Zero curl but nonzero circulation

[MHD93]

The vector field $\vec{F} = <\frac{-y}{x^2 + y^2},\frac{x}{x^2 + y^2},0>$ has a zero curl, which means its circulation is zero. However

$\int \vec{F}.d\vec{s}$ around a unit circle on the xy plane is equal to $(+/-)2\pi$ and not zero

Is it because F is undefined at (0,0)? No, because Stoke's theorem allows me to choose an arbitrary surface not including the origin(0, 0, 0)?

 

[mfb]

Is it because F is undefined at (0,0)?

Right - more general, it is undefined at (0,0,z). As result, the domain of F is not simply connected - and the surface you want to consider does not exist.