   Chapter 16.8, Problem 20E

Chapter
Section
Textbook Problem

Suppose S and C satisfy the hypotheses of Stokes’ Theorem and f, g have continuous second-order partial derivatives. Use Exercises 24 and 26 in Section 16.5 to show the following.(a) ∫c(f ∇g) · dr = ∫∫s (∇f × ∇g) · dS(b) ∫c(f∇f) · dr = 0(c) ∫c (f∇g + g∇f) · dr = 0

(a)

To determine

To show: The expression C(fg)dr=S(f×g)dS

Explanation

Given data:

Consider S and C satisfy the hypotheses of Stokes’ Theorem, f and g have continuous second-order partial derivatives.

Consider the following expressions from Exercises 24 and 26 in section 16.5.

curl(F+G)=curlF+curlG (1)

curl(fF)=fcurlF+(f)×F (2)

Formula used:

Write the expression for the Stokes’ theorem.

CFdr=ScurlFdS (3)

Write the expression for curl(fg) by using equation (2),

curl(fg)=fcurl(g)+f×g{curl(g)=0}

curl(fg)

(b)

To determine

To show: The expression C(ff)dr=0.

(c)

To determine

To show: The expression C(fg+gf)dr=0

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