Let $\omega$ be an $n$-form on an oriented $n$-manifold $M$. To integrate $\omega$, we choose an atlas $(O_\alpha, (x^1_\alpha,\dots, x^n_\alpha))_\alpha$ for $M$ and a partition of unity $\phi_\alpha$ subordinate to the atlas. Then we write $\omega|_{O_\alpha} = f_\alpha \mathrm{d}x^1 \wedge \dots \wedge \mathrm{d}x^n$ and define $\int_M \omega = \sum_\alpha \int_{O_\alpha} \phi_\alpha f_\alpha dx^1\cdots dx^n$, where now the "d"'s represent the Lebesgue measure rather than the exterior derivative of differential forms. Then we show that the result doesn't depend on the choice of atlas or partition of unity.
Is there an alternate definition that avoids the coordinates? It seems to me that one should be able to define integration of a differential form in a coordinate-independent way and then derive the above formula as a consequence.
It's not actually the partition of unity that bugs me the most. What really puzzles me is the way we use coordinates to "magically" transform our differential form into a measure. This transformation doesn't depend on a choice of coordinates, so why should we have to use coordinates to describe it?
Best Answer
There is a coordinate free approach to integration in the book Global Calculus by Ramanan, Chapter 3. In particular, the change of variable formula is deduced from abstract nonsense in Corollary 2.9. Unfortunately, I don't really understand the abstract nonsense carried out earlier. I cannot even precisely pinpoint where the "magic" is happening. I will try to summarize my understanding of what he does.
Roughly, Ramanan constructs the sheaf of densities as a subsheaf of the sheaf of Borel measures in Def. 2.6. To me, this definition is a bit unclear, because it uses a flat homomorphism from top dimensional forms to Borel measures even though it is remarked in Rem. 2.3 that this does not in general exist.
He claims that it is obvious that one has a canonical isomorphism of the sheaf of densities with the tensor product of top dimensional differential forms and the orientation sheaf (i.e. with the usual definition of densities found for example on Wikipedia. I cannot confirm that this is obvious, but if it is obvious, this suggests that the magic is happening somewhere else.
Borel measures can be integrated by definition (by pairing with the constant unit function), so viewing densities as a subsheaf of Borel measures, one can integrate densities. On the other hand, densities on $\mathbb{R}^n$ have the form $f dx_1 \dots dx_n$. I don't know where he proves that the integral of this density agrees with the Lebesgue integral of $f$.