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Spherical Coordinates

In the spherical coordinate system, $ u_1=r$ , $ u_2=\theta$ , and $ u_3=\phi$ , where $ r=\sqrt{x^{\,2}+y^{\,2}+z^{\,2}}$ , $ \theta=\cos^{-1}(z/r)$ , $ \phi=\tan^{-1}(y/x)$ , and $ x$ , $ y$ , $ z$ are standard Cartesian coordinates. Thus, $ r$ is the length of the radius vector, $ \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 C.2.)

Figure C.2: Spherical coordinates.
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A general vector $ {\bf A}$ is written

$\displaystyle {\bf A} = A_r\,{\bf e}_r+ A_\theta\,{\bf e}_\theta + A_\phi\,{\bf e}_\phi,$ (C.55)

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$ . Of course, the unit vectors $ {\bf e}_r$ , $ {\bf e}_\theta$ , and $ {\bf e}_\phi$ are mutually orthogonal, so $ A_r = {\bf A}\cdot {\bf e}_r$ , et cetera.

As is easily demonstrated, an element of length (squared) in the spherical coordinate system takes the form

$\displaystyle d {\bf x}\cdot d{\bf x} = dr^{\,2} + r^{\,2}\,d\theta^{\,2} + r^{\,2}\,\sin^2\theta\,d\phi^2.$ (C.56)

Hence, comparison with Equation (C.6) reveals that the scale factors for this system are

$\displaystyle h_r$ $\displaystyle = 1,$ (C.57)
$\displaystyle h_\theta$ $\displaystyle = r,$ (C.58)
$\displaystyle h_\phi$ $\displaystyle =r\,\sin\theta.$ (C.59)

Thus, surface elements normal to $ {\bf e}_r$ , $ {\bf e}_\theta$ , and $ {\bf e}_\phi$ are written

$\displaystyle dS_r$ $\displaystyle = r^{\,2}\,\sin\theta\,d\theta\,d\phi,$ (C.60)
$\displaystyle dS_\theta$ $\displaystyle = r\,\sin\theta\,dr\,d\phi,$ (C.61)
$\displaystyle dS_\phi$ $\displaystyle = r\,dr\,d\theta,$ (C.62)

respectively, whereas a volume element takes the form

$\displaystyle dV = r^{\,2}\,\sin\theta\,dr\,d\theta\,d\phi.$ (C.63)

According to Equations (C.13), (C.15), and (C.18), gradient, divergence, and curl in the spherical coordinate system are written

$\displaystyle \nabla \psi$ $\displaystyle = \frac{\partial \psi}{\partial r}\,{\bf e}_r + \frac{1}{r}\frac{...
...a + \frac{1}{r\,\sin\theta}\,\frac{\partial \psi}{\partial \phi}\,{\bf e}_\phi,$ (C.64)
$\displaystyle \nabla\cdot{\bf A}$ $\displaystyle =\frac{1}{r^{\,2}}\,\frac{\partial}{\partial r}\,(r^{\,2}\,A_r) +...
...eta\,A_\theta)+ \frac{1}{r\,\sin\theta}\,\frac{\partial A_\phi}{\partial \phi},$ (C.65)
$\displaystyle \nabla\times {\bf A}$ $\displaystyle = \left[\frac{1}{r\,\sin\theta}\,\frac{\partial}{\partial \theta}...
...frac{1}{r\,\sin\theta}\,\frac{\partial A_\theta}{\partial \phi}\right]{\bf e}_r$    
  $\displaystyle \phantom{=}+\left[\frac{1}{r\,\sin\theta}\,\frac{\partial A_r}{\p...
... \phi}-\frac{1}{r}\frac{\partial}{\partial r}\,(r\,A_\phi)\right]{\bf e}_\theta$    
  $\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,\ $ (C.66)

respectively. Here, $ \psi({\bf r})$ is a general scalar field, and $ {\bf A}({\bf r})$ a general vector field.

