I am currently taking an university analysis class, and the professor of my class proved the theorem below yesterday.
If $B \subset \mathbb{R}^2$ open ball, $F: B \rightarrow \mathbb{R}^2$ is smooth, and curl$F = 0$ in $B$, then $F = \nabla f $ for some smooth $f: B \rightarrow \mathbb{R}$.
He then said that the theorem is false for the annulus $B_{2} \setminus \overline{B_{1}}$ where both $B_{2} $ and $B_{1}$ are open balls and $B_{1} \subset B_{2}$; the example is the function $F(x,y) = (\frac{x}{x^2+y^2} , \frac{-y}{x^2+y^2})$. $F = \nabla \theta$ locally, but $\theta (x , y)$ has no global definition.
I just don't understand almost everything above. First of all, what is $\theta (x, y)$ function and so how is $F = \nabla \theta$ locally? Why doesn't it have any global definition?
$\theta$ is the angle in polar coordinates: $x = r \cos(\theta)$, $y = r \sin(\theta)$. This has no continuous global definition: if you rotate once around the origin, it changes by $2\pi$.