suppose that we have a function $F(x,y)=(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$ if we calculate $curlF$ we understand that $curlF=0$ ( computing $curlF$ is not my question). now the book has written that because the domain is $\mathbb{R}^2 - \{(0,0)\}$ so we can't conclude that $F$ is conservative. but F is path independent.
first question: so I concluded that if $curlF=0$ and $F$ is simply connected then $F$ is conservative but if $curlF=0$ and $F$ is not simply connected (like this example) then $F$ is path-independent. is that correct?
second question: what is the difference between conservative and path-independent?