# Is the notion of density really needed to define integration on nonorientable manifolds?

I am trying to understand, in as simple terms as possible:

1. How to define integration for non-orientable manifolds, and
2. why it is impossible to do so using only differential forms.

In particular, I've seen some discussion of using "densities" instead of $n$-forms for integration, but am not really clear on why densities are required. In other words, is it really impossible to define integration on nonorientable manifolds using forms alone?

I am of course aware that any $n$-form must vanish somewhere on a nonorientable manifold, so we cannot find a volume form, hence cannot use the standard definition of integration. I think the reason I'm not finding this answer satisfying is that it is a bit tautological: we can't define integration with respect to volume forms because there are no volume forms. But why must we define integration with respect to a (global) volume form in the first place? Is there really no other way to do it using locally-defined forms? Thinking of a manifold as a collection of local charts is common in geometry, and I'm having trouble understanding why this approach doesn't work in the case of integration.

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If you are defining an orientation as a class of $n$-forms for the equivalence relation given by multiplication by positive functions, then there are waaaay too many orientations. – Mariano Suárez-Alvarez Oct 5 '10 at 18:19
Why do regard this positive number you get as a "volume"? – Robin Chapman Oct 5 '10 at 18:39
I didn't understand how you're going to do this procedure: "We can then use our partition of unity to "sum up" these positive values over the manifold to get a positive total volume." – Ronaldo Oct 6 '10 at 2:18
The reason why people define integration with respect to forms is because they want a situation where you can generalize the fundamental theorem of calculus -- which is implicitly an oriented concept, as the interval has an initial point and a terminal point. That generalization is Stokes' theorem. There are of course all kinds of other notions of integration and they're all perfectly fine. But you use forms when you want to integrate with respect to oriented volumes, not just plain old measures. – Ryan Budney Oct 6 '10 at 17:33
Reading your question again, I notice you don't specify what you want to integrate. If you're interested in integrating real-valued functions then densities are precisely what you need. But if you're happy integrating other things (like differential forms) then differential forms are all you need. – Ryan Budney Apr 21 '11 at 23:30