# Reference for the divergence theorem for embedded $C^1$-submanifolds of $\mathbb R^d$ with boundary

I'm aware of Gauss's theorem (aka the divergence theorem) for compact subsets $$K$$ of $$\mathbb R^d$$ with "$$C^1$$-boundary"$$^1$$.

I know that there are several generalizations of this theorem, but since I'm not familiar with general differential geometry, I'd like to find a reference which particularly considers the case of the manifold boundary $$\partial M$$ of a "$$d$$-dimensional embedded $$C^1$$submanifold of $$\mathbb R^d$$ with boundary"$$^2$$.

Is there a nice reference which presents and proofs this case in a rigorous way?

Please note that my main background is abstract measure/probability theory.

$$^1$$ i.e. for each $$p\in\partial K$$, there is an open neighborhood $$U$$ of $$p$$ and a $$\psi\in C^1(U)$$ with $$K\cap U=\{\psi\le0\}$$ and $$\psi'(x)\ne0$$ for all $$x\in U$$. $$\partial K$$ is denoting the topological boundary of $$K$$ and it is a $$(d-1)$$-dimensional embedded $$C^1$$-submanifold of $$\mathbb R^d$$.

$$^2$$ i.e. each point of $$M$$ is locally $$C^1$$-diffeomorphic to $$\mathbb H^d:=\mathbb R^{d-1}\times[0,\infty)$$ and the manifold boundary $$\partial M$$ is a $$(d-1)$$-dimensional embedded $$C^1$$-submanifold of $$\mathbb R^d$$ (without boundary).

What you ask is a particular case of Stokes theorem for differential forms on (Riemannian) manifolds. I think a good reference, given what you write about your main topics, is do Carmo book on differential forms

M. Do Carmo, Differential Forms and Applications, Springer Science & Business Media, 2012

But, honestly, I think that for the working mathematician there are no good sources on this topic. Unfortunately to manage these topics (80% consist of definitions and notation) one has to workout a lot of examples and these are what textbooks lack.

• There are lots of books covering Stokes theorem. My favorite is M. Spivak's "Calculus on manifolds" en.wikipedia.org/wiki/Calculus_on_Manifolds_(book).. As for "having to work out lots of examples", does not this apply to every subject? Jul 19 '20 at 15:15
• Yes, but you can find a lot of books with solved exercises on limits or derivatives. But no one with worked out exercises on these things. The truth is that the theory is elegant in the abstract setting, but as soon as one has to go down to exercises it suddenly becomes a mess. And people avoid to put examples because it requires too much effort to write down these things, compared to examples on how to compute derivatives or limits or integrals.
– Kosh
Jul 19 '20 at 15:54
• Unfortunately, Do Carmo's book is not a good reference for this result. The book uses the term "differentiable" to mean $C^{\infty}$, and assumes this of all manifolds and forms in Stokes's theorem. You can see this when he proves that $d^2=0$ for differentiable forms, which is clearly not true for $C^1$ forms. The problem is to find a reference for the $C^1$ case. I think we need to look for something else, because I think do Carmo's proof only works for $C^1$ differential forms on $C^2$ manifolds with boundary. Jul 19 '20 at 16:28
• @BenMcKay: regarding smoothness of the manifold one can work in a $C^\infty$ atlas that is compatible with the given $C^1$ atlas. Smoothness of the form is also not a problem because one can approximate $C^1$ forms by $C^\infty$ forms, so the $C^\infty$ Stokes theorem implies the $C^1$ version. Jul 19 '20 at 18:25
• @IgorBelegradek: I agree. But if the problem is to find a reference for the final result, we might prefer a reference that carries out the steps that you indicate. Those steps, when completed in full, are on their own longer than the proof that do Carmo gives. One needs to define convolution, arrive at the relevant local results and be more careful about partitions of unity than do Carmo is to patch local results. Jul 20 '20 at 15:37

Do you read German? Several German textbooks have very thorough treatment of this; for example

O. Forster, Analysis 3; Springer