   Chapter 16.3, Problem 6E

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.6. F(x, y) = yex i + (ex + ey) 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)=yexi+(ex+ey)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)=yexi+(ex+ey)j with F(x,y)=P(x,y)i+Q(x,y)j .

P=yex (2)

Q=ex+ey (3)

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

Py=y(yex)=exy(y)=ex(1) {t(t)=1}=ex

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

Qx=x(ex+ey)=x(ex)+x(ey)=ex+0 {t(et)=eat}=ex

Substitute ex for Py and ex for Qx in equation (1),

ex=ex

Hence F(x,y)=yexyi+(1+xy)exyj 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)=yexi+(ex+ey)j

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