Chapter 15.7, Problem 31ES

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.

${\displaystyle \underset{\sigma }{\iint }\left(f\text{n}\right)\cdot \text{v}dS={\displaystyle \underset{G}{\iiint }\nabla f\cdot \text{v}dV\left(\text{v}\text{a fixed <x-custom-btb-me data-me-id="1725" class="microExplainerHighlight">vector</x-custom-btb-me>}\right)}}$