## 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 is bigger than that?

Well, paradoxically, we start by getting smaller. Specifically, I say that we can always find an orientable open cover of 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 around any point we choose. Without loss of generality we can pick the coordinates such that . There must be some open ball around whose closure is completely contained within ; this closure is itself the image of a singular cube, and the ball obviously contained in its closure. Hitting everything with we get an open set — the inverse image of the ball — contained in the image of a singular cube, all of which contains . Since we can find such a set around any point we can throw them together to get an open cover of .

So, what does this buy us? If is any compactly-supported form on an -dimensional manifold , we can cover its support with some open subsets of , each of which is contained in the image of a singular -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 .

We can then find a partition of unity subordinate to this cover of . We can decompose into a (finite) sum:

which is great because now we can define

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

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

[...] 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. [...]

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[...] 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 [...]

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[...] 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 [...]

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[...] 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 [...]

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