There is an example on DoCarmo (Differential Forms and Applications) that they show that $$ \omega = \frac{-y}{x^2+y^2} dx + \frac{x}{x^2+y^2} dy $$ is locally exact but not in their entire domain ($\mathbb{R}-\{ 0\} $).
I can see that in the path $(cos(t),sin(t)), t \in [0,2\pi]$ the integral is $2\pi$ so is not exact, but I don't understand why that form is locally exact. They show that is close (using that $d \omega = 0$). But why that implies that $\omega$ is exact?
Help with this please.