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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Section
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.

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

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Chapter 16 Solutions

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