Gradient of a dipole potential
= Gradient of a dipole potential
{title2=$\nabla[(d\cdot x)/r^3]=(I-3n\otimes n)d/r^3$}
= Dipole gradient field
{synonym}
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 $\langle n\otimes n\rangle=I/3$. That cancellation distinguishes a self-dipole field from the regular incident field that excites a small inclusion.