Let vec$(F)=(4z+4x^3)vec(i)+(2y+2z+2sin(y^3))vec(j)+(4x+2y+2e^{z^3})vec(k)$

$($a$)$ Find curl vec$(F).$

curl vec$(F)=$

$($b$)$ What does your answer to part $($a$)$ tell you about ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)$ where $C$ is the circle $(x-25{)}^2+(y-30{)}^2=1$ in the $xy-$plane, oriented clockwise?

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)=$

$($c$)$ If $C$ is any closed curve, what can you say about ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)?$

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)=$

$($d$)$ Now let $C$ be the half circle $(x-25{)}^2+(y-30{)}^2=1$ in the $xy-$plane with $y>30,$ traversed from $(26,30)$ to $(24,30).$ Find ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)$ by using your result from $($c$)$ and considering $C$ plus the line segment connecting the endpoints of $C.{\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)=$