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The purpose of this question is to gather one/multiple statements of the divergence theorem that can cover most of the cases that one might encounter, say, in a standard PDE course (see the bottom of the post for details). The statement should thoroughly clarify the terms it uses, and should include a reference too, if possible.

The reasons for this question are two:

  1. I couldn't find neither online or on MSE a precise and general enough statement, but not so advanced it would take me hours to understand it (for instance, geometric measure theory versions with sets of finite perimeter and such)

  2. Even in the books I've read (Evans' "Partial differential equations" to mention one), only the classic case is presented ($C^1$ boundary, $C^1$ functions), but then different versions are used.

Summing up, let's remove as many regularity assumptions as possible, making the statement as general as possible, with the constraint of not ending up in something not accessible (see below). Thank you!

By "statement covering the cases one might encounter while in a standard PDE course" I mean something powerful enough to hold on an emisphere for example(I guess we are talking about Lipschitz open bounded sets), and with weak derivatives (even distributional ones are welcome, if there exists a statement with it).

Lastly, I'm comfortable with Sobolev spaces and Hausdorff measures, and I guess a measure theoretic approach is necessary. I will provide feedback to communicate when a reply is "too advanced".

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    $\begingroup$ The problem is, for distributions, divergence theorem is taken as a definition of the weak derivative (or at least in pieces anyway). With distributions on a sufficiently nice space, you don't need anything more than the theorem on classic spaces anyway. $\endgroup$
    – Ninad Munshi
    Oct 20, 2020 at 8:02
  • $\begingroup$ @NinadMunshi Weak derivatives (on open sets) are defined through integration by part: is it so easy to get to the divergence theorem? (supposing this open set is bounded and $C^1$ for instance) $\endgroup$
    – Lilla
    Oct 20, 2020 at 8:14
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    $\begingroup$ Divergence theorem is integration by parts $\endgroup$
    – Ninad Munshi
    Oct 20, 2020 at 8:15
  • $\begingroup$ Here the divergence theorem I know of: $\int_{\partial \Omega} \partial_iu dH_{n-1}=\int_{\Omega} u\nu_i dm_n$. Here is the integration by parts I know: $\int_\Omega \partial_i \phi dm_n=0$. I agree they are really similar, but proving the latter is definitely less involved then proving the former (@NinadMunshi) $\endgroup$
    – Lilla
    Oct 20, 2020 at 8:32
  • $\begingroup$ @warm_fish If the function and domain are nice enough, proving the divergence theorem is involves literally only one-dimensional partial integration and a bit of bookkeeping. If the function or the domain are not nice, it is true via definition, as the trace operator can be defined as the one operator that makes the divergence theorem work. So at this level all of them start to become somewhat interchangeable. $\endgroup$
    – mlk
    Oct 23, 2020 at 15:58

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