To apply Stokes' theorem on a vector field, one needs its rotor and the normal to the surface in question. In wikipedia this normal is defined as the cross product of the gradients of a paramterization.
On the other hand, if the surface is a fiber of some regular value of a smooth function, then its normal at a point is the gradient at the same point of the smooth function.
What's the relationship between these two normals? Can I find the first one, which I need to plug into Stokes' theorem, from only knowing the gradient of the smooth function defining the surface? Are there any shortcuts here?
The normal you get from doing any surface integral is dependent on you transformation i.e;
$$\int_S \textbf{F} \cdot d\textbf{S} = \iint_D \textbf{F}(G(u,v)) \cdot \textbf{n}(u,v) \ dudv = \iint_D \left(\textbf{F}(G(u,v)) \cdot \frac{\textbf{n}(u,v)}{\|\textbf{n}(u,v)\|}\right) \|\textbf{n}(u,v)\| \ dudv$$
Here we have that;
$$\textbf{n}(u,v) = G_u \times G_v$$