I'm looking for a textbook covering vector calculus, but I haven't been able to find any that covers what I need in the generality in which I need it. For instance, Folland's Introduction to Partial Differential Equations references the divergence theorem for $C^1$ vector fields on domains with $C^1$ boundary. He refers to a book by Treves on linear partial differential equations for the proof. (Treves is perhaps a bit too brisk for my tastes, and I was hoping for something a bit more in-depth than a single chapter.) Grubb does the same in her Distributions and Operators, though she doesn't say where to find a proof of the theorem.
From what I can tell, the "correct" way of going about vector calculus these days is differential forms. But that's not sufficient is it? For instance, Lee does get Green's theorem as a corollary of Stokes' theorem, but only for smooth vector fields.
Am I wrong? If not, where do analysts go to learn vector calculus?