Chapter 16.4, Problem 27E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

Use the method of Example 5 to calculate ∫_{C}**F** · *d***r**, where

and *C* is any positively oriented simple closed curve that encloses the origin.

To determine

**To find**: The line integral of vector field
*C*

Explanation

**Given data:**

The given vector field is,

**Formula used:**

**Greenâ€™s Theorem:**

Consider a positively oriented curve *C* which is piece-wise smooth, simple closed curve in plane with domain *D*. Then,

Here,

*P*,

*Q*, and

*P* and *Q* have continuous partial derivatives.

Write the expression for circle with center origin.

Here,

*r* is radius.

Refer to Figure 11 in the textbook of section 16.4 for region *D*.

Consider a curve
*a*. The radius *a* is small and lies inside the circle *C*. The region between the circles *C* and
*D* as shown in Figure 11.

The curve *C* is positively oriented, piecewise-smooth, and simply closed curve with domain *D* and hence Greenâ€™s theorem is applicable.

Compare the two expressions

Find the value of

Take common term outside in the numerator.

Find the value of

Substitute

Find the value of

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