$(10$ points$)$ Let vec$(F)(x,y)=($:$(1+xy)e^{xy},x^2{e}^{xy}$:$)$ be a given force field and let

$C$:vec$(r)(t)=($:$cost,2sint$:$),0t1$

be a given path.

$($a$)(5$ points$)$ Find a potential function for vec$(F).$

$($b$)(5$ points$)$ Find ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r),$ the work done by vec$(F)$ in moving an object along $C.$