   Chapter 16.3, Problem 12E

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.12. F(x, y) = (3 + 2xy2) i + 2x2y j,C is the arc of the hyperbola y = 1/x from (1, 1) to (4, 1 4 )

(a)

To determine

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

Explanation

Given data:

Vector field is F(x,y)=(3+2xy2)i+2x2yj .

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)=(3+2xy2)i+2x2yj .

fx(x,y)=3+2xy2 (1)

fy(x,y)=2x2y (2)

Integrate equation (1) with respect to x.

f(x,y)=(3+2xy2)dx=3dx+2y2xdx=3x+2y2(x22)+g(y) {dt=t,tdt=t22}

f(x,y)=3x+x2y2+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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