   Chapter 16.5, Problem 33E

Chapter
Section
Textbook Problem

Use Green’s Theorem in the form of Equation 13 to prove Green’s first identity: ∬ D f ∇ 2 g   d A = ∮ C f ( ∇ g ) ⋅ n   d s − ∬ D ∇ f ⋅ ∇ g   d A where D and C satisfy the hypotheses of Green’s Theorem and the appropriate partial derivatives of f and g exist and arc continuous. (The quantity ∇g · n = Dn 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.)

To determine

To prove: The Green’s first 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+g

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