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 126.

Figure 126: Spherical coordinates.

A general vector ${\bf A}$ is written

$${\bf A} = A_r\,{\bf e}_r+ A_\theta\,{\bf e}_\theta + A_\phi\,{\bf e}_\phi,$$ (1649)

where ${\bf e}_r=\nabla r/\vert\nabla r\vert$, ${\bf e}_\theta = \nabla\theta/\vert\nabla\theta\vert$, and ${\bf e}_\phi=\nabla \phi/\vert\nabla \phi\vert$. See Figure 126. Of course, the unit vectors ${\bf e}_r$, ${\bf e}_\theta$, and ${\bf e}_\phi$ are mutually orthogonal, so $A_r = {\bf A}\cdot {\bf e}_r$, etc.

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

$$d {\bf x}\cdot d{\bf x} = dr^{\,2} + r^2\,d\theta^{\,2} + r^2\,\sin^2\theta\,d\phi^2.$$ (1650)

Hence, comparison with Equation (1600) reveals that the scale factors for this system are

$$h_r = 1,$$ (1651)

$$h_\theta = r,$$ (1652)

$$h_\phi = r\,\sin\theta.$$ (1653)

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

$$dS_r = r^2\,\sin\theta\,d\theta\,d\phi,$$ (1654)

$$dS_\theta = r\,\sin\theta\,dr\,d\phi,$$ (1655)

$$dS_\phi = r\,dr\,d\theta,$$ (1656)

respectively, whereas a volume element takes the form

$$dV = r^2\,\sin\theta\,dr\,d\theta\,d\phi.$$ (1657)

According to Equations (1607), (1609), and (1612), gradient, divergence, and curl in the spherical coordinate system are written

$$\nabla \psi = \frac{\partial \psi}{\partial r}\,{\bf e}_r + \frac{1}{r}\frac{\p... ...a + \frac{1}{r\,\sin\theta}\,\frac{\partial \psi}{\partial \phi}\,{\bf e}_\phi,$$ (1658)

$$\nabla\cdot{\bf A} = \frac{1}{r^2}\,\frac{\partial}{\partial r}\,(r^2\,A_r) + \frac{1}... ...eta\,A_\theta)+ \frac{1}{r\,\sin\theta}\,\frac{\partial A_\phi}{\partial \phi},$$ (1659)

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

$$+\left[\frac{1}{r\,\sin\theta}\,\frac{\partial A_r}{\partial \phi}-\frac{1}{r}\frac{\partial}{\partial r}\,(r\,A_\phi)\right]{\bf e}_\theta$$

$$+ \left[\frac{1}{r}\,\frac{\partial}{\partial r}\,(r\,A_\theta) - \frac{1}{r}\,\frac{\partial A_r}{\partial\theta}\right]{\bf e}_\phi,\$$ (1660)

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

According to Equation (1613), when expressed in spherical coordinates, the Laplacian of a scalar field becomes

$$\nabla^2 \psi= \frac{1}{r^2}\,\frac{\partial}{\partial r}\le... ...}{r^2\,\sin^2\theta}\,\frac{\partial^2 \psi}{\partial \phi^2}.$$ (1661)

Moreover, from Equation (1617), the components of $({\bf A}\cdot\nabla){\bf A}$ in the spherical coordinate system are

$$[({\bf A}\cdot\nabla){\bf A}]_r = {\bf A}\cdot\nabla A_r - \frac{A_\theta^{\,2}+A_\phi^{\,2}}{r},$$ (1662)

$$[({\bf A}\cdot\nabla){\bf A}]_\theta = {\bf A}\cdot\nabla A_\theta + \frac{A_r\,A_\theta-\cot\theta\,A_\phi^{\,2}}{r},$$ (1663)

$$[({\bf A}\cdot\nabla){\bf A}]_\phi = {\bf A}\cdot\nabla A_\phi+ \frac{A_r\,A_\phi+\cot\theta\,A_\theta\,A_\phi}{r}.$$ (1664)

Now, according to Equation (1620), the components of $\widetilde{\nabla {\bf A}}$ in the spherical coordinate system are

$$(\widetilde{\nabla {\bf A}})_{rr} = \frac{\partial A_r}{\partial r},$$ (1665)

$$(\widetilde{\nabla {\bf A}})_{\theta\theta} = \frac{1}{r} \frac{\partial A_\theta}{\partial \theta}+ \frac{A_r}{r},$$ (1666)

$$(\widetilde{\nabla {\bf A}})_{\phi\phi} = \frac{1}{r\,\sin\theta}\,\frac{\partial A_\phi}{\partial \phi}+ \frac{A_r}{r} + \frac{\cot\theta\,A_\theta}{r},$$ (1667)

$$(\widetilde{\nabla {\bf A}})_{r\theta}=(\widetilde{\nabla {\bf A}})_{\theta r} = \frac{1}{2}\left(\frac{1}{r}\,\frac{\partial A_r}{\partial\theta} + \frac{\partial A_\theta}{\partial r} - \frac{A_\theta}{r}\right),$$ (1668)

$$(\widetilde{\nabla {\bf A}})_{r\phi}=(\widetilde{\nabla {\bf A}})_{\phi r} = \frac{1}{2}\left(\frac{1}{r\,\sin\theta}\,\frac{\partial A_r}{\partial \phi} + \frac{\partial A_\phi}{\partial r}-\frac{A_\phi}{r}\right),$$ (1669)

$$(\widetilde{\nabla {\bf A}})_{\theta \phi}=(\widetilde{\nabla {\bf A}})_{\phi\theta} = \frac{1}{2}\left(\frac{1}{r\,\sin\theta}\,\frac{\partial A_\theta... ...{r}\frac{\partial A_\phi}{\partial \theta}-\frac{\cot\theta\,A_\phi}{r}\right).$$ (1670)

Finally, from Equation (1622), the components of $\nabla^2{\bf A}$ in the spherical coordinate system are

$$(\nabla^2{\bf A})_r = \nabla^2 A_r -\frac{2 A_r}{r^2}-\frac{2}{r^2}\,\frac{\partial A_\... ..._\theta}{r^2} -\frac{2}{r^2\,\sin\theta}\,\frac{\partial A_\phi}{\partial\phi},$$ (1671)

$$(\nabla^2{\bf A})_\theta = \nabla^2 A_\theta +\frac{2}{r^2}\,\frac{\partial A_r}{\partial\th... ...\sin^2\theta} -\frac{2}{r^2\,\sin\theta}\,\frac{\partial A_\phi}{\partial\phi},$$ (1672)

$$(\nabla^2{\bf A})_\phi = \nabla^2 A_\phi-\frac{A_\phi}{r^2\,\sin^2\theta} + \frac{2}{r^2\,... ...}+\frac{2 \cot\theta}{r^2\,\sin\theta}\,\frac{\partial A_\theta}{\partial\phi}.$$ (1673)