How to determine whether a closed form is an exact form？

Question

Let $M=R^2-(0.0)$, $\omega=\frac{xdx+ydy}{x^2+y^2}$, how does one prove $\omega$ is an exact form？
I have found a function $f=\frac{1}{2}\ln（x^2+y^2）$ such that $df=\omega$, but it is not sufficient to prove that $\omega$ is an exact form, because of the following counter example:
let $M=R^2-(0.0)$, $\eta=\frac{ydx-xdy}{x^2+y^2}$, we can also find a function $g=\arctan\frac{x}{y}$, such that $dg=\eta$ but it is well known that $\eta$ is not an exact form on $M$.
So my question is:
What is the exact condition to determine whether a closed form is an exact form?

Answer 1

The function $g$ is not defined on $M$. Just take a look at the line $x>0,y=0$.