This result is very useful in continuum mechanics involving discontinuities, for example, multiphase flows. It is basically a variant of the Stokes theorem - converting area integral into a line integral.
Consider a closed two-dimensional surface $\mathcal{A}$ with boundary $\partial\mathcal{A}$. The normal to $\partial\mathcal{A}$ is $\widehat{n}$. Prove that
$-\int_\mathcal{A}\nabla'[\delta(x-x')\delta(y-y')]\,da'= -\int_{\partial\mathcal{A}}\delta(x-x')\delta(y-y')\widehat{n}\,ds$
where $\delta$ is the Dirac delta function. Proof for a general tensor would be great, but a simplified, rigorous proof (maybe for symmetric tensors, or even just the Dirac delta case) would work as well.