   Chapter 16.8, Problem 12E

Chapter
Section
Textbook Problem

(a) Use Stokes’ Theorem to evaluate ∫c F · dr, where F( x ,  y ,  z )   =   x 2 y   i   +   1 3 x 3   j   +   x y   k and C is the curve of intersection of the hyperbolic paraboloid z = y2 - x2 and the cylinder x2 + y2 = l, oriented counterclockwise as viewed from above.(b) Graph both the hyperbolic paraboloid 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)=x2yi+13x3j+xyk (1)

Consider S is the part of the surface z=y2x2 that lies above the unit disk D.

Calculate curl F by using equation (1).

curl F=|ijkxyzx2y13x3xy|=i((xy)y(13x3)z)j((xy)x(x2y)z)+k((13x3)x(x2y)y)=xiyj+(x2x2)k

curl F=xiyj (2)

Consider the expression for g(x,y) .

g(x,y)=y2x2

By using equation (2), consider the following values since the general expression is curl F=Pi+Qj+Rk

P=x

Q=y

Consider the following general expression

(b)

To determine

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

(c)

To determine

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

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