   Chapter 16.5, Problem 13E

Chapter
Section
Textbook Problem

Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.13. F(x, y, z) = y2z3 i + 2xyz3 j + 3xy2z2 k

To determine

To find: Whether the vector field F(x,y,z)=y2z3i+2xyz3j+3xy2z2k is conservative or not.

Explanation

Given:

The vector field is F(x,y,z)=y2z3i+2xyz3j+3xy2z2k (1)

Formula used:

The equation of an curl F for F=Pi+Qj+Rk is given by,

curlF=|ijkxyzPQR| (2)

Calculation:

Substitute y2z3 for P, 2xyz3 for Q and 3xy2z2 for R in equation (2) as,

curlF=|ijkxyzy2z32xyz33xy2z2|={[y(3xy2z2)z(2xyz3)]i+[z(y2z3)x(3xy2z2)]j+[x(2xyz3)y(y2z3)]k}={[(3xz2)y(y2)(2xy)z(z3)]i+[(y2)z(z3)(3y2z2)x(x)]j+[(2yz3)x(x)(z3)y(y2)]k}={[(3xz2)(2y)(2xy)(3z2)]i+[(y2)(3z2)(3y2z2)(1)]j+[(2yz3)(1)(z3)(2y)]k}

Simplify the equation.

curlF=[(6xyz2)(6xyz2)]i+[(3y2z2)(3y2z2)]j+[(2yz3)(2yz3)]k=(0)i+(0)j+(0)k=0

Since curlF=0, the F is a conservative vector and the domain of F is 3

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

Simplify the expressions in Exercises 97106. 32/331/6

Finite Mathematics and Applied Calculus (MindTap Course List)

Show that if it a b, then aa+b2b.

Single Variable Calculus: Early Transcendentals, Volume I

In Exercises 49-62, find the indicated limit, if it exists. 51. limx0x2xx

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

Evaluate the integrals in Problems 1-26. 26. (a) (b) (c) (d)

Mathematical Applications for the Management, Life, and Social Sciences

The area of the parallelogram at the right is: 26

Study Guide for Stewart's Multivariable Calculus, 8th

True or False: converges mean exists.

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