Chapter 16.9, Problem 29E

Prove each identity, assuming that S and E satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives.

${\displaystyle \underset{S}{\iint }\left(f\nabla g\right)\cdot ndS={\displaystyle \underset{E}{\iiint }\left(f{\nabla }^{2}g+\nabla f\cdot \nabla g\right)dV}}$