= Boundary flux from a singular harmonic potential
{title2=$\int_{|x|<a}\nabla(r/a)\cdot\nabla(1/r)\,dV=-4\pi$}
Although $1/r$ is harmonic away from the origin in three dimensions, it is not a regular <harmonic function> on a ball containing the origin. Its <gradient> is $-\widehat r/r^2$, so the displayed improper volume integral follows by canceling the $r^2$ in the spherical volume element. On the annulus $\varepsilon<r<a$, <Green's first identity> gives outer boundary contribution $-4\pi$ and inner contribution $4\pi\varepsilon/a$, recovering the same limit. Its global <Laplacian> contains a <Dirac delta distribution>, so the regular-harmonic gradient-pairing theorem is inapplicable.
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