Let $\vec F = \left(3z+3x^{4}\right)\,\mathit{\vec i}+\left(5y+3z+3\sin\!\left(y^{4}\right)\right)\,\mathit{\vec j}+\left(3x+3y+5e^{z^{4}}\right)\,\mathit{\vec k}$.

(a) Find $\mbox{curl }\vec F$.
$\mbox{curl }\vec F =$

(b) What does your answer to part (a) tell you about $\int_C\vec F\cdot d\vec r$ where $C$ is the circle $(x-35)^2+(y-20)^2=1$ in the $xy$-plane, oriented clockwise?
$\int_C\vec F\cdot d\vec r =$

(c) If $C$ is any closed curve, what can you say about $\int_C \vec F\cdot d\vec r$?
$\int_C \vec F\cdot d\vec r =$

(d) Now let $C$ be the half circle $(x-35)^2+(y-20)^2=1$ in the $xy$-plane with $y > 20$, traversed from $(36, 20)$ to $(34, 20)$. Find $\int_C \vec F\cdot d\vec r$ by using your result from (c) and considering $C$ plus the line segment connecting the endpoints of $C$.
$\int_C \vec F\cdot d\vec r =$