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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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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Chapter 16 Solutions

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