According to Equation (C.19), when expressed in spherical coordinates, the Laplacian of a scalar field becomes

$\displaystyle \nabla^{\,2} \psi= \frac{1}{r^{\,2}}\,\frac{\partial}{\partial r}...
...rac{1}{r^{\,2}\,\sin^2\theta}\,\frac{\partial^{\,2} \psi}{\partial \phi^{\,2}}.$ (C.67)

Moreover, from Equation (C.23), the components of $ ({\bf A}\cdot\nabla){\bf A}$ in the spherical coordinate system are

$\displaystyle [({\bf A}\cdot\nabla)\,{\bf A}]_r$ $\displaystyle ={\bf A}\cdot\nabla A_r - \frac{A_\theta^{\,2}+A_\phi^{\,2}}{r},$ (C.68)
$\displaystyle [({\bf A}\cdot\nabla)\,{\bf A}]_\theta$ $\displaystyle = {\bf A}\cdot\nabla A_\theta + \frac{A_r\,A_\theta-\cot\theta\,A_\phi^{\,2}}{r},$ (C.69)
$\displaystyle [({\bf A}\cdot\nabla)\,{\bf A}]_\phi$ $\displaystyle = {\bf A}\cdot\nabla A_\phi+ \frac{A_r\,A_\phi+\cot\theta\,A_\theta\,A_\phi}{r}.$ (C.70)

Now, according to Equation (C.26), the components of $ \widetilde{\nabla {\bf A}}$ in the spherical coordinate system are

$\displaystyle (\widetilde{\nabla {\bf A}})_{rr}$ $\displaystyle = \frac{\partial A_r}{\partial r},$ (C.71)
$\displaystyle (\widetilde{\nabla {\bf A}})_{\theta\theta}$ $\displaystyle =\frac{1}{r} \frac{\partial A_\theta}{\partial \theta}+ \frac{A_r}{r},$ (C.72)
$\displaystyle (\widetilde{\nabla {\bf A}})_{\phi\phi}$ $\displaystyle =\frac{1}{r\,\sin\theta}\,\frac{\partial A_\phi}{\partial \phi}+ \frac{A_r}{r} + \frac{\cot\theta\,A_\theta}{r},$ (C.73)
$\displaystyle (\widetilde{\nabla {\bf A}})_{r\theta}=(\widetilde{\nabla {\bf A}})_{\theta r}$ $\displaystyle = \frac{1}{2}\left(\frac{1}{r}\,\frac{\partial A_r}{\partial\theta} + \frac{\partial A_\theta}{\partial r} - \frac{A_\theta}{r}\right),$ (C.74)
$\displaystyle (\widetilde{\nabla {\bf A}})_{r\phi}=(\widetilde{\nabla {\bf A}})_{\phi r}$ $\displaystyle =\frac{1}{2}\left(\frac{1}{r\,\sin\theta}\,\frac{\partial A_r}{\partial \phi} + \frac{\partial A_\phi}{\partial r}-\frac{A_\phi}{r}\right),$ (C.75)
$\displaystyle (\widetilde{\nabla {\bf A}})_{\theta \phi}=(\widetilde{\nabla {\bf A}})_{\phi\theta}$ $\displaystyle =\frac{1}{2}\left(\frac{1}{r\,\sin\theta}\,\frac{\partial A_\thet...
...{r}\frac{\partial A_\phi}{\partial \theta}-\frac{\cot\theta\,A_\phi}{r}\right).$ (C.76)

Finally, from Equation (C.28), the components of $ \nabla^{\,2}{\bf A}$ in the spherical coordinate system are

$\displaystyle (\nabla^{\,2}{\bf A})_r$ $\displaystyle =\nabla^{\,2} A_r -\frac{2 A_r}{r^{\,2}}-\frac{2}{r^{\,2}}\,\frac...
...{r^{\,2}} -\frac{2}{r^{\,2}\,\sin\theta}\,\frac{\partial A_\phi}{\partial\phi},$ (C.77)
$\displaystyle (\nabla^{\,2}{\bf A})_\theta$ $\displaystyle = \nabla^{\,2} A_\theta +\frac{2}{r^{\,2}}\,\frac{\partial A_r}{\...
...^2\theta} -\frac{2}{r^{\,2}\,\sin\theta}\,\frac{\partial A_\phi}{\partial\phi},$ (C.78)
$\displaystyle (\nabla^{\,2}{\bf A})_\phi$ $\displaystyle =\nabla^{\,2} A_\phi-\frac{A_\phi}{r^{\,2}\,\sin^2\theta} + \frac...
...rac{2 \cot\theta}{r^{\,2}\,\sin\theta}\,\frac{\partial A_\theta}{\partial\phi}.$ (C.79)


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Next: Exercises Up: Non-Cartesian Coordinates Previous: Cylindrical Coordinates
Richard Fitzpatrick 2016-01-22