 Gradient of a dipole potential

Away from the origin, a dipole potential $(d\cdot x)/|x{|}^3$ is harmonic. Its gradient is angularly nonuniform, decays as $r^{-3}$, and has zero spherical mean because $⟨n\otimes n⟩=I/3$. That cancellation distinguishes a self-dipole field from the regular incident field that excites a small inclusion.