylealpha=frac{{bfr}cdot{bfb}times{bfc}}{{bfa}cdot{bfb}times{bfc}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6417.png"
 ALT="$\displaystyle \alpha = \frac{{\bf r}\cdot{\bf b}\times{\bf c}}{{\bf a}\cdot{\bf b}\times{\bf c}}.$">|; 

$key = q/rho({bfr}+{bfv},deltat,t+deltat);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="$ \rho({\bf r}+{\bf v} \delta t, t+\delta t)$">|; 

$key = q/displaystyleleft(frac{partial}{partialt}+{bfV}cdotnablaright);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4073.png"
 ALT="$\displaystyle \left(\frac{\partial}{\partial t} + {\bf V}\cdot\nabla\right)$">|; 

$key = q/{M_ast};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5268.png"
 ALT="$ {M_\ast}$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{Chapter15slashcorner.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="225" BORDER="0"
 SRC="|."$dir".q|img5716.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{Chapter15/corner.eps}}
\end{figure}">|; 

$key = q/{mitDelta}P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3668.png"
 ALT="$ {\mit\Delta} P$">|; 

$key = q/beta=gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="$ \beta=\gamma$">|; 

$key = q/alpharightarrowalpha-theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6343.png"
 ALT="$ \alpha\rightarrow\alpha-\theta$">|; 

$key = q/vllu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3098.png"
 ALT="$ v\ll u$">|; 

$key = q/displaystyle=U,frac{dU}{dx}+nu,frac{partial^{,2}v_x}{partialy^{,2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2904.png"
 ALT="$\displaystyle = U \frac{dU}{dx} + \nu \frac{\partial^{ 2} v_x}{\partial y^{ 2}},$">|; 

$key = q/displaystylenablaP=-mu,nablatimes;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3784.png"
 ALT="$\displaystyle \nabla P = -\mu \nabla\times$">|; 

$key = q/displaystylefrac{z}{k}=left(frac{bar{F}}{k,V}right)^{1slash2}left(frac{bar{F}}{krmi}left(frac{bar{F}}{k,V}right)^{1slash2}+{rmi},frac{pi}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="701" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2266.png"
 ALT="$\displaystyle \frac{z}{k} = \left(\frac{\bar{F}}{k V}\right)^{1/2}\left(\frac{...
...ight]+2 {\rm i}\left(\frac{\bar{F}}{k V}\right)^{1/2}+{\rm i} \frac{\pi}{2}.$">|; 

$key = q/y=u,v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7214.png"
 ALT="$ y=u v$">|; 

$key = q/l=(a^{,2}-b^{,2})^{1slash2}slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2134.png"
 ALT="$ l=(a^{ 2}-b^{ 2})^{1/2}/2$">|; 

$key = q/deltasim(nu,xslashU_0)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3025.png"
 ALT="$ \delta\sim (\nu x/U_0)^{1/2}$">|; 

$key = q/displaystyle{bfgrad},Tequivleft(frac{partialT}{partialx},,frac{partialT}{partialy},,frac{partialT}{partialz}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6541.png"
 ALT="$\displaystyle {\bf grad} T \equiv \left(\frac{\partial T}{\partial x},  \frac{\partial T}{\partial y},  \frac{\partial T}{\partial z}\right).$">|; 

$key = q/displaystyleF(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1944.png"
 ALT="$\displaystyle F(z)$">|; 

$key = q/displaystyledS=mleft({calC}_V,frac{dT}{T}+frac{p}{T},dvright)=mleft({calC}_V,frac{dT}{T}+{calR},frac{dv}{v}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5321.png"
 ALT="$\displaystyle dS = m\left({\cal C}_V \frac{dT}{T} + \frac{p}{T} dv\right)= m\left({\cal C}_V \frac{dT}{T} + {\cal R} \frac{dv}{v}\right).$">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{Chapter09slashwing3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="527" HEIGHT="195" BORDER="0"
 SRC="|."$dir".q|img3416.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{Chapter09/wing3.eps}}
\end{figure}">|; 

$key = q/{rmMa}_1=1.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$ {\rm Ma}_1=1.2$">|; 

$key = q/5%;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5135.png"
 ALT="$ 5\%$">|; 

$key = q/displaystyle-dtheta=sqrt{{rmMa}^{,2}-1},frac{dw}{w},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5764.png"
 ALT="$\displaystyle -d\theta =\sqrt{{\rm Ma}^{ 2}-1} \frac{dw}{w},$">|; 

$key = q/displaystyle=-frac{nablap}{rho}-nabla{mitPsi}+nu,nabla^{,2}{bfv},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img544.png"
 ALT="$\displaystyle = -\frac{\nabla p}{\rho} -\nabla{\mit\Psi} +\nu \nabla^{ 2}{\bf v},$">|; 

$key = q/{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6229.png"
 ALT="$ {\bf c}$">|; 

$key = q/displaystylelambda=frac{0.47,m}{1+5,m}=-frac{0.47,theta}{pislash2-4,theta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3227.png"
 ALT="$\displaystyle \lambda = \frac{0.47 m}{1+5 m} = -\frac{0.47 \theta}{\pi/2-4 \theta}.$">|; 

$key = q/{figure}{epsfysize{=2.4in{centerline{epsffile{Chapter12slashlong2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="547" HEIGHT="271" BORDER="0"
 SRC="|."$dir".q|img4718.png"
 ALT="\begin{figure}
\epsfysize =2.4in
\centerline{\epsffile{Chapter12/long2.eps}}
\end{figure}">|; 

$key = q/x=L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2976.png"
 ALT="$ x=L$">|; 

$key = q/int{bfA}cdotd{bfr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6558.png"
 ALT="$ \int {\bf A}\cdot d{\bf r}$">|; 

$key = q/t=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6937.png"
 ALT="$ t=l$">|; 

$key = q/displaystyle=-p+2,muleft(frac{1}{r}frac{partialv_theta}{partialtheta}+frac{v_r}{r}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="$\displaystyle =-p + 2 \mu\left(\frac{1}{r}\frac{\partial v_\theta}{\partial \theta}+ \frac{v_r}{r}\right),$">|; 

$key = q/displaystyled;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6092.png"
 ALT="$\displaystyle d$">|; 

$key = q/displaystyle=n,{bfa}+m,{bfa},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6249.png"
 ALT="$\displaystyle = n {\bf a} + m {\bf a},$">|; 

$key = q/omega=c=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4120.png"
 ALT="$ \omega=c=0$">|; 

$key = q/displaystyle{calR}_n^{,(-m)}(r,theta,varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4272.png"
 ALT="$\displaystyle {\cal R}_n^{ (-m)}(r,\theta,\varphi)$">|; 

$key = q/displaystylepsi=left(frac{A}{r^{,2}},costheta+B,r^{,2}right)sin^2theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2846.png"
 ALT="$\displaystyle \psi = \left(\frac{A}{r^{ 2}} \cos\theta + B r^{ 2}\right)\sin^2\theta
$">|; 

$key = q/beta_1,sin(theta-theta_1)slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1754.png"
 ALT="$ \beta_1 \sin(\theta-\theta_1)/r$">|; 

$key = q/displaystyle{calH}={calE}+p,v.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5581.png"
 ALT="$\displaystyle {\cal H} = {\cal E} + p v.
$">|; 

