Solution
= Solution
On the <punctured plane> $x^2+y^2>0$, write $M=-y/(x^2+y^2)$, $N=x/(x^2+y^2)$. Direct differentiation gives
$$
N_x=\frac{y^2-x^2}{(x^2+y^2)^2}=M_y.
$$
There is no $z$ dependence or $z$ component, so
$$
\boxed{\nabla\times\mathbf F=(0,0,N_x-M_y)=0\qquad(x,y)\ne(0,0).}
$$
The origin is excluded: the field is undefined there. A zero classical curl away from the origin does not state that the field is smooth on an entire disk containing it.