$(15$ points$)$ Use Green's Theorem to evaluate ${\displaystyle \underset{c}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r),$ where $c$ is the boundary of the region $R$ below, oriented counterclockwise.

$($a$)(10$ points$)$ Find the curl of vec$(F),$vec$(grad)$vec$(F),$ where vec$(F)=($:$xyz,cos(z),sin(y)$:$).$

$($b$)(4$ points$)$ Is there a potential function $$ such that grad$=$vec$(F)?$ Why or why not?