Spherical polar coordinates

The spherical polar coordinates $r$, $\theta$, $\varphi$ are related to the rectangular coordinates $x$, $y$, $z$ by the formulas (see Fig. A.5):

Figure A.5: Spherical polar coordinates.

$$\left\{ \begin{array}{l} x = r \sin \theta \cos \varphi\space , \... ...\space , \quad 0 \leq \theta \leq \pi \space , \quad o \leq \varphi\leq 2 \pi )$$ (2.29)

The inverse relations are:

$$r=\sqrt{x^2 + y^2 +z^2} \space , \quad \theta= \cos^{-1}\left( \frac{z}{r} \right) \, \tan \varphi=\frac{y}{x} \space .$$ (2.30)

An infinitesimal length $dl$ is:

$$dl=\sqrt{(dr)^2 + (rd\theta)^2+(\sin \theta d\varphi)^2} \space .$$ (2.31)

An infinitesimal area $dS$ is easily calculated if one of the coordinates is constant on $DS$, by taking the product of two appropriate length elements:

$$dS = r^2 \sin \theta \, d\theta \, d\varphi\space , \quad \mbox{if $r$\space is constant} \space ,$$ (2.32)

$$= r \sin \theta \, dr \, d\varphi\space , \quad \mbox{if $\theta$\space is constant} \space ,$$ (2.33)

$$= r dr \, d\theta \space , \quad \mbox{if $\varphi$\space is constant} \space .$$ (2.34)

An infinitesimal volume element is:

$$dv=r^2 \sin \theta \, dr \, d\theta \, d\varphi\space .$$ (2.35)

The gradient, divergence and laplacian become, in spherical coordinates:

$$\nabla f = \hat{r} \frac{\partial f}{\partial r} + \hat{\theta} \frac{1}{r... ...t{\varphi} \frac{1}{r \sin \theta} \frac{\partial f}{\partial \varphi} \space ,$$ (2.36)

$$\nabla \cdot \underline{F} = \frac{1}{r^2} \frac{\partial}{\partial r}\left( r^2 F_r \right)... ... \frac{1}{r \sin \theta} \frac{\partial F_{\varphi}}{\partial \varphi} \space ,$$ (2.37)

$$\nabla \times \underline{F} = \frac{1}{r \sin \theta} \left[ \frac{\partial}{\partial \theta}... ...} \frac{\partial}{\partial r} \left( r F_{\varphi} \right) \right] \hat{\theta}$$

$$+ \frac{1}{r} \left[ \frac{\partial}{\partial r } \left( r F_{\theta} \right) - \frac{F_r}{\partial \theta} \right] \hat{\varphi} \space ,$$ (2.38)

$$\nabla^2 f = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\par... ...+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial ^2 f}{\partial \varphi^2} \space ,$$ (2.39)

where

$$\underline{F}= \hat{r} F_r + \hat{\theta} F_{\theta} + \hat{\varphi} F_{\varphi}$$ (2.40)