$key = q/displaystyle2,a_{ij}+a_{ji}=a_{ss},delta_{ij}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6904.png"
 ALT="$\displaystyle 2 a_{ij} + a_{ji} = a_{ss} \delta_{ij}.$">|; 

$key = q/displaystylefrac{Drho}{Dt}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="$\displaystyle \frac{D\rho}{Dt} = 0.$">|; 

$key = q/displaystylez+{rmi},varpi=c,exp,(xi+{rmi},eta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2723.png"
 ALT="$\displaystyle z+{\rm i} \varpi = c \exp (\xi+{\rm i} \eta),$">|; 

$key = q/z_1={rmi},d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2215.png"
 ALT="$ z_1={\rm i} d$">|; 

$key = q/displaystylesimeq-frac{k}{2},sin(2,skew{5}tilde{phi}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1212.png"
 ALT="$\displaystyle \simeq -\frac{k}{2} \sin(2 \skew{5}\tilde{\phi}),$">|; 

$key = q/u=v=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3135.png"
 ALT="$ u=v=0$">|; 

$key = q/(nabla^{,2}{bfa})_i=nabla^{,2}a_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6993.png"
 ALT="$ (\nabla^{ 2} {\bf a})_i = \nabla^{ 2} a_i$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter08slashplate1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="545" HEIGHT="252" BORDER="0"
 SRC="|."$dir".q|img3003.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter08/plate1.eps}}
\end{figure}">|; 

$key = q/{bfv}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1384.png"
 ALT="$ {\bf v}_2$">|; 

$key = q/H_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img690.png"
 ALT="$ H_1$">|; 

$key = q/0.9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$ 0.9$">|; 

$key = q/T^{,gamma}slashp^{,gamma-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5340.png"
 ALT="$ T^{ \gamma}/p^{ \gamma-1}$">|; 

$key = q/a_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6261.png"
 ALT="$ a_y$">|; 

$key = q/displaystylenablacdot(nablaT)=frac{c}{kappa}frac{partialT}{partialt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6610.png"
 ALT="$\displaystyle \nabla\cdot (\nabla T) = \frac{c}{\kappa} \frac{ \partial T}{\partial t}.$">|; 

$key = q/lim_{,dSrightarrow0}oint{bfA}cdotd{bfr}slashdS;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6668.png"
 ALT="$ \lim_{ dS\rightarrow 0}\oint{\bf A}\cdot d{\bf r} /dS$">|; 

$key = q/Rslasha_0=221.29;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1028.png"
 ALT="$ R/a_0=221.29$">|; 

$key = q/displaystyleD,{mitPhi}_r';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4840.png"
 ALT="$\displaystyle D {\mit\Phi}_r'$">|; 

$key = q/displaystyleint!int_{S_1}{bfv}cdotd{bfS}=intint_{S_2}{bfv}cdotd{bfS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6501.png"
 ALT="$\displaystyle \int\!\int_{S_1} {\bf v}\cdot d{\bf S} = \int\int_{S_2} {\bf v}\cdot d{\bf S}.$">|; 

$key = q/displaystyleF(xi)=A,{rme}^{,2,xi}+B,{rme}^{-xi}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2737.png"
 ALT="$\displaystyle F(\xi)=A {\rm e}^{ 2 \xi} + B {\rm e}^{-\xi}.$">|; 

$key = q/displaystyle(nabla{bfA})_{ij}=frac{1}{h_j},frac{partialA_i}{partialu_j}-frac{A_jj}sum_{k=1,3}frac{A_k}{h_i,h_k},frac{partialh_i}{partialu_k}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="416" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7095.png"
 ALT="$\displaystyle (\nabla{\bf A})_{ij} = \frac{1}{h_j} \frac{\partial A_i}{\partia...
...\delta_{ij}\sum_{k=1,3}\frac{A_k}{h_i h_k} \frac{\partial h_i}{\partial u_k}.$">|; 

$key = q/displaystylefrac{d^{,2}u}{dz^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img788.png"
 ALT="$\displaystyle \frac{d^{ 2}u}{dz^{ 2}}$">|; 

$key = q/displaystyleD,B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4785.png"
 ALT="$\displaystyle D B$">|; 

$key = q/displaystyle{bfv}=-nablatimes(psi,nablavarphi)+{mitOmega}_varphi,nablavarphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2559.png"
 ALT="$\displaystyle {\bf v} = -\nabla\times (\psi \nabla\varphi) + {\mit\Omega}_\varphi \nabla\varphi.$">|; 

$key = q/displaystylefrac{partialu}{partialt}-2,{mitOmega},costheta,v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4461.png"
 ALT="$\displaystyle \frac{\partial u}{\partial t} - 2 {\mit\Omega} \cos\theta v$">|; 

$key = q/displaystyle=C_{p0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6060.png"
 ALT="$\displaystyle =C_{p0},$">|; 

$key = q/DslashDt(pslashrho^{,gamma})rightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5117.png"
 ALT="$ D/Dt(p/\rho^{ \gamma})\rightarrow 0$">|; 

$key = q/displaystyletantheta=frac{2,cotbetaleft({rmMa}_1^{,2},sin^2beta-1right)}{2+{rmMa}_1^{,2}left[gamma+cos(2,beta)right]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5706.png"
 ALT="$\displaystyle \tan\theta =\frac{2 \cot\beta\left({\rm Ma}_1^{ 2} \sin^2\beta-1\right)}{2+{\rm Ma}_1^{ 2}\left[\gamma+\cos( 2 \beta)\right]}.$">|; 

$key = q/displaystyler=sqrt{x^{,2}+y^{,2}+z^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6269.png"
 ALT="$\displaystyle r = \sqrt{x^{ 2} + y^{ 2} + z^{ 2}}.$">|; 

$key = q/displaystyle=-frac{partial}{partialz}left(frac{p}{skew{3}bar{rho}}+{mitPhi}+chiright).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4466.png"
 ALT="$\displaystyle =-\frac{\partial}{\partial z}\left(\frac{p}{\skew{3}\bar{\rho}}+{\mit\Phi}+\chi\right).$">|; 

$key = q/displaystyleC_D=frac{D}{rho,U_0^{,2},a},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3094.png"
 ALT="$\displaystyle C_D = \frac{D}{\rho U_0^{ 2} a},$">|; 

$key = q/displaystyle{bfv}(P_+);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3420.png"
 ALT="$\displaystyle {\bf v}(P_+)$">|; 

$key = q/displaystylez=zeta_b(theta,phi,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4467.png"
 ALT="$\displaystyle z = \zeta_b(\theta,\phi,t),$">|; 

$key = q/l<gslashomega^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1057.png"
 ALT="$ l&lt; g/\omega^{ 2}$">|; 

$key = q/{bfa}cdot{bfb}times{bfc}={bfa}times{bfb}cdot{bfc}=-{bfb}cdot{bfa}times{bfc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6983.png"
 ALT="$ {\bf a}\cdot{\bf b}\times {\bf c}={\bf a}\times {\bf b}\cdot{\bf c}=-{\bf b}\cdot{\bf a}\times{\bf c}$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{Chapter08slashbl.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="569" HEIGHT="257" BORDER="0"
 SRC="|."$dir".q|img2857.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{Chapter08/bl.eps}}
\end{figure}">|; 

