$(25$ points total$)$ Consider the following vector field:

vec$(F)(x,y)=($:$2xy+4,x^2+3$:$)$

$($a$)(2$ points$)$ Compute the curl of vec$(F).$

$($b$)(2$ points$)$ Is this vector field path$-$independent? Explain.

$($c$)(2$ points$)$ If possible, find a potential function for vec$(F).$ Call this function $f(x,y).$ If it is not possible to do so$,$ explain why.

$($d$)(2$ points$)$ Without computing anything, determine the value of ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)dvec(r)$ if $C$ is the full circle of radius $4$ centered at $(2,0)$ oriented counterclockwise. Explain how you know the answer.

$($e$)(4$ points$)$ Compute gradg where $g(x,y)=x^{2}y+3y+4x+7.$

$($f$)(2$ points$)$ Is $g(x,y)$ a potential function for vec$(F)?$ Why or why not?

$($g$)(4$ points$)$ Explain the relationship between the slopes of the surfaces $f$ and $g$ at any point $(x,y)$ using what you've found in parts $($c$)$ and $($e$).$ Give the slope in the $x$ direction and the slope in the $y$ direction for each.

$($h$)(4$ points$)$ Find the tangent plane to the surface $f(x,y)$ at the point $(-2,3).$ Give a normal vector to this plane and state its relationship to the plane.

$($i$)(3$ points$)$ Find the tangent plane to the surface $g(x,y)$ at the point $(-2,3).$ Explain the geometric difference between the tangent planes to $f$ and $g.$