   Chapter 16.8, Problem 11E

Chapter
Section
Textbook Problem

(a) Use Stokes’ Theorem to evaluate ∫c F · dr, where F(x, y, z) = x2z i + xy2 j + z2 k and C is the curve of intersection of the plane x + y + z = 1 and the cylinder x2 + y2 = 9, oriented counterclockwise as viewed from above.(b) Graph both the plane and the cylinder with domains chosen so that you can see the curve C and the surface that you used in part (a).(c) Find parametric equations for C and use them to graph C.

(a)

To determine

To evaluate: The expression CFdr by using Stokes’ Theorem.

Explanation

Given data:

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

F(x,y,z)=x2zi+xy2j+z2k (1)

The curve of intersection is an ellipse in the plane x+y+z=1 with unit normal n , where, n=13(i+j+k) (2)

Calculate curlF by using equation (1).

curlF=|ijkxyzx2y13x3xy|=i((z2)y(xy2)z)j((z2)x(x2z)z)+k((xy2)x(x2z)y)=0i+x2j+y2k

curlF=x2j+y2k (3)

Calculate curlFn by using equations (2) and (3),

curlFn=(x2j+y2k)(13(i+j+k))=0+x213+y213=13(x2+y2)

Consider the expression for the Stokes’ theorem

(b)

To determine

To plot: the graph of both the plane and the cylinder with chosen domains.

(c)

To determine

To find: the parametric equations for the curve of intersection of the plane C and use these equations to plot the curve of intersection of plane.

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