Curvilinear Coordinates

In the cylindrical coordinate system, the Cartesian coordinates $x$ and $y$ are replaced by $r=\sqrt{x^2+y^2}$ 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.113. A general vector ${\bf A}$ is thus written

$${\bf A} = A_r\,{\bf e}_r+ A_\theta\,{\bf e}_\theta + A_z\,{\bf e}_z,$$ (1372)

where ${\bf e}_r=\nabla r/\vert\nabla r\vert$ and ${\bf e}_\theta = \nabla\theta/\vert\nabla\theta\vert$--see Figure A.113. Note that the unit vectors ${\bf e}_r$, ${\bf e}_\theta$, and ${\bf e}_z$ are mutually orthogonal. Hence, $A_r = {\bf A}\cdot {\bf e}_r$, etc. 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

$$\nabla V = \frac{\partial V}{\partial r}\,{\bf e}_r + \frac{... ...a}\,{\bf e}_\theta + \frac{\partial V}{\partial z}\,{\bf e}_z.$$ (1373)

In the spherical coordinate system, the Cartesian coordinates $x$, $y$, and $z$ are replaced by $r=\sqrt{x^2+y^2+z^2}$, $\theta = \cos^{-1}(z/r)$, 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.114. 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

$${\bf A} = A_r\,{\bf e}_r + A_\theta\,{\bf e}_\theta+ A_\phi\,{\bf e}_\phi,$$ (1374)

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, $A_r = {\bf A}\cdot {\bf e}_r$, etc. 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

$$\nabla V = \frac{\partial V}{\partial r}\,{\bf e}_r + \frac{... ...\,\sin\theta}\,\frac{\partial V}{\partial \phi}\,{\bf e}_\phi.$$ (1375)

Figure A.114: Spherical polar coordinates.