## Calculus 8th Edition

$f(x,y,z)=xy^2z^3+k$
The vector field $F$ is conservative when $curl F=0$ When $F=ai+bj+ck$, then we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$ Now, $curl F=(6x yz^2 -6xyz^2)+(3y^2z^2-3y^2z^2)+(2yz^3-2yz^3)=0$ This shows that the vector field $F$ is conservative. Consider $f(x,y,z)=xy^2z^3+g(y,z)$ $g'(y)=0$ and $F_y=2xyz^3$ Further, $f(x,y,z)=xy^2z^3+h(z)$ This implies that $h'(z)=0$ and $F_z=3xy^2z^2$ Hence, $f(x,y,z)=xy^2z^3+k$