   Chapter 16.5, Problem 15E

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

To determine

Whether the vector field F(x,y,z)=zcosyi+xzsinyj+xcosyk is conservative or not.

Explanation

Given data:

F(x,y,z)=zcosyi+xzsinyj+xcosyk (1)

Formula used:

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

curlF=|ijkxyzPQR| (2)

Substitute zcosy for P , xzsiny for Q and xcosy for R in equation (2),

curlF=|ijkxyzzcosyxzsinyxcosy|={[y(xcosy)z(xzsiny)]i[x(xcosy)z(zcosy)]j+[x(xzsiny)y(zcosy)]k}={[(x)y(cosy)(xsiny)z(z)]i

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