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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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}=[00]i[00]j+[cosycosy]k=[00]i[00]j+[0]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)

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Chapter 16 Solutions

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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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