$key = q/a=(r_+^{,2},r_-)^{1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1348.png"
 ALT="$ a=(r_+^{ 2} r_-)^{1/3}$">|; 

$key = q/0<ystackrel{_{normalsize<}}{_{normalsizesim}}delta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2855.png"
 ALT="$ 0&lt; y\stackrel {_{\normalsize &lt;}}{_{\normalsize\sim}}\delta$">|; 

$key = q/nablacdotdelta{bfv}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1611.png"
 ALT="$ \nabla\cdot\delta{\bf v}=0$">|; 

$key = q/displaystyle=c_1left[1+left(frac{gamma_1+1}{2,gamma_1}right)left(frac{p_2}{p_1}-1right)right]^{,1slash2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5646.png"
 ALT="$\displaystyle = c_1\left[1+\left(\frac{\gamma_1+1}{2 \gamma_1}\right)\left(\frac{p_2}{p_1}-1\right)\right]^{ 1/2},$">|; 

$key = q/cos({bfk}cdot{bfr}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6752.png"
 ALT="$ \cos({\bf k}\cdot {\bf r}).$">|; 

$key = q/G(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3650.png"
 ALT="$ G(x)$">|; 

$key = q/u,{bfa};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6726.png"
 ALT="$ u {\bf a}$">|; 

$key = q/displaystyle=V_z(t),frac{a^{,3}}{r^{,3}},costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2662.png"
 ALT="$\displaystyle = V_z(t) \frac{a^{ 3}}{r^{ 3}} \cos\theta,$">|; 

$key = q/displaystylev(x,y)simeqfrac{A,u_0(y)}{2sqrt{x_0-x}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3131.png"
 ALT="$\displaystyle v(x,y)\simeq \frac{A u_0(y)}{2\sqrt{x_0-x}}.$">|; 

$key = q/displaystylekappa^{,2}=kappa_x^{,2},cos^2alpha+kappa_y^{,2},sin^2alpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img658.png"
 ALT="$\displaystyle \kappa^{ 2} = \kappa_x^{ 2} \cos^2\alpha+ \kappa_y^{ 2} \sin^2\alpha.$">|; 

$key = q/displaystylep=p_g+p_r,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5179.png"
 ALT="$\displaystyle p = p_g + p_r,$">|; 

$key = q/rho_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5637.png"
 ALT="$ \rho_3$">|; 

$key = q/-3.24times10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4622.png"
 ALT="$ -3.24\times 10^{-2}$">|; 

$key = q/displaystyle{rmMa}_1,sinbetasimeq1+epsilonsqrt{{rmMa}_1^{,2}-1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5736.png"
 ALT="$\displaystyle {\rm Ma}_1 \sin\beta \simeq 1+\epsilon\sqrt{{\rm Ma}_1^{ 2}-1},$">|; 

$key = q/displaystyleint_P^Qleft(y,dx+x^{,3},dyright)=left.int_0^1x^{,3},dyright|_{x=1}+left.int_1^2y,dxright|_{y=1}=frac{7}{4}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6460.png"
 ALT="$\displaystyle \int_P^Q \left( y dx + x^{ 3} dy\right)= \left.\int_0^1 x^{ 3} dy\right\vert _{x=1} + \left.\int_1^2 y dx\right\vert _{y=1} = \frac{7}{4}.$">|; 

$key = q/|V_s|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5634.png"
 ALT="$ \vert V_s\vert$">|; 

$key = q/displaystylenabla^{,2}phi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2637.png"
 ALT="$\displaystyle \nabla^{ 2}\phi=0$">|; 

$key = q/ygeq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2494.png"
 ALT="$ y\geq 0$">|; 

$key = q/C_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1378.png"
 ALT="$ C_1$">|; 

$key = q/psi(x,0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2023.png"
 ALT="$ \psi(x,0)=0$">|; 

$key = q/displaystyleoint_S{bfh}cdotd{bfS}=-frac{partial}{partialt}left(intc,T,dVright),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6607.png"
 ALT="$\displaystyle \oint_S {\bf h}\cdot d{\bf S} = - \frac{\partial}{\partial t}\left( \int c  T dV\right),$">|; 

$key = q/displaystyle{bfB}=M_0,g,{bfe}_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img586.png"
 ALT="$\displaystyle {\bf B} = M_0 g {\bf e}_z,$">|; 

$key = q/F_1(z)=phi_1(x,y)+{rmi},psi_1(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2008.png"
 ALT="$ F_1(z)=\phi_1(x,y)+{\rm i} \psi_1(x,y)$">|; 

$key = q/displaystyle=p_0-rho,g,zeta_0,sin(omega,t-k,x)-rho,k,(V-c),A,sin(omega,t-k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4100.png"
 ALT="$\displaystyle = p_0 -\rho g \zeta_0 \sin(\omega t-k x)-\rho k (V-c) A \sin(\omega t-k x)$">|; 

$key = q/displaystyle{rmMa}_1gtreqless1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5482.png"
 ALT="$\displaystyle {\rm Ma}_1\gtreqless 1.$">|; 

$key = q/displaystyle=P,dx.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2386.png"
 ALT="$\displaystyle =P dx.$">|; 

$key = q/z_0=x_0+{rmi},y_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1983.png"
 ALT="$ z_0=x_0+{\rm i} y_0$">|; 

$key = q/X=xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4755.png"
 ALT="$ X=\xi$">|; 

$key = q/C_{L_0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6038.png"
 ALT="$ C_{L_0}$">|; 

$key = q/displaystyleb=frac{a}{cosh(k,d),(1-c^{,2}slashV^{,2})},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4128.png"
 ALT="$\displaystyle b = \frac{a}{\cosh(k d) (1-c^{ 2}/V^{ 2})},$">|; 

$key = q/displaystyleD,A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4784.png"
 ALT="$\displaystyle D A$">|; 

$key = q/displaystyle{mitDelta}C_L=frac{(1slash2),C_{L0},[(1slash2),c,d+(1slash2),c,d]}{cC_{L0}}{bslashd}=frac{(1slash2),C_{L0}}{(bslashc),(cslashd)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="542" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6077.png"
 ALT="$\displaystyle {\mit\Delta}C_L = \frac{(1/2) C_{L0} [(1/2) c d + (1/2) c d]}{c b}=\frac{(1/2) C_{L0}}{b/d} = \frac{(1/2) C_{L0}}{(b/c) (c/d)}.$">|; 

$key = q/v_t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1773.png"
 ALT="$ v_t=0$">|; 

$key = q/displaystyleK_{rmfluid};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2676.png"
 ALT="$\displaystyle K_{\rm fluid}$">|; 

$key = q/displaystyle_xcdotd{bfS},,-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$\displaystyle _x\cdot d{\bf S},  -$">|; 

$key = q/{calR}=RslashM;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5292.png"
 ALT="$ {\cal R}=R/M$">|; 

$key = q/displaystylea=left(frac{K}{pi,G,rho_c^{,2slash3}}right)^{1slash2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5211.png"
 ALT="$\displaystyle a = \left(\frac{K}{\pi G \rho_c^{ 2/3}}\right)^{1/2}.$">|; 

$key = q/k_0,dgg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4062.png"
 ALT="$ k_0 d\gg 1$">|; 

