## Purely Radial Functions With Zero Divergence

For any purely radial vector field,
$\mathbf{F} = F_r \mathbf{e_r}$
.
If
$F_r = r^n$
, are there any values of
$n$
for which
$\mathbf{\nabla} \cdot \mathbf{F} =0$
?
Yes there are.
$\mathbf{\nabla} \cdot (r^n \mathbf{e_r}) = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 r^n) =\frac{1}{r^2} \frac{\partial}{\partial r} (r^{2+n})=\frac{(2+n)r^{1+n}}{r^2} = (2+n)r^{-1+n}$

This is only zero if
$n=-2$
, then
$\mathbf{F} = \frac{K}{r^2} \mathbf{e_r}$
.
Notice that this is only true if
$r \neq 0$
/