Let vec$(F)=(4z+4x^3)vec(i)+(5y+3z+3sin(y^3))vec(j)+(4x+3y+5e^{z^3})vec(k).$

$($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$(F)*dvec(s)$ where $C$ is the circle $(x-15{)}^2+(y-5{)}^2=1$ in the $xy-$plane, oriented clockwise?

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

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

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

$($d$)$ Now let $C$ be the half circle $(x-15{)}^2+(y-5{)}^2=1$ in the $xy-$plane with $y>5,$ traversed from $(16,5)$ to $(14,5).$ Find ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(s)$ by using your result from $($c$)$ and considering $C$ plus the line segment connecting the endpoints of $C.$

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