Imagining we have a vector field $f$ for which $\operatorname{curl} f = (0,0,0)$, why can there be a potential to it on some region and not on some other. e.g. what kind of reasoning can prove that there's no potential for $$f = \left( \frac{y}{x^2+y^2}, -\frac{x}{x^2+y^2}, 0 \right)$$ on the region $$R = \{ (x,y,z) : (x,y) \neq (0,0) \}?$$ Any hint or help would be greatly appreciated, I really don't see it. Thank you!
Why does $g=\arctan\left(\dfrac{x}{y}\right)-\dfrac{x+y}{x^2+y^2}$ not work?
Hint: Integrate $f$ over the unit circle in the $xy$-plane.