$key = q/s,(1-s);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img677.png"
 ALT="$ s (1-s)$">|; 

$key = q/delta(phi)=delta_0,sinphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3489.png"
 ALT="$ \delta(\phi)=\delta_0 \sin\phi$">|; 

$key = q/displaystyleC_{pU};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5837.png"
 ALT="$\displaystyle C_{pU}$">|; 

$key = q/displaystyledT=frac{partialT}{partialx},dx+frac{partialT}{partialy},dy+frac{partialT}{partialz},dz.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6543.png"
 ALT="$\displaystyle dT = \frac{\partial T}{\partial x} dx +\frac{\partial T}{\partial y} dy+ \frac{\partial T}{\partial z} dz.$">|; 

$key = q/omega_2={mitPi}_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4585.png"
 ALT="$ \omega_2={\mit\Pi}_M$">|; 

$key = q/displaystylefrac{tan(beta-theta)}{tanbeta}=frac{u_2}{u_1}=frac{rho_1}{rho_2}=fra,{rmMa}_1^{,2},sin^2beta}{(gamma+1),{rmMa}_1^{,2},sin^2beta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5704.png"
 ALT="$\displaystyle \frac{\tan(\beta-\theta)}{\tan\beta} = \frac{u_2}{u_1}=\frac{\rho...
...-1) {\rm Ma}_1^{ 2} \sin^2\beta}{(\gamma+1) {\rm Ma}_1^{ 2} \sin^2\beta}.$">|; 

$key = q/deltatheta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6823.png"
 ALT="$ \delta\theta_2$">|; 

$key = q/displaystyletau=8pi,mu,{mitOmega}_0,a^{,3}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3888.png"
 ALT="$\displaystyle \tau = 8\pi \mu {\mit\Omega}_0 a^{ 3}.
$">|; 

$key = q/V_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img689.png"
 ALT="$ V_2$">|; 

$key = q/displaystylefrac{{mitDelta}p}{p_1}=frac{p_2-p_1}{p_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5507.png"
 ALT="$\displaystyle \frac{{\mit\Delta}p}{p_1} =\frac{p_2-p_1}{p_1}$">|; 

$key = q/displaystylec_p=left(frac{partial{calH}}{partialT}right)_v=Tleft(frac{partial{calS}}{partialT}right)_p.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5595.png"
 ALT="$\displaystyle c_p =\left(\frac{\partial {\cal H}}{\partial T}\right)_v = T\left(\frac{\partial{\cal S}}{\partial T}\right)_p.
$">|; 

$key = q/displaystyleC_p=frac{2}{gamma,{rmMa}_infty^{,2}}left(left[1+frac{1}{2},(gamma-1)c{q^{,2}}{U^{,2}}right)right]^{,gammaslash(gamma-1)}-1right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="500" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5930.png"
 ALT="$\displaystyle C_p = \frac{2}{\gamma {\rm Ma}_\infty^{ 2}}\left(\left[1+\frac{...
...2}\left(1-\frac{q^{ 2}}{U^{ 2}}\right)\right]^{ \gamma/(\gamma-1)}-1\right),$">|; 

$key = q/displaystyleC_D=frac{2}{sqrt{{rmMa}_infty^{,2}-1}},overline{left(frac{dy}{dx}right)^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6001.png"
 ALT="$\displaystyle C_D = \frac{2}{\sqrt{{\rm Ma}_\infty^{ 2}-1}} \overline{\left(\frac{dy}{dx}\right)^2}.$">|; 

$key = q/displaystyle{bfa}times({bfb}+{bfc})={bfa}times{bfb}+{bfa}times{bfc},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6365.png"
 ALT="$\displaystyle {\bf a}\times({\bf b} +{\bf c})= {\bf a} \times{\bf b}+{\bf a}\times{\bf c},$">|; 

$key = q/x>x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3134.png"
 ALT="$ x&gt;x_0$">|; 

$key = q/displaystylec(n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4706.png"
 ALT="$\displaystyle c(n)$">|; 

$key = q/displaystyle=left[frac{(m+1),U_0,x^{,m-1}}{2,nu}right]^{1slash2}left.frac{partial}{partialeta}right|_x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2926.png"
 ALT="$\displaystyle =\left[\frac{(m+1) U_0 x^{ m-1}}{2 \nu}\right]^{1/2}\left.\frac{\partial}{\partial\eta}\right\vert _x.$">|; 

$key = q/displaystylembox{hspace{0.5cm}as{rrightarrowinfty{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1800.png"
 ALT="$\displaystyle \mbox{\hspace{0.5cm}as $r\rightarrow\infty$}$">|; 

$key = q/{bfe}_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5068.png"
 ALT="$ {\bf e}_z$">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1408.png"
 ALT="$ v$">|; 

$key = q/alpha_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img990.png"
 ALT="$ \alpha_i$">|; 

$key = q/displaystyletimes{bfr}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img810.png"
 ALT="$\displaystyle \times {\bf r}),$">|; 

$key = q/displaystylev_x(x,z,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3941.png"
 ALT="$\displaystyle v_x(x,z,t)$">|; 

$key = q/xggL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3067.png"
 ALT="$ x\gg L$">|; 

$key = q/displaystylephi({bfr},t)=F(z),cos(omega,t-k,x).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3914.png"
 ALT="$\displaystyle \phi({\bf r},t) = F(z) \cos(\omega t-k x).$">|; 

$key = q/displaystyleleft[1+left(frac{gamma-1}{gamma+1}right)({rmMa}_{,e}^{,2}-1)right]=lamma+1}right)left(frac{p_0}{p_e}right)^{(gamma-1)slashgamma}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="431" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5430.png"
 ALT="$\displaystyle \left[1+\left(\frac{\gamma-1}{\gamma+1}\right)({\rm Ma}_{ e}^{ ...
...left(\frac{2}{\gamma+1}\right)\left(\frac{p_0}{p_e}\right)^{(\gamma-1)/\gamma}.$">|; 

$key = q/h_2>h_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1497.png"
 ALT="$ h_2&gt;h_1$">|; 

$key = q/displaystyleddot{p}_r+2,{mitOmega}sum_{s=1,infty}(beta_{r,,s},dot{p}_s+beta_{r,,-s},dot{p}_{-s})+frac{g,d}{a^{,2}},lambda_r,p_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4912.png"
 ALT="$\displaystyle \ddot{p}_r + 2 {\mit\Omega}\sum_{s=1,\infty} (\beta_{r, s} \dot{p}_s+\beta_{r, -s} \dot{p}_{-s}) + \frac{g d}{a^{ 2}} \lambda_r p_r$">|; 

$key = q/displaystyle=frac{2}{a_1^{,3},sin^3gamma},frac{F(gamma,alpha)-E(gamma,alpha)}{sin^2alpha},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1001.png"
 ALT="$\displaystyle =\frac{2}{a_1^{ 3} \sin^3\gamma} \frac{F(\gamma,\alpha)-E(\gamma,\alpha)}{\sin^2\alpha},$">|; 

$key = q/{calO}(sin^2delta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4311.png"
 ALT="$ {\cal O}(\sin^2\delta)$">|; 

$key = q/omega_M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4592.png"
 ALT="$ \omega_M$">|; 

