5 Spherical polar coordinates

In spherical polar coordinates $\left(r,\theta ,\varphi \right)$ , the 3 unit vectors are $\underset{̲}{\stackrel{̂}{r}}$ , $\underset{̲}{\stackrel{̂}{\theta }}$ and $\underset{̲}{\stackrel{̂}{\varphi }}$ with scale factors $h_r=1$ , $h_{\theta }=r$ , $h_{\varphi }=rsin\theta$ . The quantities $r$ , $\theta$ and $\varphi$ are related to $x$ , $y$ and $z$ by $x=rsin\theta cos\varphi$ , $y=rsin\theta sin\varphi$ and $z=rcos\theta$ . In spherical polar coordinates,

grad $f=\underset{̲}{\nabla }f=\frac{\partial f}{\partial r}\underset{̲}{\stackrel{̂}{r}}+\frac{1}{r}\frac{\partial f}{\partial \theta }\underset{̲}{\stackrel{̂}{\theta }}+\frac{1}{rsin\theta }\frac{\partial f}{\partial \varphi }\underset{̲}{\stackrel{̂}{\varphi }}$



If $\underset{̲}{F}=F_r\underset{̲}{\stackrel{̂}{r}}+F_{\theta }\underset{̲}{\stackrel{̂}{\theta }}+F_{\varphi }\underset{̲}{\stackrel{̂}{\varphi }}$

then

div $\underset{̲}{F}=\underset{̲}{\nabla }\cdot \underset{̲}{F}=\frac{1}{r^{2}sin\theta }\left[\frac{\partial }{\partial r}\left(r^{2}sin\theta F_r\right)+\frac{\partial }{\partial \theta }\left(rsin\theta F_{\theta }\right)+\frac{\partial }{\partial \varphi }\left(r{F}_{\varphi }\right)\right]$

curl $\underset{̲}{F}=\underset{̲}{\nabla }\times \underset{̲}{F}=\frac{1}{r^{2}sin\theta }\left|\begin{array}{ccc}\hfill \underset{̲}{\stackrel{̂}{r}}\hfill & \hfill r\underset{̲}{\stackrel{̂}{\theta }}\hfill & \hfill rsin\theta \underset{̲}{\stackrel{̂}{\varphi }}\hfill \\ \hfill \hfill \\ \hfill \frac{\partial }{\partial r}\hfill & \hfill \frac{\partial }{\partial \theta }\hfill & \hfill \frac{\partial }{\partial \varphi }\hfill \\ \hfill \hfill \\ \hfill F_r\hfill & \hfill r{F}_{\theta }\hfill & \hfill rsin\theta F_{\varphi }\hfill \end{array}\right|$

Example 25

In spherical polar coordinates, find $\underset{̲}{\nabla }f$ for

1. $f=r$
2. $f=\frac{1}{r}$
3. $f=r^{2}sin\left(\varphi +\theta \right)$

   [Note: parts 1. and 2. relate to Exercises 2(a) and 2(c) on page 22.]

Solution

1. $$\begin{array}{rcll}\underset{̲}{\nabla }f& =& \frac{\partial f}{\partial r}\underset{̲}{\stackrel{̂}{r}}+\frac{1}{r}\frac{\partial f}{\partial \theta }\underset{̲}{\stackrel{̂}{\theta }}+\frac{1}{rsin\theta }\frac{\partial f}{\partial \varphi }\underset{̲}{\stackrel{̂}{\varphi }}& \\ & =& \frac{\partial \left(r\right)}{\partial r}\underset{̲}{\stackrel{̂}{r}}+\frac{1}{r}\frac{\partial \left(r\right)}{\partial \theta }\underset{̲}{\stackrel{̂}{\theta }}+\frac{1}{rsin\theta }\frac{\partial \left(r\right)}{\partial \varphi }\underset{̲}{\stackrel{̂}{\varphi }}& \\ & =& 1\underset{̲}{\stackrel{̂}{r}}=\underset{̲}{\stackrel{̂}{r}}& \end{array}$$
2. $$\begin{array}{rcll}\underset{̲}{\nabla }f& =& \frac{\partial f}{\partial r}\underset{̲}{\stackrel{̂}{r}}+\frac{1}{r}\frac{\partial f}{\partial \theta }\underset{̲}{\stackrel{̂}{\theta }}+\frac{1}{rsin\theta }\frac{\partial f}{\partial \varphi }\underset{̲}{\stackrel{̂}{\varphi }}& \\ & =& \frac{\partial \left(\frac{1}{r}\right)}{\partial r}\underset{̲}{\stackrel{̂}{r}}+\frac{1}{r}\frac{\partial \left(\frac{1}{r}\right)}{\partial \theta }\underset{̲}{\stackrel{̂}{\theta }}+\frac{1}{rsin\theta }\frac{\partial \left(\frac{1}{r}\right)}{\partial \varphi }\underset{̲}{\stackrel{̂}{\varphi }}& \\ & =& -\frac{1}{r^2}\underset{̲}{\stackrel{̂}{r}}& \end{array}$$
3. $$\begin{array}{rcll}\underset{̲}{\nabla }f& =& \frac{\partial f}{\partial r}\underset{̲}{\stackrel{̂}{r}}+\frac{1}{r}\frac{\partial f}{\partial \theta }\underset{̲}{\stackrel{̂}{\theta }}+\frac{1}{rsin\theta }\frac{\partial f}{\partial \varphi }\underset{̲}{\stackrel{̂}{\varphi }}& \\ & =& \frac{\partial \left(rsin\left(\varphi +\theta \right)\right)}{\partial r}\underset{̲}{\stackrel{̂}{r}}+\frac{1}{r}\frac{\partial \left(rsin\left(\varphi +\theta \right)\right)}{\partial \theta }\underset{̲}{\stackrel{̂}{\theta }}+\frac{1}{rsin\theta }\frac{\partial \left(r^{2}sin\left(\varphi +\theta \right)\right)}{\partial \varphi }\underset{̲}{\stackrel{̂}{\varphi }}& \\ & =& 2rsin\left(\varphi +\theta \right)\underset{̲}{\stackrel{̂}{r}}+\frac{1}{r}{r}^{2}cos\left(\varphi +\theta \right)\underset{̲}{\stackrel{̂}{\theta }}+\frac{1}{rsin\theta }{r}^{2}cos\left(\varphi +\theta \right)\underset{̲}{\stackrel{̂}{\varphi }}& \\ & =& 2rsin\left(\varphi +\theta \right)\underset{̲}{\stackrel{̂}{r}}+rcos\left(\varphi +\theta \right)\underset{̲}{\stackrel{̂}{\theta }}+\frac{rcos\left(\varphi +\theta \right)}{sin\theta }\underset{̲}{\stackrel{̂}{\varphi }}& \end{array}$$