Chapter 16.4, Problem 18E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

A particle starts at the origin, moves along the *x*-axis to (5, 0), then along the quarter-circle *x*^{2} + *y*^{2} = 25, *x* ⩾ 0, *y* ⩾ 0 to the point (0, 5), and then down the *y*-axis back to the origin. Use Green’s Theorem to find the work done on this particle by the force field **F**(*x*, *y*) = (sin *x*, sin *y* + *xy*^{2} +
*x*^{3}⟩.

To determine

**To find**: The work done by force field using Green’s Theorem.

Explanation

**Given data:**

Force is
*x*-axis to
*y*-axis to origin.

**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 work done by force
*C* is,

Here,

*P*,

*Q*, and

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

The curve *C* is typically a quarter-disk, which is piecewise-smooth, and simply closed curve with domain
*C* is positively oriented. Therefore, the Greenâ€™s theorem is applicable.

Consider curve parametric equations as

Compare the two vector fields

Find the value of

Find the value of

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