Chapter 16.8, Problem 12E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Use Stokes’ Theorem to evaluate ∫_{c}**F** · *d*r*,* where
*C* is the curve of intersection of the hyperbolic paraboloid *z = y*^{2}* - x*^{2} and the cylinder *x*^{2} + *y*^{2} = 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

Explanation

**Given data:**

Consider the expression for the vector field

Consider *S* is the part of the surface
*D*.

Calculate

Consider the expression for

By using equation (2), consider the following values since the general expression is

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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