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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

Verify that Stokes’ Theorem is true for the vector field F(x, y, z) = x2 i + y2 j + z2 k, where S is the part of the paraboloid z = 1 − x2y2 that lies above the xy-plane and S has upward orientation.

To determine

To verify: The Stokes’ Theorem is true for the vector field F(x,y,z)=x2i+y2j+z2k .

Explanation

Given data:

F(x,y,z)=x2i+y2j+z2k , z=1x2y2 .

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)=x2i+y2j+z2k (3)

Write the expression for F(x,y,z) .

F(x,y,z)=Pi+Qj+Rk (4)

Compare equations (3) and (4).

P=x2Q=y2R=z2

Substitute x2 for P , y2 for Q and z2 for R in equation (2),

curlF=|ijkxyzx2y2z2|={i[(z2)y(y2)z]j[(z2)x(x2)z]+k[(y2)x(x2)y]}=i[00]j[00]+k[00]=0

Find the value of ScurlFdS .

ScurlFdS=S(0)dS

ScurlFdS=0 (5)

Since, the paraboloid lies above the xy-plane , the value of z=0 .

Consider the expression for part of the paraboloid as follows.

z=1x2y2

Substitute 0 for z ,

0=1x2y2

x2+y2=1 (6)

Write the expression for circle.

x2+y2=r2 (7)

Compare equations (6) and (7)

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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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