Chapter 16.9, Problem 31E

Suppose S and E satisfy the conditions of the Divergence Theorem and ƒ is a scalar function with continuous partial derivatives. Prove that

${\displaystyle \underset{S}{\iiint }fndS={\displaystyle \underset{E}{\iiint }\nabla fdV}}$

These surface and triple integrals of vector functions are vectors defined by integrating each component function. [Hint: Start by applying the Divergence Theorem to $F=fc$, where c is an arbitrary constant vector.]