$key = q/displaystyleleft.phantom{=}-{bfA}times(nablatimes{bfB})-{bfB}times(nablatimes{bfA})right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7083.png"
 ALT="$\displaystyle \left.\phantom{=} - {\bf A}\times(\nabla\times {\bf B})-{\bf B}\times (\nabla\times {\bf A})\right].$">|; 

$key = q/displaystyleint_{mitOmega}({mitPhi}_s,D,{mitPhi}_r-{mitPhi}_r,D,{mitPhi}_s),d{mis-lambda_r)int_{mitOmega}{mitPhi}_s,{mitPhi}_r,d{mitOmega}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="455" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4813.png"
 ALT="$\displaystyle \int_{\mit\Omega} ({\mit\Phi}_s D {\mit\Phi}_r-{\mit\Phi}_r D\...
...da_s-\lambda_r)\int_{\mit\Omega} {\mit\Phi}_s {\mit\Phi}_r d{\mit\Omega} = 0.$">|; 

$key = q/gamma=5slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5195.png"
 ALT="$ \gamma=5/3$">|; 

$key = q/nablatimes{bfE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7212.png"
 ALT="$ \nabla\times {\bf E}$">|; 

$key = q/ksim{calO}(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3725.png"
 ALT="$ k\sim {\cal O}(1)$">|; 

$key = q/dy=y-y_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6534.png"
 ALT="$ dy=y-y_0$">|; 

$key = q/displaystyle{bfv}_{mitSigma}=-w(z),{bfe}_perp;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3453.png"
 ALT="$\displaystyle {\bf v}_{\mit\Sigma} = - w(z) {\bf e}_\perp$">|; 

$key = q/displaystyledot{W}=int_Vv_i,F_i,dV+oint_Sv_i,sigma_{ij},dS_j=int_Vleft[v_i,F_i+frac{partial(v_i,sigma_{ij})}{partialx_j}right]dV,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="485" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="$\displaystyle \dot{W} = \int_V v_i F_i dV + \oint_S v_i \sigma_{ij} dS_j = \int_V\left[v_i F_i+ \frac{\partial(v_i \sigma_{ij})}{\partial x_j}\right]dV,$">|; 

$key = q/displaystyleg=frac{2}{A},frac{W}{L_0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3530.png"
 ALT="$\displaystyle g = \frac{2}{A} \frac{W}{L_0}.$">|; 

$key = q/displaystylea_{ijkl}=alpha,delta_{ij},delta_{kl}+beta,delta_{ik},delta_{jl}+gamma,delta_{il},delta_{jk},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6972.png"
 ALT="$\displaystyle a_{ijkl} = \alpha \delta_{ij} \delta_{kl} + \beta \delta_{ik} \delta_{jl} +\gamma \delta_{il} \delta_{jk},$">|; 

$key = q/displaystyle=-a,frac{cosh[k,(z+d)]}{sinh(k,d)},cos(omega,t-k,x),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3968.png"
 ALT="$\displaystyle = -a \frac{\cosh[k (z+d)]}{\sinh(k d)} \cos(\omega t-k x),$">|; 

$key = q/xslasha=pmsqrt{1-zeta^{,2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2001.png"
 ALT="$ x/a=\pm\sqrt{1-\zeta^{ 2}}$">|; 

$key = q/P=p+rho,{mitPsi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3791.png"
 ALT="$ P = p + \rho {\mit\Psi}$">|; 

$key = q/displaystyle=h^{-1},epsilon_{ijk}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7062.png"
 ALT="$\displaystyle = h^{-1} \epsilon_{ijk}.$">|; 

$key = q/xi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2470.png"
 ALT="$ \xi=0$">|; 

$key = q/displaystyleD=rho,V^{,2},b-rho,V^{,2},kint_0^1left[frac{1}{gamma^{1slash2}}-left(frac{1}{gamma}-1right)^{1slash2}right]dgamma,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2283.png"
 ALT="$\displaystyle D= \rho V^{ 2} b-\rho  V^{ 2} k\int_0^1\left[\frac{1}{\gamma^{1/2}}-\left(\frac{1}{\gamma}-1\right)^{1/2}\right]d\gamma,$">|; 

$key = q/displaystyleB=-left[frac{V,a^{,2},e^{,2}}{e,(1-e^{,2})^{1slash2}-sin^{-1}e}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2774.png"
 ALT="$\displaystyle B = - \left[\frac{V a^{ 2} e^{ 2}}{e (1-e^{ 2})^{1/2}-\sin^{-1}e}\right].$">|; 

$key = q/displaystyle{mitPsi}''=-frac{1}{2},omega^2left[left(x_1-frac{M'}{M+M'},Rright)^2+x_2^{,2}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img980.png"
 ALT="$\displaystyle {\mit\Psi}'' = -\frac{1}{2} \omega^2\left[\left(x_1-\frac{M'}{M+M'} R\right)^2+x_2^{ 2}\right]$">|; 

$key = q/displaystyle=-frac{g}{a},frac{partial}{partialtheta},(zeta-skew{5}bar{zeta}'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5054.png"
 ALT="$\displaystyle =- \frac{g}{a} \frac{\partial}{\partial\theta} (\zeta-\skew{5}\bar{\zeta}'),$">|; 

$key = q/nu=1.5times10^{-5},{rmm^2slashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3822.png"
 ALT="$ \nu=1.5\times 10^{-5} {\rm m^2/s}$">|; 

$key = q/duslashdz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img793.png"
 ALT="$ du/dz$">|; 

$key = q/z_H=-h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img649.png"
 ALT="$ z_H=-h$">|; 

$key = q/displaystyle=nabla^{,2}A_r-frac{2A_r}{r^{,2}}-frac{2}{r^{,2}},frac{partialA_thet,2}}-frac{2}{r^{,2},sintheta},frac{partialA_phi}{partialphi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7195.png"
 ALT="$\displaystyle =\nabla^{ 2} A_r -\frac{2 A_r}{r^{ 2}}-\frac{2}{r^{ 2}} \frac...
...{r^{ 2}} -\frac{2}{r^{ 2} \sin\theta} \frac{\partial A_\phi}{\partial\phi},$">|; 

$key = q/displaystylea_{ij}=a_{ji}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6905.png"
 ALT="$\displaystyle a_{ij} = a_{ji}.$">|; 

$key = q/displaystyler=b,cosh(zslashb).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1264.png"
 ALT="$\displaystyle r = b \cosh(z/b).$">|; 

$key = q/displaystylenabla^{,2}{bfv}=frac{nablaP}{mu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3647.png"
 ALT="$\displaystyle \nabla^{ 2}{\bf v} = \frac{\nabla P}{\mu}.$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{Chapter15slashflat.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="516" HEIGHT="277" BORDER="0"
 SRC="|."$dir".q|img6065.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{Chapter15/flat.eps}}
\end{figure}">|; 

$key = q/intnablacdot{bfv},,dV=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6595.png"
 ALT="$ \int \nabla\cdot{\bf v}  dV = 0$">|; 

$key = q/nabla^{,2}widetilde{phi}=nabla^{,2}widetilde{psi}=nablawidetilde{phi}cdotnablawidetilde{psi}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2088.png"
 ALT="$ \nabla^{ 2}\widetilde{\phi}=\nabla^{ 2}\widetilde{\psi}=\nabla\widetilde{\phi}\cdot\nabla\widetilde{\psi}=0$">|; 

