   Chapter 16.8, Problem 19E

Chapter
Section
Textbook Problem

If S is a sphere and F satisfies the hypotheses of Stokes’ Theorem, show that ∫∫s curl F · dS = 0.

To determine

To show: The surface integral of field F over a sphere surface is ScurlFdS=0 .

Explanation

Formula Used:

Write the expression for the Stokes’ theorem for field F(x,y,z)=Pi+Qj+Rk .

CFdr=ScurlFdS (1)

Here,

S is surface.

Consider the center of sphere S is located at origin with radius a. Consider the upper hemisphere of S is H1 and lower hemisphere of S is H2 . Then the surface integral is,

ScurlFdS=H1curlFdS+H2curlFdS (2)

Consider the curve of upper hemisphere as C1 and lower hemisphere as C2 .

Re-modify equation (2) by using equation (1).

ScurlFdS=C1Fdr+C2Fdr (3)

The curves C1 and C2 are circles with x2+y2=a2 and the orientation of C1 is counterclockwise whereas C2 is in clockwise direction

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