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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

Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.

18. F(x, y, z) = ex sin yz i + zex cos yz j + yex cos yz k

To determine

Whether the vector field F(x,y,z)=exsinyzi+zexcosyzj+yexcosyzk is conservative or not.

Explanation

Given data:

F(x,y,z)=exsinyzi+zexcosyzj+yexcosyzk (1)

Formula used:

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

curlF=|ijkxyzPQR| (2)

Substitute exsinyz for P , zexcosyz for Q and yexcosyz for R in equation (2),

curlF=|ijkxyzexsinyzzexcosyzyexcosyz|={[y(yexcosyz)z(zexcosyz)]i[x(yexcosyz)z(exsinyz)]j+[x(zexcosyz)y(exsinyz)]k}={[(ex)y(ycosyz)(ex)z(zcosyz)]i[(ycosyz)x(ex)(ex)z(sinyz)]j+[(zcosyz)x(ex)(ex)y(sinyz)]k}={{ex[y(zsinyz)+(cosyz)]ex[z(ysinyz)+(cosyz)]}i[ycosyz(ex)exycosyz]j+[(zcosyz)(ex)(ex)(zcosyz)]k} {t(uv)=ut(v)+vt(u)}

Simplify the equation.

curlF={(exyzsinyz+excosyz+exyzsinyzexcosyz)i[ex(ycosyz)ycosyzex]j+[(zcosyz)(ex)(ex)(zcosyz)]k}=(0)i(0)j+(0)k=0

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

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Chapter 16 Solutions

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