Find the circulation of $F = \frac{-y}{x^2+y^2}i + \frac{x}{x^2+y^2}j$ along the unit circle.
I decided to express $F$ in cylindrical coordinates: $$F = \frac{1}{r}{\hat{\theta}}.$$ However, I found $\nabla \times F = 0$ So, I assume that would imply, $$\oint F\cdot dr = \iint_S (\nabla \times F)\cdot n\,dA = \iint_S\vec{0} \cdot n\,dA = 0.$$ Later, I attempted to solve this by parametizing F and the path along the circle and found a non-zero answer.
Why does the double integral yield a different answer?
This contradiction is pretty subtle and is usually used as a learning exercise for exactly this reason. You are correct that $\nabla\times F=0$, but we have two contradictory statements: $$\oint F\cdot dr = 2\pi, \text{ and} \\ \iint_{R}\nabla\times FdA=0.$$
Green's Theorem says that these two integrals should be equal, but a careful look at the assumptions for Green's Theorem will show that $F$ must be defined over the entire interior of the curve, even if the curl is identically $0$ everywhere.