$$\int\int_S (\nabla \times F) \cdot n \, dS$$
Solving Stokes' theorem using Curl form, I'm having an issue with the calculation of the normal vector when using parameterization of the surface and a cross product vs. $f(x,y,z)=z-g(x,y)$ form.
Curl is given as $\langle-2z, -2x, -2y\rangle$ and C as the intersection of $z=\sqrt{5-x^2 -y^2}$ and $z=1$
Parameterizing results in $r(x,y)=\langle x, y, 1\rangle$. The cross product of the partial with respect to x and y gives a normal vector of $\langle 0,0,1 \rangle$
However if I want to use the form $z-g(x,y)$ with $z=\sqrt{5-x^2 -y^2} $ it is not evident to me how I get to the same normal vector after taking the gradient. I get $\left\langle \frac{x}{\sqrt{5-x^2 -y^2}}，\frac{y}{\sqrt{5-x^2 -y^2}}, 1 \right\rangle$
Can someone point me in the right direction?