Stokes theorem on $C^1$ manifolds?

I have recently encountered Stokes theorem on embedded submanifolds of $\mathbb{R}^n$, and I didn't manage to find a proof for $C^1$ vector fields over $C^1$ manifolds, infact I have only seen that proof for $C^1$ vector fields on $C^2$ manifolds.

Now, it seems to me that lots of the special cases of stokes theorem (such as the divergence theorem) can be generalized to $C^1$ manifolds and I'm starting to wondering if the general case works also for such manifolds.

Does anyone have a reference, or maybe is so kind to write a proof himself, for such a generalization of the theorem, or, conversely, knows a counterexample?

Thanks in advance and excuse me for my english, I hope the question is clear enough.