Math 396. How to compute integralsIn the homework, you developed the theory of absolute integrability for top-degree differentialforms on smooth manifolds M with constant positive dimension n > 0, in both the oriented andnon-oriented cases (though the integration operatorRM|ω| in the non-oriented case is not linearin ω). You also proved that in the oriented case, if ω is such an absolutely integrable form then|RM,µω| ≤RM|ω|. These definitions are well-suited to theoretical considerations but not to actualcomputation, since nobody can (or wants to) write down partitions of unity. In this handout, wewish to take up a few refinements to the theory, essentially to make the task of actually computingsuch integrals be much like the case of integration for functions on Euclidean space (and even reduceto such calculations when we understand the geometry of our domain sufficiently well).For example, we certainly want to say that for n > 0 and an n-form ω on the sphere Sn(with achosen orientation),RSnω =RH+ω|H++RH−ω|H−where H±are a pair of “complementary” hemi-spheres viewed as closed smooth submanifolds with boundary in Sn(sharing a common manifold-boundary). Ignoring the equator, these hemispheres are parameterized by open unit n-balls, so ourintegration problems should shift to old-fashioned function integrals on such n-balls.Also, it should surely be the case that for any top-degree form ω ∈ ΩnM(M) on a manifold withcorners M, if ω0= ω|M−∂Mdenotes the restriction of ω over the open submanifold compleme ntaryto the singular locus then absolute integrability for ω over M and for ω0over M −∂M are equivalent,and moreover that the resulting integrals agree (in both the non-oriented and oriented senses). Lestone dismiss this as “obvious”, it does require a bit of thought because (in view of how we definedintegration of differential forms) if {φi} is a C∞partition of unity with compact supports (containedin coordinate domains) on M then {φi|M−∂M} is a C∞partition of unity on M −∂M with a locallyfinite collection of supports but these supports are generally non-compact! Hence, strictly speaking,this latter collection of functions is not the sort used in the definition of integration over M − ∂M.Briefly put, to handle these and other related matters in a straightforward manner we need torevisit how partitions of unity are used in the calculation of integrals. More specifically, since we atleast now have a general concept of integration of differential forms on M (in addition to the theoryfor functions on Euclidean space), we are now in p osition to try to use a wider class of partitionsof unity than were permitted in the initial definition of such integrals. Once we develop some moreefficient computational techniques, we will be able to prove everything that we expect to hold forany reasonable theory of integration.1. Partitions of unityLet M be a smooth manifold with corners and constant dimension n > 0. Let {φi} be acollection of non-negative smooth functions on M whose supports are locally finite and such thatPφi= 1. We do not assume that the φi’s have compact support and we do not assume that theirsupports lie in coordinate domains. It is reasonable to expect that for any n-form ω on M, ω isabsolutely integrable if and only if two conditions hold: (i) the φiω’s are absolutely integrable and(ii)PiRM|φiω| is finite. In such cases, we expectRM|ω| =PiRM|φiω|. If moreover M is oriented,the sumPiRMφiω is absolutely convergent (as it is termwise bounded above in absolute value bythe terms of the convergent sumPiRM|φiω|) and we expect that this sum should e qualRMω. (Weshall generally suppress explicit mention of the choice of orientation in our integration “withoutabsolute values”.)Note that the preceding desired properties do not just repeat the definition of integration ofdifferential forms, since we are specifically avoiding two key assumptions on the φi’s that were usedin the definition of such integrals (via partitions of unity), namely we allow that φi’s may have12non-compact support and we allow that these supports m ay fail to lie in coordinate domains. Inthe introductory discussion we saw why it was desired to allow for non-compact supports.Theorem 1.1. Let {φi} be a collection of non-negative smooth functions on M whose closed sup-ports form a locally finite collection of closed sets in M, and assumePφi= 1. For any top-degreedifferential form ω on M, ω is absolutely integrable if and only if all φiω’s are absolutely integrableandPiRM|φiω| is finite, in which case this sum is equal toRM|ω|.If M is oriented and ω is absolutely integrable thenPiRMφiω is absolutely convergent and equaltoRMω.This theorem says that the “recipe” used to initially define integration of differential forms is aposteriori applicable for the widest class of partitions of unity {φi} that one could hope for. Wecouldn’t state a result such as this prior to the initial definition of global integration of differentialforms because to make sense ofRMφiω for φi’s as general as in this theorem requires defining aconcept of absolute integrability for general differential forms in the first place! (The point beingthat φiω is possibly not supported inside of a coordinate domain.)Proof. Let {ψj} be a smooth partition of unity with locally finite and compact supports containedin coordinate domains. First we assume that ω is absolutely integrable and we seek to show that theφiω’s are absolutely integrable andPiRM|φiω| =RM|ω| (in particular, this summation is finite).Since ω is absolutely integrable,RM|ω| =PjRM|ψjω| by definition. Since each ψjis compactlysupported with support contained in a coordinate domain, ψjφiω = 0 for all but finitely many i(depending on j) andRM|ψjω| =PiRM|ψjφiω| due to finite additivity for integration of functionson opens in sectors in Euclidean space (here we also use thatPφi= 1 and all φi≥ 0). Hence,the double sumPjPiRM|ψjφiω| is convergent and equal toRM|ω|. We may rearrange it to getPiPjRM|ψjφiω| =RM|ω|. In particular, for each i the inner summation over j is convergent, but(due to the choice of the ψj’s) this is exactly the definition ofRM|φiω|. Hence, we conclude thateach φiω is absolutely integrable and thatPiRM|φiω| is convergent and equal toRM|ω|.Now suppose that each φiω is absolutely integrable withPiRM|φiω| convergent. We