   Chapter 16.3, Problem 10E

Chapter
Section
Textbook Problem

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇ f.10. F(x, y) = (ln y + y/x) i + (ln x + x/y) j

To determine

Whether F is a conservative vector field and find corresponding function f such that F=f .

Explanation

Given data:

Vector field is F(x,y)=(lny+yx)i+(lnx+xy)j .

Formula used:

Consider a vector field as F(x,y)=P(x,y)i+Q(x,y)j . The condition for vector field F being a conservative field is,

Py=Qx (1)

Here,

Py is continuous first-order partial derivative of P, and

Qx is continuous first-order partial derivative of Q,

Compare the vector field F(x,y)=(lny+yx)i+(lnx+xy)j with F(x,y)=P(x,y)i+Q(x,y)j .

P=lny+yx (2)

Q=lnx+xy (3)

Apply partial differentiation with respect to y on both sides of equation (2).

Py=y(lny+yx)=ylny+y(yx)=1y+1x(1) {t(t)=1,tlnt=1t}=1y+1x

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

Qx=x(lnx+xy)=x(lnx)+1yx(x)=1x+1y(1) {t(t)=1,t(lnt)=1t}=1y+1x

Substitute 1y+1x for Py and 1y+1x for Qx in equation (1),

1y+1x=1y+1x

Hence F(x,y)=(lny+yx)i+(lnx+xy)j is conservative vector field.

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)=(lny+yx)i+(lnx+xy)j

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