(a) Show that each of the vector fields $\vec F = 2y\,\vec i + 2x\,\vec j$, $\vec G = \frac{3 y}{x^2 + y^2}\,\vec i + \frac{-3 x}{x^2 + y^2}\,\vec j$, and $\vec H = \frac{ x}{\sqrt{x^2 + y^2}}\,\vec i + \frac{ y}{\sqrt{x^2 + y^2}}\,\vec j$ 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 $h(x,y) =$

(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.
$\int_C\vec F\cdot d\vec r =$
$\int_C\vec G\cdot d\vec r =$
$\int_C\vec H\cdot d\vec r =$

(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 ? only F only G only H F and G F and H G and H F, G and H
(Be sure that you are able to explain why or why not.)