The Unapologetic Mathematician

Mathematics for the interested outsider

Integrals over Manifolds (part 2)

Okay, so we can now integrate forms as long as they’re supported within the image of an orientation-preserving singular cube. But what if the form \omega is bigger than that?

Well, paradoxically, we start by getting smaller. Specifically, I say that we can always find an orientable open cover of M such that each set in the cover is contained within the image of a singular cube.

We start with any orientable atlas, which gives us a coordinate patch (U,x) around any point p we choose. Without loss of generality we can pick the coordinates such that x(p)=0. There must be some open ball around 0 whose closure is completely contained within x(U); this closure is itself the image of a singular cube, and the ball obviously contained in its closure. Hitting everything with x^{-1} we get an open set — the inverse image of the ball — contained in the image of a singular cube, all of which contains p. Since we can find such a set around any point p\in M we can throw them together to get an open cover of M.

So, what does this buy us? If \omega is any compactly-supported n form on an n-dimensional manifold M, we can cover its support with some open subsets of M, each of which is contained in the image of a singular n-cube. In fact, since the support is compact, we only need a finite number of the open sets to do the job, and throw in however many others we need to cover the rest of M.

We can then find a partition of unity \Phi=\{\phi\} subordinate to this cover of M. We can decompose \omega into a (finite) sum:

\displaystyle\omega=\sum\limits_{\phi\in\Phi}\phi\omega

which is great because now we can define

\displaystyle\int\limits_M\omega=\sum\limits_{\phi\in\Phi}\int\limits_M\phi\omega

But now we must be careful! What if this definition depends on our choice of a suitable partition of unity? Well, say that \Psi=\{\psi\} is another such partition. Then we can write

\displaystyle\sum\limits_{\phi\in\Phi}\int\limits_M\phi\omega=\sum\limits_{\phi\in\Phi}\int\limits_M\sum\limits_{\psi\in\Psi}\psi\phi\omega=\sum\limits_{\psi\in\Psi}\int\limits_M\sum\limits_{\phi\in\Phi}\phi\psi\omega=\sum\limits_{\psi\in\Psi}\int\limits_M\psi\omega

so we get the same answer no matter which partition we use.

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September 7, 2011 - Posted by | Differential Topology, Topology

4 Comments »

  1. […] Without loss of generality, we may assume that is supported within the image of a singular cube . If not, we break it apart with a partition of unity as usual. […]

    Pingback by Switching Orientations « The Unapologetic Mathematician | September 8, 2011 | Reply

  2. […] we can assume that the support of fits within some singular cube , for if it doesn’t we can chop it up into pieces that do fit into cubes , and similarly chop up into pieces that fit within […]

    Pingback by Integrals and Diffeomorphisms « The Unapologetic Mathematician | September 12, 2011 | Reply

  3. […] is such a manifold of dimension , and if is a compactly-supported -form, then as usual we can use a partition of unity to break up the form into pieces, each of which is supported within […]

    Pingback by Stokes’ Theorem on Manifolds « The Unapologetic Mathematician | September 16, 2011 | Reply

  4. […] many such singular cubes, and the integral on each is well-defined. Using a partition of unity as usual this shows us that the integral over all of exists and, further, must be strictly positive. In […]

    Pingback by Compact Oriented Manifolds without Boundary have Nontrivial Homology « The Unapologetic Mathematician | November 24, 2011 | Reply


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