## Finding a Vector Field With Divergence Equal to a Power of r

To solve the equation
$\mathbf{\nabla} \cdot \mathbf{F} =r^n$
in spherical polar coordinates, use the facet that fore a radial vector field
$F=F_r \mathbf{e_r}$
,
$\mathbf{\nabla} \cdot \mathbf{F} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 F_r ) =r^n$

Hence
$\frac{\partial}{\partial r} (r^2 F_r ) =r^{n+2}$

Integrate both sides to get
$r^2 F_r = \frac{r^{n+3}}{n+3}+A$

Then
$F_r = \frac{r^{n+1}}{n+3} + \frac{A}{r^2}$
if
$n \neq 0$
and
$F_r = \frac{r^{n+1}}{n+3}$
if
$r=0$
is possible, since
$\mathbf{\nabla} \cdot \mathbf{F}=0 \rightarrow F_r =\frac{1}{r^2}$
using this result.