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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 ∫C F · dr, where F(x, y, z) = xy i + yz j + zx k, and C is the triangle with vertices (1, 0. 0), (0, 1, 0), and (0, 0, 1), oriented counterclockwise as viewed from above.

To determine

To evaluate: The value of CFdr , where F(x,y,z)=xyi+yzj+zxk using Stokes’ Theorem.

Explanation

Given data:

The field is F(x,y,z)=xyi+yzj+zxk and C is triangle with vertices are (1,0,0) , (0,1,0) and (0,0,1) .

Formula used:

Write the expression for curl of F(x,y,z)=Pi+Qj+Rk .

curlF=|ijkxyzPQR|

curlF=(RyQz)i(RxPz)j+(QxPy)k (1)

Write the expression for the Stokes’ theorem.

CFdr=ScurlFdS (2)

Here,

S is surface.

Consider surface S, z=g(x,y) is in upward orientation. Write the expression for surface integral of F over surface S.

ScurlFdS=D(PgxQgy+R)dA (3)

Here,

A is area.

Consider the general expression for field as follows.

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

Consider the field expression as follows.

F(x,y,z)=xyi+yzj+zxk (5)

Compare equations (4) and (5).

P=xyQ=yzR=zx

Find the value of curlF by using equation (1).

curlF=[((zx)y(yz)z)i((zx)x(xy)z)j+((yz)x(xy)y)k]=(0y)i(z0)j+(0x)k=yizjxk

Consider the surface S is a planar region which is enclosed by curve C. The expression for surface S, D={(x,y)|0x1,0y1x} since it is a plane is,

x+y+z=1z=1xy

Hence the equation is in the form of z=g(x,y) .

The curve C is oriented in counter-clockwise, so the orient the surface S upward

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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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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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