Problem: conservative and not conservative $F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right)$ I don't know how I can solve this problem: Consider $$F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right).$$ Proving that $F$ is not conservative in $\mathbb{R}^2-(0,0)$ but is conservative in the domain $$\Omega=\\{(x,y)\in\mathbb{R}^2:y>x^2\\}$$ * * * To prove that $F$ is conservative I calculated its rotational, but $\operatorname{rot}(F)=0$ so I did not prove nothing because if $\operatorname{rot}(F)=0$ then $F$ could be conservative or not conservative. So? what is what I have to do?. To prove that $F$ is conservative in that domain... I don't know... help please?

To show that $F$ is not conservative on $\mathbb{R}^2 \setminus \\{ 0 \\}$, compute the curve integral of $F$ along the unit circle. It will turn out to be non-zero, which shows $F$ can't be conservative.

For b), the domain $\Omega$ is simply connected, so the fact that $\operatorname{rot}(F) = 0$ on $\Omega$ is enough to guarantee that $F$ is convervative here. (You can also compute a potential function if you'd like.)