$$\mathbf{u}=\frac{\mathbf{r}}{r^\alpha},\;div\,\mathbf{u}=\frac{3-\alpha}{r^\alpha},\;\alpha\le3$$ The divergence theorem holds on the unit ball $r\leq 1$ for $\alpha\lt 3$ but not for $\alpha=3$. Why is this?
I can't see how this would break the hypotheses of the divergence theorem as the field is still differentiable on the unit ball.