(adsbygoogle = window.adsbygoogle || []).push({}); 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...

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# Homework Help: Integrating on Compact Manifolds

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