$(25$ points$)$ Consider the vector field $F(x,y)=($:$4x^3-12ycos(xy),9y^2-12xcos(xy)$:$).$

$($a$)(5$ points$)$ Check to see if $F$ satisfies the Cross$-$Partial Test for conservative fields.

$($b$)(15$ points$)$ Find a potential function $f$ for the vector field $F.$ Check your work by finding gradf.

$($c$)(5$ points$)$ Using an appropriate method, find ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr$ where $C$ is the quarter circle of radius $2$ from $(-2,0)$ to $(0,2).$