   Chapter 16.3, Problem 17E

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.17. F(x, y, z) = yzexz i + exz j + xyexz k,C: r(t) = (t2 + 1) i + (t2 − 1) j + (t2 – 2t) k, 0 ⩽ t ⩽ 2

(a)

To determine

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

Explanation

Given data:

Vector field is F(x,y,z)=yzexzi+exzj+xyexzk .

Consider f=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k .

Write the relation between the potential function f and vector field F .

f=F

Substitute fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k for f ,

F=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k

Compare the equation F=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k with F(x,y,z)=yzexzi+exzj+xyexzk .

fx(x,y,z)=yzexz (1)

fy(x,y,z)=exz (2)

fz(x,y,z)=xyexz (3)

Integrate equation (1) with respect to x.

f(x,y,z)=(yzexz)dx=yzexzdx=yz(exzz)+g(y,z) {eatdt=eata}

f(x,y,z)=yexz+g(y,z) (4)

Apply partial differentiation with respect to y on both sides of equation (4)

(b)

To determine

The value of Cfdr along the curve C.

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

Show that sin x x if 0 x 2.

Single Variable Calculus: Early Transcendentals, Volume I

6x2(3x2+2)+14(x+2)

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

Sketch the parabola that has directrix d and focus F.

Elementary Geometry For College Students, 7e

Find the limit. limx12x2x45+x3x4

Single Variable Calculus: Early Transcendentals

True or False: The graph of x = 5 is a cylinder.

Study Guide for Stewart's Multivariable Calculus, 8th 