Divergence of a radial function

Question

Is there any proof of the relation

$$\nabla\cdot f(r)= \frac{1}{r^2}\frac{\partial}{\partial r}({r^2}f(r))$$

Is it true for any radially directed function or for some specific function?

Answer 1

I suppose you are referring to this formula: $$\operatorname{div}\mathbf{F} = \nabla\cdot\mathbf F = \frac1{r^2} \frac{\partial}{\partial r}\left(r^2 F_r\right) + \frac1{r\sin\theta} \frac{\partial}{\partial \theta} (\sin\theta\, F_\theta) + \frac1{r\sin\theta} \frac{\partial F_\varphi}{\partial \varphi}$$ Note that the terms involving the angular derivatives are zero due to the fact that the function is radially symmetric, i.e. independent of these components.