$key = q/displaystylealpha=frac{partialh}{partialx},~~~~~beta=frac{partialh}{partialy}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6532.png"
 ALT="$\displaystyle \alpha = \frac{\partial h}{\partial x},&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\beta= \frac{\partial h}{\partial y}.$">|; 

$key = q/x=cslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$ x=c/2$">|; 

$key = q/1-{rmMa}_infty^{,2}equivm^{,2}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5957.png"
 ALT="$ 1-{\rm Ma}_\infty^{ 2}\equiv m^{ 2}&gt;0$">|; 

$key = q/displaystylefrac{dv_y}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3558.png"
 ALT="$\displaystyle \frac{dv_y}{dt}$">|; 

$key = q/displaystyleT,cos(alpha+delta)=D,cosdelta+L,sindelta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3521.png"
 ALT="$\displaystyle T \cos(\alpha+\delta) = D \cos\delta + L \sin\delta.$">|; 

$key = q/(zeta-a)slasharightarrow-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2171.png"
 ALT="$ (\zeta-a)/a\rightarrow -1$">|; 

$key = q/displaystyle{mitPhi}_{n>0}(rleqa,theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4366.png"
 ALT="$\displaystyle {\mit\Phi}_{n&gt;0}(r\leq a,\theta,\phi)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{Chapter05slashwedge3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1884.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter05/wedge3.eps}}
\end{figure}">|; 

$key = q/{mitPsi}_r(theta,phi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4826.png"
 ALT="$ {\mit\Psi}_r(\theta,\phi)$">|; 

$key = q/deltastackrel{_{normalsize<}}{_{normalsizesim}}y<infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2856.png"
 ALT="$ \delta\stackrel {_{\normalsize &lt;}}{_{\normalsize\sim}}y&lt;\infty$">|; 

$key = q/displaystyle=frac{4}{sqrt{{rmMa}_1^{,2}-1}}left[,overline{left(frac{dh}{dx}right)^2}+overline{alpha^{,2}}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5859.png"
 ALT="$\displaystyle = \frac{4}{\sqrt{{\rm Ma}_1^{ 2}-1}}\left[ \overline{\left(\frac{dh}{dx}\right)^2}+\overline{\alpha^{ 2}}\right]$">|; 

$key = q/{bfv}_{mitSigma}(P_+)={bfv}_{mitSigma}(P_-)={bfv}_{mitSigma}(P);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3427.png"
 ALT="$ {\bf v}_{\mit\Sigma}(P_+)={\bf v}_{\mit\Sigma}(P_-)=
{\bf v}_{\mit\Sigma}(P)$">|; 

$key = q/delta(-z)=delta(z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3398.png"
 ALT="$ \delta(-z)=\delta(z)$">|; 

$key = q/displaystylephi(x,y)=F(x),G(y).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5959.png"
 ALT="$\displaystyle \phi(x,y)= F(x) G(y).$">|; 

$key = q/({bfa}times{bfb})cdot{bfd}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6716.png"
 ALT="$ ({\bf a}\times {\bf b})\cdot {\bf d} = 0$">|; 

$key = q/displaystyleQ=int_0^dv_x,dy=frac{G,d^{,3}}{12,mu}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3584.png"
 ALT="$\displaystyle Q = \int_0^d v_x dy = \frac{G d^{ 3}}{12 \mu}.$">|; 

$key = q/displaystyleleft[alpha_1-frac{3,lambda}{(3slash2),G,M}right]a_1^{,2}=alpha_2,a_2^{,2}=left[alpha_3+frac{lambda}{(3slash2),G,M}right]a_3^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img994.png"
 ALT="$\displaystyle \left[\alpha_1-\frac{3 \lambda}{(3/2) G M}\right]a_1^{ 2} =\alpha_2 a_2^{ 2} = \left[\alpha_3+\frac{\lambda}{(3/2) G M}\right] a_3^{ 2}.$">|; 

$key = q/{bfe}_3=h_3,nablau_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7035.png"
 ALT="$ {\bf e}_3=h_3 \nabla u_3$">|; 

$key = q/z_H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img704.png"
 ALT="$ z_H$">|; 

$key = q/displaystyle=frac{c,sinheta}{cosheta-cosxi},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2461.png"
 ALT="$\displaystyle =\frac{c \sinh \eta}{\cosh\eta-\cos\xi},$">|; 

$key = q/rho'=M'slash[(4slash3),pi,{a_0'}^{,3}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1020.png"
 ALT="$ \rho'=M'/[(4/3) \pi {a_0'}^{ 3}]$">|; 

$key = q/displaystylefrac{w_2}{w_1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5742.png"
 ALT="$\displaystyle \frac{w_2}{w_1}$">|; 

$key = q/displaystylea_{ik}'={calR}_{il},{calR}_{km},a_{lm},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6866.png"
 ALT="$\displaystyle a_{ik}' = {\cal R}_{il} {\cal R}_{km} a_{lm},$">|; 

$key = q/displaystyleequivint_1^{{rmMa}}frac{sqrt{M^{,2}-1}}{1+[(gamma-1)slash2],M^{,2}},frac{dM}{M};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5772.png"
 ALT="$\displaystyle \equiv \int_1^{{\rm Ma}}\frac{\sqrt{M^{ 2}-1}}{1+[(\gamma-1)/2] M^{ 2}} \frac{dM}{M}$">|; 

$key = q/x+lambda,y={rmconst.};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5989.png"
 ALT="$ x+\lambda y={\rm const.}$">|; 

$key = q/displaystyle(1slash4),gamma,a^{,2},k^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1344.png"
 ALT="$\displaystyle (1/4) \gamma a^{ 2} k^{ 2}.
$">|; 

$key = q/displaystyle=-V,costheta+frac{3}{2},V,cos,theta,frac{a}{r}+{calO}left(frac{a}{r}right)^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3827.png"
 ALT="$\displaystyle = -V \cos\theta + \frac{3}{2} V \cos \theta \frac{a}{r} + {\cal O}\left(\frac{a}{r}\right)^2,$">|; 

$key = q/displaystylefrac{delta}{L}=frac{U_1}{U_0}=frac{1}{{rmRe}^{1slash2}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2891.png"
 ALT="$\displaystyle \frac{\delta}{L} =\frac{U_1}{U_0}= \frac{1}{{\rm Re}^{1/2}},$">|; 

$key = q/displaystylem_{rmadded}=frac{8}{3},rho,a^{,3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2791.png"
 ALT="$\displaystyle m_{\rm added} = \frac{8}{3} \rho a^{ 3},$">|; 

$key = q/45^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2847.png"
 ALT="$ 45^\circ$">|; 

$key = q/f''(0)>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2966.png"
 ALT="$ f''(0)&gt;0$">|; 

$key = q/z';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6566.png"
 ALT="$ z'$">|; 

$key = q/displaystyle={bfA}cdotnablaA_theta+frac{A_r,A_theta-cottheta,A_phi^{,2}}{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7186.png"
 ALT="$\displaystyle = {\bf A}\cdot\nabla A_\theta + \frac{A_r A_\theta-\cot\theta A_\phi^{ 2}}{r},$">|; 

