I am trying to solve a classic problem of showing that the form $\omega = \frac{xdy - ydx}{x^2+y^2}$ on $\mathbb{R}^2 \setminus \{ (0,0) \}$ is not exact. However, I have some questions regarding the standard solutions.
Typically when I try to show that a closed form is not exact, I try to show that the integral is nonzero: $\int_{\mathbb{R}^2 \setminus \{ (0,0) \}} \omega \neq 0$. Why is it sufficient to show that $\int_{S^1} \omega \neq 0$? In other words, is it not possible for a differential form $\omega$ to have nonzero integral but a restriction to a subspace $\omega|_U$ to have an integral equal to $0$?
To compute $\int_{S^1} \omega$ can I apply Stokes' theorem via $\partial D^2 = S^1$ even though $\omega$ has a problem on $(x,y) = (0,0)$?
The Fundamental Theorem of Calculus (or the very first case of Stokes’s Theorem) tells you that if $C$ is a curve from $P$ to $Q$, then $$\int_C df = f(Q)-f(P).$$ Hence, for any closed curve $C$, $\displaystyle\int_C df =0$.