Chapter 16.3, Problem 28E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

Let **F** = ∇* f*, where *f*(*x*, *y*) = sin(*x* − 2*y*)*.* Find curves *C*_{1} and *C*_{2} that are not closed and satisfy the equation.

(a) ∫
**F** · *d***r** = 0 (b) ∫
**F** · *d***r** = 1

**(a)**

To determine

**To find**: The curve

Explanation

**Given data:**

Vector function

**Formula used:**

Consider vector function
*C*. Consider *f* is a differentiable function two or three variables of gradient function
*C*. Then,

Here,

*a* is starting point, and

*b* is ending point.

Write the expression for vector function
*C* from

Here,

*t* is parameter.

Write the expression for vector function.

Write the expression for

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

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

**(b)**

To determine

**To find**: The curve

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