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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. In each case C is oriented counterclockwise as viewed from above.

8. F(x, y, z) = i + (x + yz) j + (xy -z) k, C is the boundary of the part of the plane 3x + 2y + z = 1 in the first octant

To determine

To evaluate: The value of CFdr by the use of Stokes’ theorem.

Explanation

Given data:

The field is F(x,y,z)=i+(x+yz)j+(xyz)k and C is boundary of part of plane 3x+2y+z=1 in first octant.

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.

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

curlF=((xyz)y(x+yz)z)i((xyz)x(1)z)j+((x+yz)x(1)y)k=((x(1)0)(0+y(1)))i((y(1)0)0)j+((1+0)0)k=(xy)iyj+k

The surface S is a planar region which is enclosed by plane C. The expression for surface S, D={(x,y)|0x13,0y12(13x)} is,

3x+2y+z=1z=13x2y

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.

Compare the equations curlF=Pi+Qj+Rk and curlF=(xy)iyj+k .

P=xyQ=yR=1

Find the value of ScurlFdS by the use of equation (3)

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Chapter 16 Solutions

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