Given is the surface S which is the part of the sphere $$x^2+y^2+z^2=1 $$above the XY-plane. (z>0)
$$u(x,y,z)=x^3-y^3+z^2$$ $$v(x,y,z)=x+y+z$$
Calculate the surface integral: $$ \int_S^\ (\nabla u\times\nabla v)\cdot \vec{n}\cdot d\sigma $$
Stokes theorem gives $$\int_S^\ (\nabla u\times\nabla v)\cdot \vec{n}\cdot d\sigma\ =\dfrac{3}{2}\pi$$ (with n = unity vector perpendicular to the surface facing outward)
Divergention theorem states the following:
$$\int_S^\ \vec{G}\cdot \vec{n}\cdot d\sigma\ =\int_V^\ \nabla \vec{G}\cdot dV $$
But then we get:
$$ \nabla G = \nabla(\nabla u\times \nabla v)=\vec{0}$$
$$\int_S^\ \vec{G}\cdot \vec{n}\cdot d\sigma\ =\int_V^\ \nabla \vec{G}\cdot dV = 0 $$
This is not the same result as Stokes theorem, while the conditions are met for both theorems. What is the problem here?