I'll try to summarize what I'm having problem to understand without confusion.
In vector analysis right now I learned about 4 equivalent theorems:
"Let $F$ be a vector field $C^1$ defined in a simply connected domain $U \subset \mathbb{R}^2$. The following conditions are equivalent:
(i) $\oint F.dr=0$ for every closed loop $C$ ($C^1)$ inside $U$.
(ii) The line integral of $F$ from $A$ to $B$ is independent of the curve $C$ ($C^1$) inside $U$.
(iii) $F$ is conservative.
(iv) $\frac{\partial F_2}{\partial x}=\frac{\partial F_1}{\partial y}$ in $U$.
There are two problems I'm facing:
(1) What is $U$?
Is $U$ the region where F is defined? Therefore $U$ is simply the domain of $F$? Or $U$ is a region and the domain of $F$ is another region inside $U$ (or not)?
Why I ask this? There was an exercise where it is asked for all possible values of $F$ and the domain of $F$ didn't include the origin $(0,0)$. In the answer it was stated that there was two possible scenarios: "$U$ is a simply connected region that contains $(0,0)$ or "$U$ is a simply connected region that doesn't contains $(0,0)$."
If $U$ is the domain of $F$ there shouldn't be two possibilities, $U$ just doesn't contain $(0,0)$. So I start to suspect that $U$ is another region that has nothing to do with the domain of $F$. Then I decided to ask my professor, and he said that $U$ is the domain of $F$. But he was in a bit of a hurry so I still don't know exactly the answer.
$U$ is is a part of $\mathbb{R}^2$ with no holes (link). Otherwise your closed path integrals get complicated.
On $U$ for your vector field $F$ the properties $(i)$ to $(iv)$ are equivalent.
I would need to see your exercise text to get an idea what was meant.