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Spherical Coordinates
In the spherical coordinate system,
,
, and
,
where
,
,
, and
,
,
are standard Cartesian coordinates.
Thus,
is the length of the radius vector,
the angle subtended between the radius vector and the
-axis, and
the angle subtended between the projection of the radius vector
onto the
-
plane and the
-axis. (See Figure C.2.)
Figure C.2:
Spherical coordinates.
![\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixC/figC.02.eps}}
\end{figure}](img7165.png) |
A general vector
is written
![$\displaystyle {\bf A} = A_r\,{\bf e}_r+ A_\theta\,{\bf e}_\theta + A_\phi\,{\bf e}_\phi,$](img7166.png) |
(C.55) |
where
,
, and
. Of course, the unit vectors
,
, and
are mutually orthogonal, so
, et cetera.
As is easily demonstrated, an element of length (squared) in the spherical coordinate system takes the form
![$\displaystyle d {\bf x}\cdot d{\bf x} = dr^{\,2} + r^{\,2}\,d\theta^{\,2} + r^{\,2}\,\sin^2\theta\,d\phi^2.$](img7169.png) |
(C.56) |
Hence, comparison with Equation (C.6) reveals that the scale factors for this system are
Thus, surface elements normal to
,
, and
are
written
respectively, whereas
a
volume element takes the form
![$\displaystyle dV = r^{\,2}\,\sin\theta\,dr\,d\theta\,d\phi.$](img7174.png) |
(C.63) |
According to Equations (C.13), (C.15), and (C.18), gradient, divergence, and curl in the spherical
coordinate system are written
respectively. Here,
is a general scalar field, and
a general vector field.
According to Equation (C.19), when expressed in spherical coordinates, the Laplacian of a scalar field becomes
![$\displaystyle \nabla^{\,2} \psi= \frac{1}{r^{\,2}}\,\frac{\partial}{\partial r}...
...rac{1}{r^{\,2}\,\sin^2\theta}\,\frac{\partial^{\,2} \psi}{\partial \phi^{\,2}}.$](img7180.png) |
(C.67) |
Moreover, from Equation (C.23), the components of
in the spherical coordinate system are
Now, according to Equation (C.26), the components of
in the spherical
coordinate system are
![$\displaystyle (\widetilde{\nabla {\bf A}})_{rr}$](img7144.png) |
![$\displaystyle = \frac{\partial A_r}{\partial r},$](img7145.png) |
(C.71) |
![$\displaystyle (\widetilde{\nabla {\bf A}})_{\theta\theta}$](img7146.png) |
![$\displaystyle =\frac{1}{r} \frac{\partial A_\theta}{\partial \theta}+ \frac{A_r}{r},$](img7147.png) |
(C.72) |
![$\displaystyle (\widetilde{\nabla {\bf A}})_{\phi\phi}$](img7186.png) |
![$\displaystyle =\frac{1}{r\,\sin\theta}\,\frac{\partial A_\phi}{\partial \phi}+ \frac{A_r}{r} + \frac{\cot\theta\,A_\theta}{r},$](img7187.png) |
(C.73) |
![$\displaystyle (\widetilde{\nabla {\bf A}})_{r\theta}=(\widetilde{\nabla {\bf A}})_{\theta r}$](img7150.png) |
![$\displaystyle = \frac{1}{2}\left(\frac{1}{r}\,\frac{\partial A_r}{\partial\theta} + \frac{\partial A_\theta}{\partial r} - \frac{A_\theta}{r}\right),$](img7151.png) |
(C.74) |
![$\displaystyle (\widetilde{\nabla {\bf A}})_{r\phi}=(\widetilde{\nabla {\bf A}})_{\phi r}$](img7188.png) |
![$\displaystyle =\frac{1}{2}\left(\frac{1}{r\,\sin\theta}\,\frac{\partial A_r}{\partial \phi} + \frac{\partial A_\phi}{\partial r}-\frac{A_\phi}{r}\right),$](img7189.png) |
(C.75) |
![$\displaystyle (\widetilde{\nabla {\bf A}})_{\theta \phi}=(\widetilde{\nabla {\bf A}})_{\phi\theta}$](img7190.png) |
![$\displaystyle =\frac{1}{2}\left(\frac{1}{r\,\sin\theta}\,\frac{\partial A_\thet...
...{r}\frac{\partial A_\phi}{\partial \theta}-\frac{\cot\theta\,A_\phi}{r}\right).$](img7191.png) |
(C.76) |
Finally, from Equation (C.28), the components of
in the
spherical coordinate system are
Next: Exercises
Up: Non-Cartesian Coordinates
Previous: Cylindrical Coordinates
Richard Fitzpatrick
2016-01-22