Chapter 16.3, Problem 7E

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.7. F(x, y) = (yex + sin y) i + (ex + x cos 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)=(yex+siny)i+(ex+xcosy)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)=(yex+siny)i+(ex+xcosy)j with F(x,y)=P(x,y)i+Q(x,y)j .

P=yex+siny (2)

Q=ex+xcosy (3)

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

Py=y(yex+siny)=exyy+ysiny=ex(1)+(cosy) {t(t)=1,tsint=cost}=ex+cosy

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

Qx=x(ex+xcosy)=x(ex)+cosyx(x)=ex+cosy(1) {t(et)=eat}=ex+cosy

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

ex+cosy=ex+cosy

Hence F(x,y)=(yex+siny)i+(ex+xcosy)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)=(yex+siny)i+(ex+xcosy)j

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