Past exam of the mathematical methods II course of the University of Oxford / 2016 resit / 3 / iii

Away from the origin, the product rule and $\nabla r=\mathbf{r}/r$ giveThe field is undefined at $r=0$. It has outward flux $4\pi$ through any sphere centred there, so its global distributional divergence is $4\pi {\delta }^{(3)}(\mathbf{r})$, not the zero distribution. The pointwise result must not be extended through the singularity when applying the divergence theorem.