Distributional divergence ($⟨\mathrm{div}z,\eta ⟩=-\int z\cdot \nabla \eta$)

For a locally integrable vector field, this formula defines its divergence as a distribution. For a piecewise smooth field, a jump of its normal component across an interface adds a surface measure to the smooth divergences. A continuous normal component eliminates that measure, as follows by integration by parts on the two sides.

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 $ϵ$. The inward-oriented inner boundary gives $-\int \mathbf{F}\cdot \nabla \eta ={\int }_{S^2}\eta (ϵ\hat{\mathbf{r}})d\Omega$, which tends to $4\pi \eta (0)$. This proves the displayed distributional divergence, with the three-dimensional Dirac delta function representing the point source.