Since all components are $0,we$ conclude that curl$(F)=0$ and, therefore, $Fis$ conservative. Thus, a potential function $f(x,y,z)$ exists for which

$f_x(x,y,z)=$

This means that

$f(x,y,z)=,+g(y,z)$

Step $4$

$We$ have

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}[4x{y}^2{z}^3+g(y,z)]=,+g_y(y,z)$

Also, since $Fis$ conservative, $we$ know that

$f_y(x,y,z)=$

Therefore,

$g_y(y,z)=$