Chapter 15.7, Problem 32E

Proof In Exercises 31 and 32, prove the identity, assuming that Q, S, and N meet the conditions of the Divergence Theorem and that the required partial derivatives of the scalar functions f and g are continuous. The expressions $D_{N}f$ and $D_{N}g$ are the derivatives in the direction of the vector N and are defined by $D_{\text{N}}f=\nabla f\cdot N\text{and}{D}_{\text{N}}g=\nabla g\cdot N.$

${\displaystyle \underset{Q}{\iiint }\left(f{\nabla }^{2}g-g{\nabla }^{2}f\right)dV={\displaystyle {\int }_S{\displaystyle \int \left(f{D}_{\text{N}}g-g{D}_{\text{N}}f\right)dS}}}$

[ Hint: Use Exercise 31 twice.]