$($Sections $7\mathrm{.}10,8\mathrm{.}2,8\mathrm{.}3$ and $8\mathrm{.}4)$ Consider the $3-$dimensional vector field $F_{?}$ defined by

$F_{x,y,z}=(18x^2{y}^3+2x,18x^3{y}^2+4y{z}^3+2y,6y^2{z}^2-4z).$

$($e$)$ Find a potential function for F$,$ using the method of Example $7\mathrm{.}9\mathrm{.}1$ Note, however, that the example

concerns a $2-$dimensional vector field, so you will have to adapt the method to be suitable for

a $3-$dimensional vector field. Pay special attention to the notation that you use for derivatives

of functions of more than one variable.