   Chapter 16.3, Problem 14E

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.14. F(x, y) = (1 + xy)exy i + x2exy j,C: r(t) = cos t i + 2 sin t j, 0 ⩽ t ⩽ π/2

(a)

To determine

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

Explanation

Given data:

Vector field is F(x,y)=(1+xy)exyi+x2exyj .

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)=(1+xy)exyi+x2exyj .

fx(x,y)=(1+xy)exy (1)

fy(x,y)=x2exy (2)

Integrate equation (1) with respect to x.

f(x,y)=((1+xy)exy)dx=(1+xy)exydx(ddx(1+xy)exydx)dx {uvdt=uvdt(uvdt)dt}=(1+xy)(exyy)((0+y)(exyy))dx <

(b)

To determine

The value of Cfdr along the curve C.

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