= Inverse-square radial point source
{title2=$\nabla\cdot(\hat{\mathbf r}/r^2)=4\pi\delta^{(3)}(\mathbf r)$}
The inverse-square <radial vector field> has zero ordinary <divergence> at every nonzero radius but flux $4\pi$ through every centred sphere. It is locally integrable in three dimensions. For a smooth compactly supported test function $\eta$, integrate by parts outside a ball of radius $\epsilon$. The inward-oriented inner boundary gives $-\int \mathbf F\cdot\nabla\eta=\int_{S^2}\eta(\epsilon\hat{\mathbf r})d\Omega$, which tends to $4\pi\eta(0)$. This proves the displayed <distributional divergence>, with the three-dimensional <Dirac delta function> representing the point source.
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