   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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Finding Derivatives In Exercises 51-62, find f(x). f(x)=x2+4x1x

Calculus: An Applied Approach (MindTap Course List)

In Exercises 1520, simplify the expression. 15. 4(x2+y)3x2+y

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Using a binomial series, the Maclaurin series for is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 