Finding potential of a given vector field

Question

I am trying to solve the following problem:

Let $ \textbf{F}=f(r) (x,y,z)$ where $r=(x^{2}+y^{2}+z^{2})^{1/2} $. Find an expression for a potential for $ \textbf{F}$. Find an expression also for the divergence of $ \textbf{F}$ in terms of the potential.

I would normally just intregrate coordinates of the vector and adjust it so that the arbitrary functions agree, but in this case I am not quite sure what to do and how. Any clue will be much appreciated :)

Answer 1

Hint for the potential: Compute the gradient of $\Phi(\vert \mathbf{x} \vert)$ for $\Phi$ differentiable on a suitable interval, be it $\mathbb{R}_{\geq 0}$ or $\mathbb{R}_{> 0}$.