Let vec$(F)=(7z+7x^2)vec(i)+(2y+5z+5sin(y^2))vec(j)+(7x+5y+2e^{z^2})vec(k).$

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

curlvec$(F)=$

$($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 $(x-30{)}^2+(y-10{)}^2=1$ 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 $(x-30{)}^2+(y-10{)}^2=1$ in the $xy-$plane with $y>10,$ traversed from $(31,10)$ to $(29,10).$ 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$(F)*dvec(r)=$

$$