Show that $\displaystyle F=(\displaystyle \frac{x}{x^2+y^2},\displaystyle \frac{y}{x^2+y^2})$ and $\displaystyle G=(\displaystyle \frac{y}{x^2+y^2},\displaystyle \frac{x}{x^2+y^2})$ are such that one of $\displaystyle F.dP$, $\displaystyle G.dP$ is exact but the other is not.

(dP=(dx,dy)).

Let C be the shorter arc of $\displaystyle x^2+y^2=1$ from $\displaystyle (1/\sqrt{2},-1/\sqrt{2})$ to $\displaystyle (1/\sqrt{2},1/\sqrt{2})$. Evaluate $\displaystyle \int _c\,\!F.dP and \int _c\,\!G.dP$.