= Solution
Away from the origin, the <product rule> and $\nabla r=\mathbf r/r$ give
$$
\nabla\cdot(r^{-3}\mathbf r)
=r^{-3}(3)+\mathbf r\cdot(-3r^{-5}\mathbf r)
=\boxed{0\quad(r\ne0).}
$$
The 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>.
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