   Chapter 16.3, Problem 13E

Chapter
Section
Textbook Problem

(a) Find a function f such that F = ∇ f and (b) use part (a) to evaluate ∫C F · dr along the given curve C.13. F(x, y) = x2y3 i + x3y2 j,C: r(t) = ⟨t3 – 2t, t3 + 2t⟩, 0 ⩽ t ⩽ 1

(a)

To determine

To find: The potential function f such that F=f .

Explanation

Given data:

Vector field is F(x,y)=x2y3i+x3y2j .

Consider f=fx(x,y)i+fy(x,y)j .

Write the relation between the potential function f and vector field F .

f=F

Substitute fx(x,y)i+fy(x,y)j for f ,

F=fx(x,y)i+fy(x,y)j

Compare the equation F=fx(x,y)i+fy(x,y)j with F(x,y)=x2y3i+x3y2j .

fx(x,y)=x2y3 (1)

fy(x,y)=x3y2 (2)

Integrate equation (1) with respect to x.

f(x,y)=(x2y3)dx=y3x2dx=y3(x33)+g(y) {t2dt=t33}

f(x,y)=13x3y3+g(y) (3)

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

(b)

To determine

The value of Cfdr along the curve C.

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