As far as i can understand, Stokes Theorem says that it doesn't matter which surface $S$ with boundary $C$ (closed curve) i take, the circulation of a vector field $F$ through that surface is $\int_C F \cdot dr$
But, let say i have the curve $x^2+y^2=1$ and $F(x,y,z)=(y,x,z)$ with $curl\ F=0$ so $\int_C F \cdot dr=\iint_S \ curl\ F\cdot dS=0$. This is correct for the circle $x^2+y^2\le1$ but for the paraboloid $z=1-x^2-y^2,\ \ z \ge 0$ the surface integral is $\pi/2$ and not $0$ so Stokes Theorem is not correct?
Is there something that I don't understand correctly or maybe i'm missing a condition for the theorem to work? I appreciate your time to help me.