$($a$)$ Show that each of the vector fields vec$(F)=4$yvec$(i)+4$xvec$(j),$vec$(G)=\frac{2y}{x^2+y^2}$vec$(i)+\frac{-2x}{x^2+y^2}$vec$(j),$ and vec$(H)=\frac{2x}{\sqrt[\phantom{2}]{x^2+y^2}}$vec$(i)+\frac{2y}{\sqrt[\phantom{2}]{x^2+y^2}}$vec$(j)$ are gradient vector fields on some domain $($not necessarily the whole plane$)$ by finding a potential function for each.

For vec$(F),$ a potential function is $f(x,y)=$

For vec$(G),$ a potential function is $g(x,y)=$

For vec$(H),$ a potential function is $h(x,y)=q,$

$($b$)$ Find the line integrals of vec$(F),$vec$(G),$vec$(H)$ around the curve $C$ given to be the unit circle in the $xy-$plane, centered at the origin, and traversed counterclockwise.

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

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(G)*dvec(r)=$

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(H)*dvec(r)=$

$($c$)$ For which of the three vector fields can Green's Theorem be used to calculate the line integral in part $($b$)?$

It may be used for $$

$($Be sure that you are able to explain why or why not.$)$