   Chapter 16.5, Problem 17E

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.17. F(x, y, z) = eyz i + xzeyz j + xyeyz k

To determine

Whether the vector field F(x,y,z)=eyzi+xzeyzj+xyeyzk is conservative or not.

Explanation

Given data:

F(x,y,z)=eyzi+xzeyzj+xyeyzk (1)

Formula used:

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

curlF=|ijkxyzPQR| (2)

Substitute eyz for P , xzeyz for Q , and xyeyz for R in equation (2),

curlF=|ijkxyzeyzxzeyzxyeyz|={[y(xyeyz)z(xzeyz)]i[x(xyeyz)z(eyz)]j+[x(xzeyz)y(eyz)]k}={[(x)y(yeyz)(x)z(zeyz)]i[(yeyz)x(x)z(eyz)]j+[(zeyz)x(x)y(eyz)]k}={{x[y(zeyz)+eyz]x[z(yeyz)+eyz]}i[(yeyz)(1)(yeyz)]j+[(zeyz)(1)(zeyz)]k} {t(uv)=ut(v)+vt(u)}

Simplify the equation.

curlF=(xyzeyz+xeyzxzyeyzxeyz)i(yeyzyeyz)j+[zeyzzeyz]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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