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?
• How do you get $\pi/2$ for the surface integral? $\text{curl}(F)=0$ so its surface integral through any surface must be zero. – kccu Dec 6 '17 at 22:05
• Also your first sentence is not quite correct. Stokes' Theorem says that the surface integral of $\text{curl}(F)$ (not $F$) through $S$ is equal to $\int_C F \cdot dr$. Perhaps this is the source of your confusion. – kccu Dec 6 '17 at 22:07