Using spherical polar coordinates ($r, \theta, \phi$) verify that the vector $F = r^{-2}e_r$ is solenoidal. Find the function $\psi(r, \theta)$ such that $A = \frac{\psi(r, \theta)}{rsin \theta}e_\phi$ is the vector potential of $F$ satisfying $ \nabla \cdot A = 0$ and $F = \nabla \wedge A.$
I've verified that F is solenoidal but can't find the function $\psi(r,\theta)$
Using
$x = rsin\theta cos\phi$
$y = rsin\theta sin\phi$
$z=rcos\theta$
$F = r^{-2}e_r = r^{-2}[\frac{1}{h_r}(\frac{\partial x}{\partial r}i + \frac{\partial y}{\partial r}j + \frac{\partial z}{\partial r}k)]$
$F = (\frac{\partial x}{\partial r}\cdot\frac{1}{r^2\sqrt{sin^2\theta + cos^2\theta}} + \frac{\partial y}{\partial r}\cdot\frac{1}{r^2\sqrt{sin^2\theta + cos^2\theta}} + \frac{\partial z}{\partial r}\cdot\frac{1}{r^2\sqrt{sin^2\theta + cos^2\theta}})$
Then using $F \cdot \nabla = 0$
$(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}) \cdot (\frac{\partial x}{\partial r}\cdot\frac{1}{r^2\sqrt{sin^2\theta + cos^2\theta}} + \frac{\partial y}{\partial r}\cdot\frac{1}{r^2\sqrt{sin^2\theta + cos^2\theta}} + \frac{\partial z}{\partial r}\cdot\frac{1}{r^2\sqrt{sin^2\theta + cos^2\theta}})$
=...
$= \frac{-6}{r^3\sqrt{sin^2\theta + cos^2 \phi}}$
At $(0,0,0)$ we get 0 which is required.
But then finding the function $\psi (r,\theta)$ I am unsure what to do.
When trying to find out $e_\phi$ in
$A = \frac{\psi(r, \theta)}{rsin \theta}e_\phi$
is it: $e_\phi$ = ($\frac{\frac{\partial x}{\partial \phi}i + \frac{\partial y}{\partial \phi}j + \frac{\partial z}{\partial \phi}k)}{rsin\theta})$?
then doing $\nabla \cdot A = 0$
I get $\frac{\partial}{\partial \phi} \frac{\psi (r,\theta)}{r^2sin^2\theta} = 0$
Then working out $F = \nabla \wedge A$
I get
$\frac{\psi (r,\theta)}{r^2sin^2\theta}[\frac{\partial}{\partial x}(\frac{\partial y}{\partial \phi}+\frac{\partial z}{\partial \phi}) + \frac{\partial}{\partial y}(\frac{\partial z}{\partial \phi}-\frac{\partial x}{\partial \phi} )- \frac{\partial}{\partial z}(\frac{\partial y}{\partial \phi}+\frac{\partial x}{\partial \phi})] = \frac{1}{r^2\sqrt{sin^2\theta + cos^2 \phi}}[\frac{\partial x}{\partial r} + \frac{\partial y}{\partial r} + \frac{\partial z}{\partial r}]$
and now I have no idea what to do.