TITLE: What is "calculus" on a manifold?
QUESTION [19 upvotes]: I guess my question is about interpretation.  My (very rough) understanding of the way we do calculus on a smooth manifold is as follows...
A smooth manifold is a topological space without necessarily any metrical structure- therefore, if we speak of "calculus" being done on smooth manifolds, then it is a calculus on spaces which are not generally metric.  How then do we define a "derivative" for such a calculus?  (In analysis we learn that a derivative is the (limit of) the ratio of distance functions evaluated on elements of two (functionally related) metric spaces.)  And how then do we define the derivative's inverse- the integral?  Now, I know that the objects which are differentiated and integrated on smooth manifolds are not functions, but differential forms.  The theory is developed in such a way that this calculus of forms reduces - via coordinates - to good old calculus of functions on R^n endowed with the Euclidean metric.  So a metric does sneak its way into this calculus on manifolds.  What's going on here?  Would it be fair to call calculus on manifolds a non-metrical calculus or not? 
Perhaps my question is ill-defined, but here's my attempt to answer it...
The Euclidean metric only comes into play to allow us to do calculations (via the pullback).  The existence of many different (diffeomorphic) coordinate spaces in which to do these calculations means that the Euclidean metric in any one coordinate space cannot be interpreted as carrying metrical information about the underlying manifold.  Forms are the objects of calculus on a manifold precisely because the calculations we do with them (e.g. integration, exterior differentiation,...) can be defined without reference to coordinates (R^n) and are therefore invariant under coordinate transformations.  So, yes, the calculus on a manifold - which apparently is the calculus of forms - is a non-metrical calculus.  
I'm on my own here so any guidance is very much appreciated.

REPLY [3 votes]: Willie Wong has already said most of what needs to be said (though if you'd like a reference, see Lang's Differential and Riemannian Manifolds). But for a nice summary of those ideas in the context of the most general setting in which one can do calculus, you might also like to read the introduction of the book Calculus in Vector Spaces without Norm, by Frölicher and Bucher.
Let me also say that you seem to have an impoverished view of the notion of manifold. You are thinking of smooth manifolds modeled on Euclidean spaces. But manifolds can be modeled on more general spaces, such as Banach spaces and even Fréchet spaces. The resulting manifolds are called -- drumroll -- Banach manifolds and Fréchet manifolds. Since differentiation (and, in particular, the chain rule) makes sense in Banach spaces and in Fréchet spaces, we can lift the idea of differentiation to manifolds modeled on such spaces. Of course, given the greater generality, this project encounters some difficulties: it's not true that all the nice features of high school differential calculus carry over. But some semblance of the core notion of derivative does.