   Chapter 16, Problem 12RE

Chapter
Section
Textbook Problem

Show that F is a conservative vector field. Then find a function f such that F = ∇f.12. F(x,y,z) = sin y i + x cos y j − sin z k

To determine

To show: F is a conservative vector field and function f such that F=f .

Explanation

Given data:

F(x,y,z)=sinyi+xcosyjsinzk (1)

Formula used:

Consider the standard equation of an curl F for F=Pi+Qj+Rk

curlF=|ijkxyzPQR| (2)

Find the value of curlF .

Substitute siny for P , xcosy for Q and sinz for R in equation (2),

curlF=|ijkxyzsinyxcosysinz|={[y(sinz)z(xcosy)]i[x(sinz)zsiny]j+[x(xcosy)ysiny]k}=ij+[cosycosy]k=ij+k

Simplify expression as follows.

curlF=0

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

Thus, F is a conservative vector field is shown.

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 (3)

Compare the equation (3) and equation (1)

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

Find more solutions based on key concepts 