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Integrating on Compact Manifolds

  1. Mar 22, 2012 #1
    1. The problem statement, all variables and given/known data
    This problem is in Analysis on Manifolds by Munkres in section 25. [itex] R [/itex] means the reals
    Suppose [itex]M \subset R^m[/itex] and [itex]N \subset R^n[/itex] be compact manifolds and let [itex]f: M \rightarrow R[/itex] and [itex]g: N \rightarrow R[/itex] be continuous functions.

    Show that [itex] \int_{M \times N} fg = [\int_M f] [ \int_N g ] [/itex]

    The hint states to consider the case where supports of f and g are contained in coordinate patches.


    2. Relevant equations
    I know the general formulation of integration over a compact manifold when the support of the function is contained in a single coordinate patch. Also, I know how to integrate functions on manifolds using partitions of unity.


    3. The attempt at a solution
    I know the general idea. First I must prove that this holds when the support of f and g are in a single coordinate patch. Then using partitions of unity, I want to notice the integral is just the sum of integrals over finitely many coordinate patches. What I can't get is the part where the support of f and g are in a single coordinate patch:

    Suppose [itex] supp(f) \subset V [/itex] with [itex] \phi : U \rightarrow V \subset M [/itex] and [itex] supp(g) \subset V' [/itex] with [itex] \psi : U' \rightarrow V' \subset N [/itex].

    Then [itex] \int_{M\times N} fg = \int_{U \times U'} (fg) \circ (\phi\times\psi) V_k(\phi \times \psi) [/itex] but I don't know how to deal with this...
     
  2. jcsd
  3. Mar 22, 2012 #2

    Dick

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    I think you just want to use Fubini's theorem. The integral of |fg| exists because f and g are continuous on compact sets, so they are uniformly continuous and bounded. Doesn't that work?
     
  4. Mar 22, 2012 #3
    The problem is that I don't know how to split up the k-volume term and I don't know how to deal with phi x psi. Unless what I've done so far is completely wrong. I've been stuck for days now and I'm just frustrated and missing something obvious, I think.
     
  5. Mar 22, 2012 #4

    Dick

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    I think you are over complicating it. If you are working on a single coordinate patch and you've applied a partition of unity then the functions you are integrating are continuous and the domains over which you are integrating have compact support. The Jacobean is differentiable. Everything is bounded. I think you can just say it's Fubini's theorem on R^m x R^n. I don't think you need to do anything complicated.
     
    Last edited: Mar 22, 2012
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