   Chapter 16.3, Problem 18E

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.18. F(x, y, z) = sin y i + (x cos y + cos z) j − y sin z k,C: r(t) = sin t i + t j + 2t k, 0 ⩽ t ⩽ π/2

(a)

To determine

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

Explanation

Given data:

Vector field is F(x,y,z)=sinyi+(xcosy+cosz)jysinzk .

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)=sinyi+(xcosy+cosz)jysinzk .

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

fy(x,y,z)=xcosy+cosz (2)

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

Integrate equation (1) with respect to x.

f(x,y,z)=(siny)dx=sinydx=siny(x)+g(y,z) {dt=t}

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

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

fy(x,y,z)=y(xsiny+g(y,z))=y(xsiny)+y(g(y,z))=xy(siny)+gy(y,z)=x(cosy)+gy(y,z) {t(sint)=1}

fy(x,y,z)=xcosy+gy(y,z) (5)

Compare the equations (2) and (5)

(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

In Exercises 14. find the values of x that satisfy the inequality (inequalities). 3. x 3 2 or x + 3 1

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

Find for x = 3t2 + 1, y = t6 + 6t5. t4 + 5t3 4t3 + 15t2

Study Guide for Stewart's Multivariable Calculus, 8th

True or False: If a function is not increasing on an interval, then it is decreasing on the interval.

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