   Chapter 16.5, Problem 34E

Chapter
Section
Textbook Problem

Use Green’s first identity (Exercise 33) to prove Green’s second identity: ∬ D ( f ∇ 2 g − g ∇ 2 f )   d A = ∮ C ( f ∇ g − g ∇ f ) ⋅ n   d s where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and g exist and are continuous.

To determine

To prove: The Green’s second identity.

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

Simplify the equation.

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