Chapter 16.9, Problem 30E

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.

30. ${\displaystyle \underset{S}{\iint }(f\nabla g-g\nabla f)\cdot \text{n}dS\text{=}{\displaystyle \underset{E}{\iiint }(f{\nabla }^{2}g-g{\nabla }^{2}f)dV}}$