(Sorry if that isn't the formal name for it)
The vector field $F=(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$ has a curl of $0$, but when I calcuate the line integral $\int F\cdot dr$ over the unit circle, I get a value of the arclength, $2\pi$. According to Green's Theorem, if the curl is $0$, the line integral over a closed curve should also be $0$. Why is this?