Curvilinear Coordinates

Figure A.27: Cylindrical coordinates.

In the cylindrical coordinate system, the 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.27. A general vector ${\bf A}$ is thus written

$${\bf A} = A_r\,{\bf e}_r+ A_\theta\,{\bf e}_\theta + A_z\,{\bf e}_z,$$ (1.166)

where ${\bf e}_r=\nabla r/\vert\nabla r\vert$ and ${\bf e}_\theta=\nabla\theta/\vert\nabla\theta\vert$. See Figure A.27. Note that the unit vectors ${\bf e}_r$, ${\bf e}_\theta$, and ${\bf e}_z$ are mutually orthogonal. Hence, $A_r = {\bf A}\cdot {\bf e}_r$, et cetera. The volume element in this coordinate system is $d^{\,3}{\bf r} = r\,dr\,d\theta\,dz$. Moreover, gradient, divergence, and curl take the forms

$$\nabla V = \frac{\partial V}{\partial r}\,{\bf e}_r + \frac{1}{r}\frac{\pa... ... V}{\partial\theta}\,{\bf e}_\theta + \frac{\partial V}{\partial z}\,{\bf e}_z,$$ (1.167)

$$\nabla\cdot {\bf A} =\frac{1}{r}\,\frac{\partial}{\partial r}\,(r\,A_r) + \frac{1}{r}\,\frac{\partial A_\theta}{\partial\theta} + \frac{\partial A_z}{\partial z},$$ (1.168)

$$\nabla\times{\bf A} = \left(\frac{1}{r}\,\frac{\partial A_z}{\partial \theta}-\frac{\... ...{\partial A_r}{\partial z}-\frac{\partial A_z}{\partial r}\right){\bf e}_\theta$$

$$\phantom{=}+ \left(\frac{1}{r}\,\frac{\partial}{\partial r}\,(r\,A_\theta) - \frac{1}{r}\,\frac{\partial A_r}{\partial\theta}\right){\bf e}_z,$$ (1.169)

respectively. Here, $V({\bf r})$ is a general vector field, and ${\bf A}({\bf r})$ a general scalar field. Finally, the Laplacian is written

$$\nabla^2 V = \frac{1}{r}\,\frac{\partial}{\partial r}\left(r\,\fr... ...^2}\,\frac{\partial^2 V}{\partial\theta^2} + \frac{\partial^2 V}{\partial z^2}.$$ (1.170)

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.28. 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

$${\bf A} = A_r\,{\bf e}_r + A_\theta\,{\bf e}_\theta+ A_\phi\,{\bf e}_\phi,$$ (1.171)

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, $A_r = {\bf A}\cdot {\bf e}_r$, et cetera. The volume element in this coordinate system is $d^{\,3}{\bf r} = r^{\,2}\,\sin\theta\,dr\,d\theta\,d\phi$. Moreover, gradient, divergence, and curl take the forms

$$\nabla V = \frac{\partial V}{\partial r}\,{\bf e}_r + \frac{1}{r}\frac{\pa... ...heta + \frac{1}{r\,\sin\theta}\,\frac{\partial V}{\partial \phi}\,{\bf e}_\phi,$$ (1.172)

$$\nabla\cdot {\bf A} =\frac{1}{r^2}\,\frac{\partial}{\partial r}\,(r^2\,A_r) + \frac{1}{r\,\sin\theta}\,\frac{\partial }{\partial\theta} \,(\sin\theta\,A_\theta)$$

$$\phantom{=}+ \frac{1}{r\,\sin\theta}\,\frac{\partial A_\phi}{\partial \phi},$$ (1.173)

$$\nabla\times{\bf A} = \left(\frac{1}{r\,\sin\theta}\,\frac{\partial}{\partial \theta}... ...frac{1}{r\,\sin\theta}\,\frac{\partial A_\theta}{\partial \phi}\right){\bf e}_r$$

$$\phantom{=}+\left(\frac{1}{r\,\sin\theta}\,\frac{\partial A_r}{\p... ... \phi}-\frac{1}{r}\frac{\partial}{\partial r}\,(r\,A_\phi)\right){\bf e}_\theta$$

$$\phantom{=}+ \left(\frac{1}{r}\,\frac{\partial}{\partial r}\,(r\,A_\theta) - \frac{1}{r}\,\frac{\partial A_r}{\partial\theta}\right){\bf e}_\phi,$$ (1.174)

respectively. Here, $V({\bf r})$ is a general vector field, and ${\bf A}({\bf r})$ a general scalar field. Finally, the Laplacian is written

$$\nabla^2 V = \frac{1}{r^2}\,\frac{\partial}{\partial r}\left(r^2\... ...ta}\right) + \frac{1}{r^2\,\sin^2\theta}\,\frac{\partial^2 V}{\partial \phi^2}.$$ (1.175)

Figure A.28: Spherical coordinates.