I am self studying Divergence, Curl and All That: An Informal Text On Vector Calculus
I have some confusion regarding problem 22 from chapter 2, which uses information given in problem 21:
My problem is understanding how to even work with $f(r)$, and how to translate the vector function into spherical coordinates.
I do not think I want to just write $f(r) = f(\sqrt{x^2+y^2+z^2})$ and write $F(r)$ as $(ix+jy+kz)/r * f(\sqrt{x^2+y^2+z^2})$ because I do not know how to evaluate $r^2F_r$ from here. I thought perhaps $\partial/\partial{\phi} (sin{\phi}F_{\phi}) = 0$ and similarly for $\theta$, but then I thought I need to translate the vector function into spherical coordinates before I determine what the partial of phi and theta is. I do not feel confident about my ability to translate this into spherical coordinates.
I know I am a bit all over the place with this one, but some hints, a push forward or some insight would be greatly appreciated.
In problem $21$, $\mathbf{F}$ is represented as $F_r \mathbf{\hat{e}}_r + F_\theta \mathbf{\hat{e}}_\theta + F_\phi \mathbf{\hat{e}}_\phi$. So in problem $22$, $\mathbf{F}$ has only a radial component, $f(r)$. The divergence is then $\frac{1}{r^2}\frac{d}{dr}(r^2 f)$, which is zero iff $r^2 f(r)$ is constant. Thus $f(r) = A/r^2$ for some constant $A$.