Let vec$($F$)=(2$z$+2$x$^(3))$vec$($i$)+(5$y$+3$z$+3$sin$($y$^(3)))$vec$($j$)+(2$x$+3$y$+5$e$^($z$^(3)))$vec$($k$).$

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

curlvec$($F$)=$

$($b$)$ What does your answer to part $($a$)$ tell you about $\backslash$int$\_$C vec$($F$)*$dvec$($r$)$ where C is the circle $($x$-15)^(2)+($y$-30)^(2)=1$ in the xy$-$plane, oriented

clockwise?

$\backslash$int$\_$C vec$($F$)*$dvec$($r$)=$

$($c$)$ If C is any closed curve, what can you say about $\backslash$int$\_$C vec$($F$)*$dvec$($r$)?$

$\backslash$int$\_$C vec$($F$)*$dvec$($r$)=$

$($d$)$ Now let C be the half circle $($x$-15)^(2)+($y$-30)^(2)=1$ in the xy$-$plane with y$>30,$ traversed from $(16,30)$ to $(14,30).$ Find $\backslash$int$\_$C vec$($F$)*$dvec$($r$)$ by

using your result from $($c$)$ and considering C plus the line segment connecting the endpoints of C$.$

$\backslash$int$\_$C vec$($F$)*$dvec$($r$)=$