Chapter 16, Problem 33RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use Stokes’ Theorem to evaluate ∫_{C}**F** · *d***r**, where **F**(*x*, *y*, *z*) *= xy* **i** + *yz* **j** + *zx* **k**, and *C* is the triangle with vertices (1, 0. 0), (0, 1, 0), and (0, 0, 1), oriented counterclockwise as viewed from above.

To determine

**To evaluate:** The value of

Explanation

**Given data:**

The field is
*C* is triangle with vertices are

**Formula used:**

Write the expression for curl of

Write the expression for the Stokesâ€™ theorem.

Here,

*S* is surface.

Consider surface *S*,
*S*.

Here,

*A* is area.

Consider the general expression for field as follows.

Consider the field expression as follows.

Compare equations (4) and (5).

Find the value of

Consider the surface *S* is a planar region which is enclosed by curve *C*. The expression for surface *S*,

Hence the equation is in the form of

The curve *C* is oriented in counter-clockwise, so the orient the surface *S* upward

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