Let $\displaystyle\mathbf{v}_2=\frac{1}{r^2}\hat{\mathbf{r}}$.
I found that $\displaystyle\nabla.\mathbf{v}_2=0$ everywhere except at the origin, where it is not defined. So, we cannot use the divergence theorem for a sphere of radius $R$ centered at the origin.
I also found that $\displaystyle\iint_S\mathbf{v}_2\cdot d\mathbf{a}=4\pi$ where $S$ is the surface of the sphere.
My professor said that the divergence theorem can still be used provided we gouge out the origin from the sphere but I don't quite remember how he proceeded. Could someone please explain it to me? Thanks.