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Curvilinear coordinates

In the cylindrical coordinate system, the standard Cartesian coordinates $ x$ and $ y$ are replaced by $ r=(x^{\,2}+y^{\,2})^{1/2}$ and $ \theta=\tan^{-1}(y/x)$ . Here, $ r$ is the perpendicular distance from the $ z$ -axis, and $ \theta $ the angle subtended between the perpendicular radius vector and the $ x$ -axis. (See Figure A.2.) A general vector $ {\bf A}$ is thus written

$\displaystyle {\bf A} = A_r\,{\bf e}_r+ A_\theta\,{\bf e}_\theta + A_z\,{\bf e}_z,$ (A.97)

where $ {\bf e}_r=\nabla r/\vert\nabla r\vert$ and $ {\bf e}_\theta = \nabla\theta/\vert\nabla\theta\vert$ . (See Figure A.2.) The unit vectors $ {\bf e}_r$ , $ {\bf e}_\theta$ , and $ {\bf e}_z$ are mutually orthogonal. Hence, $ A_r = {\bf A}\cdot {\bf e}_r$ , and so on. The volume element in this coordinate system is $ d^{\,3}{\bf r} = r\,dr\,d\theta\,dz$ . Moreover, the gradient of a general scalar field $ V({\bf r})$ takes the form

$\displaystyle \nabla V = \frac{\partial V}{\partial r}\,{\bf e}_r + \frac{1}{r}...
... V}{\partial\theta}\,{\bf e}_\theta + \frac{\partial V}{\partial z}\,{\bf e}_z.$ (A.98)

Figure A.2: Cylindrical coordinates.
\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixA/figA.02.eps}}
\end{figure}

In the spherical coordinate system, the Cartesian coordinates $ x$ , $ y$ , and $ z$ are replaced by $ r=(x^{\,2}+y^{\,2}+z^{\,2})^{1/2}$ , $ \theta = \cos^{-1}(z/r)$ , and $ \phi=\tan^{-1}(y/x)$ . Here, $ r$ is the radial distance from the origin, $ \theta $ the angle subtended between the radius vector and the $ z$ -axis, and $ \phi $ the angle subtended between the projection of the radius vector onto the $ x$ -$ y$ plane and the $ x$ -axis. (See Figure A.3.) Note that $ r$ and $ \theta $ in the spherical system are not the same as their counterparts in the cylindrical system. A general vector $ {\bf A}$ is written

$\displaystyle {\bf A} = A_r\,{\bf e}_r + A_\theta\,{\bf e}_\theta+ A_\phi\,{\bf e}_\phi,$ (A.99)

where $ {\bf e}_r=\nabla r/\vert\nabla r\vert$ , $ {\bf e}_\theta = \nabla\theta/\vert\nabla\theta\vert$ , and $ {\bf e}_\phi = \nabla\phi/\vert\nabla\phi\vert$ . The unit vectors $ {\bf e}_r$ , $ {\bf e}_\theta$ , and $ {\bf e}_\phi$ are mutually orthogonal. Hence, $ A_r = {\bf A}\cdot {\bf e}_r$ , and so on. The volume element in this coordinate system is $ d^{\,3}{\bf r} = r^{\,2}\,\sin\theta\,dr\,d\theta\,d\phi$ . Moreover, the gradient of a general scalar field $ V({\bf r})$ takes the form

$\displaystyle \nabla V = \frac{\partial V}{\partial r}\,{\bf e}_r + \frac{1}{r}...
...heta + \frac{1}{r\,\sin\theta}\,\frac{\partial V}{\partial \phi}\,{\bf e}_\phi.$ (A.100)

Figure A.3: Spherical coordinates.
\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixA/figA.03.eps}}
\end{figure}


next up previous
Next: Conic sections Up: Useful mathematics Previous: Precession
Richard Fitzpatrick 2016-03-31