   Chapter 16.5, Problem 16E

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.16. F(x, y, z) = i + sin z j + y cos z k

To determine

Whether the vector field F(x,y,z)=i+sinzj+ycoszk is conservative or not.

Explanation

Given data:

F(x,y,z)=i+sinzj+ycoszk (1)

Formula used:

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

curlF=|ijkxyzPQR| (2)

Substitute 1 for P , sinz for Q and ycosz for R in equation (2),

curlF=|ijkxyz1sinzycosz|={[y(ycosz)z(sinz)]i[x(ycosz)z(1)]j+[x(sinz)y(1)]k}={[(cosz)y(y)z(sinz)]i[(ycosz)x(1)z(1)]j+[(sinz)x(1)y(1)]k}={[(cosz)(1)(cosz)]i[(ycosz)(0)0]j+[(sinz)(0)(0)]k}

Simplify the equation.

curlF=[(cosz)(cosz)]ij+[(0)(0)]k=(0)i(0)j+(0)k=0

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

Consider f=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k

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