From Stokes theorem, it is easy to prove the folowing proposition:
$\int\int_\vec{S}\vec{F}d\vec{S}=0$ if $\vec{F}=curl$ $\vec{G}$ for vector fields $\vec{F}$ and $\vec{G}$ and a closed parametrized surface $\vec{S}$.
Are there any important results that can be derived from this proposition?
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If you didn't know the divergence of the curl of any vector field is identically zero, then your proposition plus the Divergence Theorem would show you that this must be the case:
$$\iint_{S}\nabla\times\vec{G}\cdot d\vec{S}=0\implies \iiint_{V}\left(\nabla\cdot\nabla\times\vec{G}\right)dV=0\\ \implies \nabla\cdot\nabla\times\vec{G}=0.$$
One application that immediately springs to mind is classical electrodynamics. The magnetic field $B$ can be described as the curl of the magnetic vector potential $A$.
So, for instance, this application of Stokes' theorem tells you that the net flux of a magnetic field through a surface is always zero: There are no magnetic monopoles. In physics, this is called Gauss's law for magnetism. The differential form (obtained by integrating by parts over arbitrary surfaces) is perhaps more familiar: $$\operatorname{div} B = 0$$