   Chapter 16.5, Problem 36E

Chapter
Section
Textbook Problem

Use Green’s first identity to show that if f is harmonic on D, and if f(x, y) = 0 on the boundary curve C, then ∬D |∇f|2 dA = 0. (Assume the same hypotheses as in Exercise 33.)

To determine

To show: The D|f|2dA=0 .

Explanation

Consider the expression of second vector form of Green’s Theorem.

CFnds=DdivF(x,y)dA (1)

As the gn occurs in the line integral, then the equation (1) can be modified as follows.

Cf(g)nds=Ddivf(g)dA=D[fdiv(g)+gf]dA {div(fF)=fdivF+Ff}=D[f2g+gf]dA {div(g)=2g}=Df2gdA+DgfdA

Rearrange the equation.

Df2gdA=Cf(g)ndsDgfdA (2)

Consider g=f

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