Let vec$(F)=(2z+2x^3)vec(i)+(7y+2z+2sin(y^3))vec(j)+(2x+2y+7e^{z^3})vec(k).$

$($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 $(x-10{)}^2+(y-25{)}^2=1$ 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 $(x-10{)}^2+(y-25{)}^2=1$ in the $xy-$plane with $y>25,$ traversed from $(11,25)$ to $(9,25).$ 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$(F)*dvec(r)=$

$$