I have the following problem and I'm not able to solve it. Given $F(x,y,z) = (1-2z, 0, 2y)$, calculate the line integral of C, where C is the contour of the surface. $S=\{(x,y,z) / x\geq 0, y \geq 0, z\geq 0, z=1-x^2 - y^2\}$
Using Stokes Theorem:
$$\int _S \nabla x F\cdot dS = \int_C F\cdot dS $$
I have calculated the Curl $\nabla x F = (2,-2,0)$
I also parametrized the surface with $T(u,v)=(u,v,1-u^2-v^2)$
So my normal vector is $T_u x T_v=(2u,2v,1)$
$$\int_CF\cdot dS=\int _S \nabla x F\cdot dS =\int_0^1\int_0^1 (4u-4v)dudv=0$$
But C is a closed curve!! So if the line integral over a closed curve is cero, by the conservative vector field theorem, the curl of the field should ALSO be cero!! WTF, is going on? PLEASE HELP.