I want to consider $C^k$ manifolds because, in PDE, people want to consider $C^k$ boundaries etc, and a version of divergence theorem for $C^k$ manifold is needed. Is it necessary to repeat all the proofs for $C^k$ manifolds? How to think of it in terms of the theory of smooth manifolds?
I think there can be a lot of fundamental differences between the two. For starters, the tangent space of a $C^k$ manifold is not of finite dimension if we use derivation to define it. Also, I am not sure whether there is a $C^k$ partition of unity for $C^k$ a manifold. So that makes me wonder whether it is a good idea to think of $C^k$ manifolds in terms of smooth ones, then just skip all the proofs.
Can someone familiar with PDE give me some hints on how people think of divergence theorems for $C^k$ boundaries? I know that there are ways that you can derive the divergence theorem without defining tangent spaces etc. but ignoring the similarity between it and the theory of smooth manifolds makes me feel so uncomfortable.