Proof of Kelvin-Stokes' theorem

Question

The Kelvin-Stokes' Theorem states that for a $C^1$ vector field on a smooth surface $S$, $$\oint_{\partial S}\mathbf{F}\cdot ds=\iint_S\nabla\times\mathbf{F}\cdot d\sigma.$$ All proofs I can find (including this Wikipedia article) uses the mixed derivative theorem on the parametrization of the surface, which is not assumed in the statement of the theorem. Is there any way to prove the theorem without invoking the mixed derivative theorem?