# Why functions can't be integrated on manifolds

I'm trying to teach myself about differential forms, and my book says that functions can't be integrated on manifolds because the integral isn't coordinate independent, but if the manifold has dimension $n$, you can integrated $n$ forms in a coordinate independent way.

The example given is simple, but I'm struggling with it. when the manifold is just $(\mathbb{R},1_{i.d.})$. Let $x$ and $y=Cx$ be the coordinate, and coordinate change, and $f(x)=1$. Then $\int_a^bf(x)dx=\int_a^bdx=\int_{Ca}^{Cb}\frac{dx}{dy}dy=(b-a)\neq C(b-a)=\int_{Ca}^{Cb} f(y)dy$. So I guess you would say that the Riemann integral is not coordinate invariant. But the claim is that you can integrate one-forms on a one dimensional manifold in a coordinate independent way. But $f(x)dx$ is a one-form, so I'm a little confused.

When considering coordinate changes, one-forms and functions are different things, as you transform them differently when changing coordinates. The Riemann integral of a one-form does not change after a coordinate change in this context, which is what allows classical "u-substitution", but it does change if you try to transform the one-form $f(x)\,dx$ as the function $f(x)$.
Explicitly, the one-form $f(x)\,dx$ should be pulled back along the transformation $x=a(u)$ as $f(x)\,dx \mapsto f(a(u)) a'(u)\, du$, in correspondence with the chain rule, rather than simply $f(a(u)) du$.
• I probably don't understand what is meant by coordinate independent. Correct me if I'm wrong, but it seems like you're saying it means if you transform $f(x)dx$ and the limits of integration to the new coordinates, nothing changes. I was thinking it meant you have an integral $\int f(x)dx$ and if you formally just substitute $x$ for $y$, you get the same value for $\int f(y)dy$ Oct 13, 2015 at 6:36
• I mean to say, $\int f(x)dx\to\int f(y)dy$ would be considered changing coordinates of the function, $\int f(y(x))y'(x)dx\to\int f(y)dy$ would be considered changng coordinates of the one form? Oct 13, 2015 at 6:41