Matrix trick for pushing integrals onto the boundary

I now present an algebraic technique for expressing integrals over $\mathcal{D}$ as surface integrals over $\Gamma$. This was invented, as far as I am aware, by Mike Haggerty (unpublished), in response to the ugly vector algebra performed by Boasman [33] needed to achieve the result (H.14) in $d=2$. I have generalized this technique to the case of singular matrices, and to the tensor divergence theorem.