Conservative field? Let a vector field $F$ what it is defined by $F(x,y)=(\frac{-y}{(x-1)^2+y^2},\frac{x^2+y^2-x}{(x-1)^2+y^2})\  \forall \ (x,y)\epsilon\mathbb{R}^2$\ {$(1,0)$} then... is the vector field $F$ a conservative field in $\mathbb{R}^2$\ { $(1,0)$}? Why? I dont know how I can prove that, I know that if I calculate its rotational I cant say nothing, so what can I do?

If you shift $x$ by $1$ to get $G(x,y)=F(x+1,y)=(\frac{-y}{x^2+y^2},1+\frac{x}{x^2+y^2})$ then it is easy to see that $G$ is not conservative, e.g. calculate the curve integral along the unit circle to see it is not zero. Neglecting the constant component $(0,1)$ it is the electromagnetic field around $z$-axis.
