Chapter 15.7, Problem 28ES

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.

$\begin{array}{l}{\displaystyle \underset{\sigma }{\iint }\nabla f\cdot \text{n}dS={\displaystyle \underset{G}{\iiint }{\nabla }^{2}fdV}}\\ \left({\nabla }^{2}f=\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}\right)\end{array}$