Consider the vector field $F=($:$x^4,xy$:$)$ and the oriented curve $C$ that traverses the triangle with vertices $(0,0),(1,0)$ and $(0,1)$ in the counter$-$clockwise direction.

$($a$)$ Use the first form of Green's theorem to calculate the circulation $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}F-dr$ of $F$ around $C.$

$($b$)$ Use the second form of Green's theorem to calculate the flux $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*nds$ of $F$ across $C.$

Calculate the circulation of $F=(3y-e^{sinx},7x\sqrt[\phantom{2}]{y^{4}1})$ around the circle $x^2{y}^2=9$ oriented counterclockwise.

Let $F=($:$y,x^2-y$:$),$ let $R$ be the region in the first quadrant bounded by the concentric circles $x^2{y}^2=1$ and $x^2{y}^2=9,$ and the $x$ and $y-$axes. Let $C$ be the boundary curve of $R,$ oriented counterclockwise.

$($a$)$ Calculate the circulation $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*Tds.$

$($b$)$ Calculate the flux $o{\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*nds.$

Let $F=$xzi $xyzj-y^{2}k.$ Calculate the curl grad$F$ and divergence grad$*F$ at the point $(1,1,-1).$ Please answer all $4$ questions without using chatgpt