$key = q/displaystyleU_0,frac{partialu}{partialx}simeqnu,frac{partial^{,2}u}{partialy^{,2}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3022.png"
 ALT="$\displaystyle U_0 \frac{\partial u}{\partial x} \simeq \nu \frac{\partial^{ 2} u}{\partial y^{ 2}}.$">|; 

$key = q/displaystyleomega^2=left(g,k+frac{gamma,k^{,3}}{rho}right)tanh(k,d),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4147.png"
 ALT="$\displaystyle \omega^2 =\left(g k+\frac{\gamma k^{ 3}}{\rho}\right) \tanh(k d),$">|; 

$key = q/displaystyle({bfv}cdotnabla),{bfv}equivnabla(v^{,2}slash2)-{bfv}times;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5865.png"
 ALT="$\displaystyle ({\bf v}\cdot\nabla) {\bf v} \equiv\nabla (v^{ 2}/2) -{\bf v}\times$">|; 

$key = q/rho_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5891.png"
 ALT="$ \rho_\infty$">|; 

$key = q/nablacdot({mitOmega}_varphi,nablavarphi)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2555.png"
 ALT="$ \nabla\cdot ({\mit\Omega}_\varphi \nabla\varphi)=0$">|; 

$key = q/displaystylefrac{v_x(x,y)}{U_0}=1-0.375left(frac{L}{x}right)^{1slash2}exp(-y^{,2}slashdelta^{,2}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3063.png"
 ALT="$\displaystyle \frac{v_x(x,y)}{U_0} = 1 -0.375\left(\frac{L}{x}\right)^{1/2}\exp(-y^{ 2}/\delta^{ 2}),$">|; 

$key = q/displaystyleh_2,h_1,(h_2+h_1)=frac{2,Q^{,2}}{g}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1480.png"
 ALT="$\displaystyle h_2 h_1 (h_2+h_1)= \frac{2 Q^{ 2}}{g}.$">|; 

$key = q/({bfe}_i)_j={bfe}_icdot{bfe}_j=delta_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7094.png"
 ALT="$ ({\bf e}_i)_j = {\bf e}_i\cdot {\bf e}_j=\delta_{ij}$">|; 

$key = q/displaystyle=-frac{1}{2},V,c^{,2},cosh^2xi_0,sin^2eta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2758.png"
 ALT="$\displaystyle = -\frac{1}{2} V c^{ 2} \cosh^2\xi_0 \sin^2\eta,$">|; 

$key = q/displaystyle{mitGamma}_C=oint_C{bfv}cdotd{bfr},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1549.png"
 ALT="$\displaystyle {\mit\Gamma}_C = \oint_C {\bf v}\cdot d{\bf r},$">|; 

$key = q/displaystylerho,frac{Dv_i}{Dt}=F_i-frac{partialp}{partialx_i}+frac{partial}{partalx_i}!left(frac{2}{3},mu,frac{partialv_j}{partialx_j}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="507" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="$\displaystyle \rho  \frac{D v_i}{Dt}= F_i - \frac{\partial p}{\partial x_i} + ...
...partial x_i}\!\left(\frac{2}{3} \mu \frac{\partial v_j}{\partial x_j}\right).$">|; 

$key = q/c-b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1901.png"
 ALT="$ c-b$">|; 

$key = q/displaystylerightarrowV,sintheta.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3758.png"
 ALT="$\displaystyle \rightarrow V \sin\theta.$">|; 

$key = q/displaystyle=frac{y}{delta},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2878.png"
 ALT="$\displaystyle = \frac{y}{\delta},$">|; 

$key = q/displaystylephi(r,theta)=sum_{l=0,infty}left[alpha_l,r^{,l}+beta_l,r^{-(l+1)}right]P_l(costheta),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2594.png"
 ALT="$\displaystyle \phi(r,\theta) = \sum_{l=0,\infty} \left[\alpha_l r^{ l} + \beta_l r^{-(l+1)}\right]P_l(\cos\theta),$">|; 

$key = q/displaystyle{bfs}=rho,{calS},{bfv}+frac{{bfq}}{T},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img524.png"
 ALT="$\displaystyle {\bf s} = \rho {\cal S} {\bf v} + \frac{{\bf q}}{T},
$">|; 

$key = q/f'(z)=dfslashdz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2073.png"
 ALT="$ f'(z)= df/dz$">|; 

$key = q/j=5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$ j=5$">|; 

$key = q/gamma_{13}>gamma_{23}+gamma_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1132.png"
 ALT="$ \gamma_{13}&gt; \gamma_{23}+\gamma_{12}$">|; 

$key = q/displaystylesigma_{ij}=-p,delta_{ij}+muleft(frac{partialxi_i}{partialx_j}+frac{partialxi_j}{partialx_i}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4372.png"
 ALT="$\displaystyle \sigma_{ij} = -p \delta_{ij} + \mu\left(\frac{\partial \xi_i}{\partial x_j}+\frac{\partial \xi_j}{\partial x_i}\right),$">|; 

$key = q/intnablacdot{bfA},,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6586.png"
 ALT="$ \int \nabla\cdot{\bf A}  dV$">|; 

$key = q/displaystyle=frac{partial^{,2}T}{partialx^{,2}}+frac{partial^{,2}T}{partialy^{,2}}+frac{partial^{,2}T}{partialz^{,2}}equivnabla^{2}T.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6614.png"
 ALT="$\displaystyle =\frac{\partial^{ 2} T}{\partial x^{ 2}}+\frac{\partial^{ 2} T...
...rtial y^{ 2}} + \frac{\partial^{ 2} T}{\partial z^{ 2}} \equiv \nabla^{2} T.$">|; 

$key = q/deltaslashc=0.1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$ \delta /c=0.1$">|; 

$key = q/displaystyle|V_s|=left(frac{gamma_1+1}{gamma_4-1}right)c_4,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5654.png"
 ALT="$\displaystyle \vert V_s\vert = \left(\frac{\gamma_1+1}{\gamma_4-1}\right)c_4,
$">|; 

$key = q/ABC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2189.png"
 ALT="$ ABC$">|; 

$key = q/{rmRe}=U_0,xslashnu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3212.png"
 ALT="$ {\rm Re}=U_0 x/\nu$">|; 

$key = q/displaystyle=-left(frac{m-m'}{m+m'}right),g,t^{,2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1857.png"
 ALT="$\displaystyle = -\left(\frac{m-m'}{m+m'}\right) g t^{ 2}.$">|; 

$key = q/t=i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6912.png"
 ALT="$ t=i$">|; 

$key = q/displaystyle=-frac{nablap}{rho}-nabla{mitPsi}+frac{mu}{rho}left[nabla^{,2}{bfv}+frac{1}{3},nabla(nablacdot{bfv})right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="$\displaystyle = -\frac{\nabla p}{\rho} - \nabla{\mit\Psi}+\frac{\mu}{\rho}\left[\nabla^{ 2}{\bf v} + \frac{1}{3} \nabla(\nabla\cdot{\bf v})\right],$">|; 

$key = q/Oxyz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6252.png"
 ALT="$ Oxyz$">|; 

$key = q/displaystylez^{,2}-omega^{-1},z+1=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img799.png"
 ALT="$\displaystyle z^{ 2}-\omega^{-1} z+1=0.$">|; 

$key = q/y=y_L(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5817.png"
 ALT="$ y=y_L(x)$">|; 

