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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use Stokes’ Theorem to evaluate ∫∫s curl F · dS.

4. F(x, y, z) = tan-1(x2yz2) i + x2y j + x2z2 k. S is the cone x = x 2 + z 2 , 0 x 2 , oriented in the direction of the positive x-axis

To determine

To evaluate: The expression ScurlFdS by the use of Stokes’ theorem.

Explanation

Given data:

Consider the expression for the vector field F(x,y,z) ,

F(x,y,z)=tan1(x2yz2)i+x2yj+x2z2k (1)

And S is the cone that is,

x=y2+z2,0x2 (2)

Formula Used:

Consider the expression for the Stokes’ theorem,

ScurlFdS=CFdr (3)

Consider the boundary curve C is the circle y2+z2=4,x=2 . This boundary curve C must be oriented in the counter-clockwise direction. Therefore, the vector equation of C is,

r(t)=2i+2costj+2sintk,0t2π (4)

Differentiate equation (4) with respect to t.

r(t)=ddt(2i+2costj+2sintk)=2sintj+2costk

Find the expression for F(r(t)) .

Substitute 2 for x, 2cost for y and 2sint for z in equation (1) to find F(r(t)) ,

F(r(t))=tan1(22(2cost)(2sint)2)i+22(2cost)j+22(2sint)2k=tan1(32costsin2t)i+8costj+16sin2tk

Write the expression for the Stokes’ theorem in equation (3),

ScurlFdS=CFdr=02πF(r(t))r(t)dt

Substitute tan1

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