# Divergence theorem for non-compact manifolds

I wonder whether there is a generalization of the divergence theorem or more generally of Stokes' theorem to non-compact domains or manifolds, much like the improper Riemann integrals. Consider the function $$f(x, y) = \frac{1}{x^2 y^2}$$ integrated over the domain $$D = [1, \infty)^2$$. This can be written as a nested improper Riemann integral and turns out as $$1$$. Now this did not really use the divergence theorem, but consider a similar case for the divergence theorem where all integrals would be finite since the integrands decay quickly enough at infinity.

If I understand it correctly, the generalized Stokes' theorem (and as a special case also the divergence theorem) still apply if the domain is not fully bounded in all directions as long as the integrand has compact support. But $$\frac{1}{x^2 y^2}$$ has no compact support in $$D$$.

First question: Is there a generalization relaxing the compact support requirement? Such that it is enough that all integrals exist?

So far I found (Karp - On Stoke's Theorem For Noncompact Manifolds, https://www.jstor.org/stable/2043967), but that requires complete manifolds. (Edit: It was pointed out that $$D$$ is cauchy complete. Although the article does not explicitly state it, I assume it refers to geodesical completeness, see https://en.wikipedia.org/wiki/Geodesic_manifold. This assumption makes sense for the article as it deals with geodesic balls and grows them towards infinity. For clearness I'll add "geodesically" to every occurrence of "complete" in the following.) $$[1, \infty)^2$$ is not geodesically complete, right? It appears to me that something similar should hold if the manifold or the domain is geodesically complete in some directions but bounded in other directions, possibly in more complicated manner than $$[1, \infty)^2$$. E.g. consider a domain like $$\{(x, y) \in \mathbb{R}^2 \colon y > x^2\}$$.

My idea so far: If we had a nested sequence of compact domains $$C_0 \subset C_1 \subset \ldots \subset D$$ such that $$\lim\limits_{j \to \infty} C_j = D$$, one can apply Stokes' Theorem to $$f|_{C_j}$$ and observe if the result converges as $$j \to \infty$$. This is much like the technique behind improper Riemann integrals, but growing a compact domain instead of an interval towards an improper boundary part. Here it is probably helpful that "in good cases" (https://en.wikipedia.org/wiki/Support_(mathematics)#Compact_support) compactly supported functions lie densely in the set of functions that vanish at infinity.

Second question: This construction is so simple it must be well known. I kindly ask for references where this is studied, material about this. Is there some pitfall that prevents us using it to apply Stokes' or the divergence theorem to manifolds or domains that are not compact or possibly neither compact nor geodesically complete? (Otherwise, why is it so hard to find a formulation for cases that are neither compact nor geodesically complete?)

Third question: I am not much interested in pathological cases and assume that the integrand has no singularities. For my usecase, it can be assumed to be strictly positive, possibly zero at the boundary or at infinity. However, there certainly are pathological cases and it would be good to know conditions (on $$f$$ and $$D$$) to exclude them. For $$D$$ this should be somewhat weaker than geodesically complete or compact.

Note: There are a couple of similar questions but I did not find a satisfying(ly answered) one. One of the best answers I found is in Requirements for integration by parts/ Divergence theorem, however it does not cover manifolds. My examples of non-manifold domains are illustrative, I am interested in a (Riemannian) manifold version.

Also this one has an answer but also for a subset of $$\mathbb{R}^n$$ and required some justification in the comments: Divergence Theorem/Integration by Parts on Unbounded Domains

These questions do not ask the same as my question although the title suggests some overlap: Divergence theorem on special unbounded domains Stokes theorem for non-compact case Domains for which the divergence theorem holds

Sorry if I overlooked an actual duplicate.

• Hmmm, I assumed "complete manifolds" refers to geodesically complete (en.wikipedia.org/wiki/Geodesic_manifold). $[1, \infty)^2$ is not geodesically complete, right? I will have a closer look at the article to see what he means by "complete". Apr 27 '20 at 0:37
• The case of non-compact support is discussed, albeit briefly, in Lang's Real Analysis in the section called Stokes' theorem on a manifold. Apr 27 '20 at 0:57
• For the record: My comment above is a reply to a deleted comment that pointed out that there is also cauchy-completeness, and $[1, \infty)^2$ is cauchy complete. Indeed the article I linked does not specify what is meant by "complete". I still believe it's "geodesically complete" because it is working with geodesic balls, growing them to infinity for limit operations. Apr 27 '20 at 13:07
• @TedShifrin Thanks for the reference! That is indeed helpful. Apr 27 '20 at 13:08