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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?

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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;

$$\textbf{n}(u,v) = G_u \times G_v$$

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    How is this normal related to the gradient? – 2017-02-06
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    Here $G: D \subset \mathbb{R}^2 \to S$ and so we have; $$\nabla G = \langle G_u, G_v \rangle$$ Let $p$ be the image of some point $(x_0,y_0)$ in $D$ under $G$. Then $\{G_v(p), G_u(p)\}$ span the tangent plane at $p$ i.e $G_u \times G_v$ is a normal vector at $p$. – 2017-02-06