I am reading Evans's appendix, which includes the following Gauss-Green formula,
In the setting of Riemannian manifolds, we can consider divergence etc. and we have the abstract divergence theorem. However, for the case Evans stated, I think it is hard to consider it via the languages of differential geometry. First of all, I assume that $dS$ means the "volume" element on $\partial U$, but $\partial U$ is not necessarily a compact Riemannian manifold here, how can we refer to its volume element?
So I guess my question should be what is the right set-up for Gauss-Green theorem with respect to $C^1$ boundaries. Is any reference?