Chapter 15.4, Problem 52E

Proof In Exercises 51 and 52, prove the identity, where R is a simply connected region with piecewise smooth boundary C. Assume that the required partial derivatives of the scalar functions f and g are continuous. The expressions $D_{\text{N}}f$ and $D_{\text{N}}g$ are the derivatives in the direction of the outward normal vector N of C and are defined by $D_{\text{N}}f=\nabla f\cdot \text{N}$ and $D_{\text{N}}g=\nabla g\cdot \text{N}$.

Green’s second identity:

${\displaystyle {\int }_R{\displaystyle \int (f{\nabla }^{2}g}}-g{\nabla }^{2}f)dA={\displaystyle {\int }_C\left(f{D}_{\text{N}}g-g{D}_{\text{N}}f\right)ds}$

(Hint: Use Green’s first identity from Exercise 51 twice.)