Chapter 16.4, Problem 29E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

If **F** is the vector field of Example 5, show that ∫_{C}**F** · *d***r** = 0 for every simple closed path that does not pass through or enclose the origin.

To determine

**To show**: The value of

Explanation

**Given data:**

Field vector is

**Formula used:**

**Greenâ€™s Theorem:**

Consider a positively oriented curve *C* which is piece-wise smooth, simple closed curve in plane with region *D*. Then the line integration of vector field
*C* is,

Here,

*A* is area,

*P*,

*Q*, and

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

Consider the field as

The curve *C* is simple and it is not passing through the origin or enclosing origin. Hence, the region of curve *C* does not contain origin, but Greenâ€™s Theorem is applicable.

Compare the vector field

Apply partial differentiation with respect to *y* on both sides of equation (2).

Simplify the equation.

Apply partial differentiation with respect to *x* on both sides of equation (3)

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