Chapter 15.7, Problem 29ES

Prove the identity, assuming that F, $\sigma$ , and G satisfy the hypotheses of the Divergence Theorem and that all necessary differentiability requirements for the functions $f\left(x,y,z\right)\text{and}g\left(x,y,z\right)$ are met.

${\displaystyle \underset{\sigma }{\iint }\left(f\nabla g\right)\cdot \text{n}dS={\displaystyle \underset{G}{\iiint }\left(f{\nabla }^{2}g+\nabla f\cdot \nabla g\right)dV}}$