$key = q/displaystyle-{mitDelta}P=12,mu,Q,l,langled^{,-3}rangle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3672.png"
 ALT="$\displaystyle -{\mit\Delta} P = 12 \mu Q l \langle d^{ -3}\rangle,$">|; 

$key = q/displaystylepsi(x,y)=-frac{Q}{2pi},tan^{-1}left[frac{4,x,y,(x^{,2}-y^{,2})}{x^{,4}-6,x^{,2},y^{,2}+y^{,4}+4,a^{,4}}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="410" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2119.png"
 ALT="$\displaystyle \psi(x,y) = -\frac{Q}{2\pi} \tan^{-1}\left[\frac{4 x y (x^{ 2}-y^{ 2})}{x^{ 4}-6 x^{ 2} y^{ 2}+y^{ 4}+4 a^{ 4}}\right],$">|; 

$key = q/p-p_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img722.png"
 ALT="$ p-p_0$">|; 

$key = q/displaystylephantom{=}+left(frac{gamma+1}{gamma-1}right)^{1slash2}(z-z_1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6136.png"
 ALT="$\displaystyle \phantom{=}+\left(\frac{\gamma+1}{\gamma-1}\right)^{1/2}(z-z_1),$">|; 

$key = q/displaystylesimeq-frac{G,M}{r}-frac{G,M}{5,r^{,5}},a_1^{,2}left[x_1^{,2}-frac{1}{2},(x_2^{,2}+x_3^{,2})right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7291.png"
 ALT="$\displaystyle \simeq -\frac{G M}{r} - \frac{G M}{5 r^{ 5}} a_1^{ 2}\left[x_1^{ 2}-\frac{1}{2} (x_2^{ 2}+x_3^{ 2})\right]$">|; 

$key = q/{bfb}slashbeta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6738.png"
 ALT="$ {\bf b}/\beta$">|; 

$key = q/displaystyle=sum_{j,k=1,3}frac{partial(h_k,A_k)}{partialu_j},nablau_jtimesnablau_k,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7075.png"
 ALT="$\displaystyle =\sum_{j,k=1,3}\frac{\partial (h_k A_k)}{\partial u_j} \nabla u_j\times\nabla u_k,$">|; 

$key = q/displaystyleK=frac{1}{4},rho,a^{,2},omega^{,2}int_{-d}^0frac{cosh[2,k,(z+d)]}{si},dz=frac{1}{4},rho,g,a^{,2},frac{omega^{,2}}{g,k,tanh(k,d)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="515" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3976.png"
 ALT="$\displaystyle K = \frac{1}{4} \rho a^{ 2} \omega^{ 2}\int_{-d}^0 \frac{\co...
...,d)} dz=\frac{1}{4} \rho g a^{ 2} \frac{\omega^{ 2}}{g k \tanh(k d)}.$">|; 

$key = q/displaystyle=frac{sinalpha,sinomega}{cosalpha-cosomega}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2311.png"
 ALT="$\displaystyle =\frac{\sin\alpha \sin\omega}{\cos\alpha-\cos\omega}.$">|; 

$key = q/displaystyle{calF}={calE}-T,{calS}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5585.png"
 ALT="$\displaystyle {\cal F} = {\cal E} - T {\cal S}.
$">|; 

$key = q/V_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="$ V_0$">|; 

$key = q/displaystyleP_0(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4239.png"
 ALT="$\displaystyle P_0(x)$">|; 

$key = q/displaystyle{mitPhi}+chi=left(g-frac{2}{3},{mitOmega}^{,2},aright)(z-d)+delta{mitPhi}(theta,phi,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4479.png"
 ALT="$\displaystyle {\mit\Phi} + \chi = \left(g-\frac{2}{3} {\mit\Omega}^{ 2} a\right)(z-d) + \delta{\mit\Phi}(\theta,\phi,t),$">|; 

$key = q/c=x_R-x_L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5851.png"
 ALT="$ c=x_R-x_L$">|; 

$key = q/{bfa}slashalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6737.png"
 ALT="$ {\bf a}/\alpha$">|; 

$key = q/z_2>z_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5167.png"
 ALT="$ z_2&gt;z_1$">|; 

$key = q/displaystyle=gamma,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5611.png"
 ALT="$\displaystyle = \gamma,$">|; 

$key = q/displaystylefrac{|dzeta''|}{|dzeta'|}=frac{|dz''|}{|dz'|},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2075.png"
 ALT="$\displaystyle \frac{\vert d\zeta''\vert}{\vert d\zeta'\vert} = \frac{\vert dz''\vert}{\vert dz'\vert},$">|; 

$key = q/displaystyleleft[m_j-n,(n+1),f_jright]{mitPsi}_n+left(frac{n+1}{n},frac{n-m_j}{2_{n-1}+left(frac{n}{n+1},frac{n+m_j+1}{2,n+3}right)z_{n+1}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="649" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4693.png"
 ALT="$\displaystyle \left[m_j-n (n+1) f_j\right]{\mit\Psi}_n + \left(\frac{n+1}{n}\...
...n-1}\right)z_{n-1}+\left(\frac{n}{n+1} \frac{n+m_j+1}{2 n+3}\right)z_{n+1}=0,$">|; 

$key = q/displaystylefrac{p_4}{p_1}=frac{p_2}{p_1}left[1-frac{(gamma_4-1),(c_1slashc_4),(ma_1+1),(p_2slashp_1-1)}}right]^{-2,gamma_4slash(gamma_4-1)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="502" HEIGHT="84" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5642.png"
 ALT="$\displaystyle \frac{p_4}{p_1}=\frac{p_2}{p_1}\left[1-\frac{(\gamma_4-1) (c_1/c...
...t{2 \gamma_1
+(\gamma_1+1) (p_2/p_1-1)}}\right]^{-2 \gamma_4/(\gamma_4-1)}.
$">|; 

$key = q/sigma_{ii}=sigma_{11}+sigma_{22}+sigma_{33};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="$ \sigma_{ii}=\sigma_{11}
+\sigma_{22}+\sigma_{33}$">|; 

$key = q/displaystyle{bfa}cdot{bfb}times{bfc}=-{bfa}cdot{bfc}times{bfb}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6410.png"
 ALT="$\displaystyle {\bf a} \cdot {\bf b}\times{\bf c} = - {\bf a} \cdot {\bf c}\times{\bf b}.$">|; 

$key = q/y=b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2314.png"
 ALT="$ y=b$">|; 

$key = q/displaystyle{mitGamma}_infty=pi,V,(c+delta),sinalpha.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3346.png"
 ALT="$\displaystyle {\mit\Gamma}_\infty =\pi V (c+\delta) \sin\alpha.$">|; 

$key = q/displaystyle=frac{delta_2}{U}left.frac{partialu}{partialy}right|_{y=0}=left(2+fr5}-frac{{mitLambda}}{945}-frac{{mitLambda}^{,2}}{9072}right),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3194.png"
 ALT="$\displaystyle =\frac{\delta_2}{U}\left.\frac{\partial u}{\partial y}\right\vert...
...rac{37}{315}-\frac{{\mit\Lambda}}{945}-\frac{{\mit\Lambda}^{ 2}}{9072}\right),$">|; 

$key = q/(-0.5,X_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4223.png"
 ALT="$ (-0.5 X_1$">|; 

1;

