   Chapter 16, Problem 32RE

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∬s curl F · dS, where F(x, y, z) = x2yz i + yz2 j + z3exy k, S is the part of the sphere x2 + y2 + z2 = 5 that lies above the plane z=1, and S is oriented upward.

To determine

To evaluate: ScurlFdS , where F(x,y,z)=x2yzi+yz2j+z3exyk .

Explanation

Given data:

F(x,y,z)=x2yzi+yz2j+z3exyk , x2+y2+z2=5 , z=1 .

Formula used:

Write the expression for Stokes’ Theorem.

ScurlFdS=CFdr (1)

Write the expression for curlF .

curlF=|ijkxyzPQR| (2)

Consider the expression as follows.

F(x,y,z)=x2yzi+yz2j+z3exyk (3)

Consider the expression for part of the sphere as follows.

x2+y2+z2=5

Substitute 1 for z ,

x2+y2+12=5x2+y2=51

x2+y2=4 (4)

Write the expression for circle.

x2+y2=r2 (5)

Compare equations (4) and (5).

r2=4r=2

Write the expression for polar co-ordinates.

x=rcosty=rsint

Substitute 2 for r ,

x=2costy=2sint

Consider the value of r(t) as follows.

r(t)=xi+yj+zk

Substitute 2cost for x , 2sint for y and 1 for z ,

r(t)=2costi+2sintj+k , 0t2π

Differentiate r(t) with respect to t .

drdt=(2sinti+2costj)dr=(2sinti+2costj)dt

Find the value of CFdr

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