Integrals and Diffeomorphisms
Let’s say we have a diffeomorphism from one -dimensional manifold to another. Since is both smooth and has a smooth inverse, we must find that the Jacobian is always invertible; the inverse of at is at . And so — assuming is connected — the sign of the determinant must be constant. That is, is either orientation preserving or orientation-reversing.
Remembering that diffeomorphism is meant to be our idea of what it means for two smooth manifolds to be “equivalent”, this means that is either equivalent to or to . And I say that this equivalence comes out in integrals.
So further, let’s say we have a compactly-supported -form on . We can use to pull back from to . Then I say that
where the positive sign holds if is orientation-preserving and the negative if is orientation-reversing.
In fact, we just have to show the orientation-preserving side, since if is orientation-reversing from to then it’s orientation-preserving from to , and we already know that integrals over are the negatives of those over . Further, 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 corresponding singular cubes .
But now it’s easy! If is supported within the image of an orientation-preserving singular cube , then must be supported within , which is also orientation-preserving since both and are, by assumption. Then we find
In this sense we say that integrals are preserved by (orientation-preserving) diffeomorphisms.