   Chapter 16.5, Problem 14E

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.14. F(x, y, z) = xyz4 i + x2z4 j + 4x2yz3 k

To determine

Whether the vector field F(x,y,z)=xyz4i+x2z4j+4x2yz3k is conservative or not.

Explanation

Given data:

F(x,y,z)=xyz4i+x2z4j+4x2yz3k (1)

Formula used:

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

curlF=|ijkxyzPQR| (2)

Substitute xyz4 for P , x2z4 for Q and 4x2yz3 for R in equation (2),

curlF=|ijkxyzxyz4x2z44x2yz3|={[y(4x2yz3)z(x2z4)]i+[z(xyz4)x(4x2yz3)]j+[x(x2z4)y(xyz4)]k}={[(4x2z3)y(y)(x2)z(z4)]i+[(x

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