Chapter 16.4, Problem 13E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use Green’s Theorem to evaluate ∫_{C}**F** · *d***r**. (Check the orientation of the curve before applying the theorem.)

**13. F**(*x*, *y*) = ⟨*y* − cos *y*, *x* sin *y*⟩, *C* is the circle (*x* − 3)^{2} + (*y* + 4)^{2} = 4 oriented clockwise

To determine

**To evaluate**: The line integral using Green’s Theorem.

Explanation

**Given data:**

Vector field is
*C* is a circle

**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 the line integration of vector field
*C* is,

Here,

*P*,

*Q*, and

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

Write the equation of a circle.

Here,

*r* is radius.

Compare the equations

The curve *C* is a disk with center
*C* is piecewise-smooth, and simply closed curve with domain *D* and curve *C* is in transverse clockwise direction and hence

Compare the two vector fields

Find the value of

Find the value of

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