Chapter 16.5, Problem 35E

Recall from Section 14.3 that a function g is called harmonic on D if it satisfies Laplace’s equation, that is ${\nabla }^{2}g=0$ on D. Use Green’s first identity (with the same hypotheses as in Exercise 33) to show that if g is harmonic on D, then ${\displaystyle {\oint }_C{D}_{n}gds=0}$. Here $D_{n}g$ is the normal derivative of g defined in Exercise 33.

Use Green’s Theorem in the form of Equation 13 to prove Green’s first identity:

${\displaystyle \underset{D}{\iint }f{\nabla }^{2}gdA={\displaystyle {\oint }_{c}f\left(\nabla g\right)\cdot \text{n }ds-{\displaystyle \underset{D}{\iint }\nabla f\cdot \nabla gdA}}}$

Where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity $\nabla g\cdot n=D_{n}g$ occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)