Chapter 16.3, Problem 13E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Find a function *f* such that **F** = ∇* f* and (b) use part (a) to evaluate ∫_{C}**F** · *d***r** along the given curve *C*.

**13. F**(*x*, *y*) = *x*^{2}*y*^{3} **i** + *x*^{3}*y*^{2} **j**,

*C*: **r**(*t*) = `⟨`

*t*^{3} – 2*t*, *t*^{3} + 2*t*`⟩`

`,`

0 ⩽ *t* ⩽ 1

**(a)**

To determine

**To find**: The potential function *f* such that

Explanation

**Given data:**

Vector field is

Consider

Write the relation between the potential function *f* and vector field

Substitute

Compare the equation

Integrate equation (1) with respect to *x*.

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

**(b)**

To determine

The value of
*C*.

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