Chapter 16.8, Problem 11E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Use Stokes’ Theorem to evaluate ∫_{c}**F** · *d*r*,* where **F**(*x*, *y*, *z*) = *x*^{2}*z* **i** + *xy*^{2} **j** *+ z*^{2} **k** and *C* is the curve of intersection of the plane *x* + *y* + *z =* 1 and the cylinder *x*^{2} + *y*^{2} *=* 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

Explanation

**Given data:**

Consider the expression for the vector field

The curve of intersection is an ellipse in the plane

Calculate

Calculate

Consider the expression for the Stokesâ€™ 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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