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
\[r \neq 0\]
/