TITLE: Why do we care about differential forms? (Confused over construction)
QUESTION [28 upvotes]: So it's said that differential forms provide a coordinate free approach to multivariable calculus. Well, in short I just don't get this, despite reading from many sources. I shall explain how it all looks to me. 
Let's just stick to $\mathbb{R}^2$ for the sake of simplicity (maybe this is the down fall..). Picking some point $P=(x,y)\in\mathbb{R}^2$, we could ask about the directional derivative of some function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, in direction $v=a\vec{i}+b\vec{j}$.  This will be $$(\nabla \cdot v)|_P(f)=a\dfrac{\partial f}{\partial x}|_P+b\dfrac{\partial f}{\partial y}|_P =\underbrace{ (a\dfrac{\partial }{\partial x}|_P+b\dfrac{\partial }{\partial y}|_P)}_\text{$w_P$ }(f)$$ Where we can think of $w_P$ as an element of the tangent space at $P$. Now this in itself is a little weird; why have differential operators as a basis for something geometrical like a tangent space to a manifold? In any case, we apply these vectors to a function defined on our manifold, and we get the value we wanted out. 
So who cares about differential forms? We just did all this without them. We could've done this by calculating $\mathrm{d}f$, in some basis $\mathrm{d}x, \mathrm{d}y$ (which is quite confusing), and then calculating $\mathrm{d}f(w_P)$, but what do we gain in doing it this way? 
I mentioned I think the $\mathrm{d}x$'s are confusing. Well, $\mathrm{d}x$ is just the function $\mathrm{d}x(\frac{\partial}{\partial x})=1$ and 0 for any other differential operator -  why write this as $\mathrm{d}x$, which had always previously been used to mean an infinitesimal increase in x? 
Now I can understand caring about the dual of the tangent space. We are combining a vector in the tangent space with something and we're getting a scalar out of it - this something should then probably belong to the dual space. But if we're thinking of just the vector, then the function $f$ on the manifold needs to be encoded by the 1-form, right? Well, we can have 1-forms which aren't derivatives of any function on the manifold - what should it mean to combine such forms with tangent vectors?
And lastly, if we're writing all our forms in terms of $\mathrm{d}x$'s etc., where the $x$'s are exactly the coordinates of the manifold, then how exactly have we escaped dependence on coordinates? We're still essentially calculating with respect to a given coordinate system as in usual multivariable calculus!

REPLY [3 votes]: Differential forms are an appropriate generalisation of derivation and integration (i.e of calculus) for arbitary (within the scope of integration) manifolds and spaces.  
In order to arrive at such a generalisation (e.g like E. Cartan did) one starts with the basic definitions and operations of derivation and integration, how they affect the space they are part of and how they affect each other. Let's say a "methodological shift" from a coordinate-formula-based definition, which is restricted in its choice of representation, to a operational-based definition. (Note that in such a scheme the fundamental theorem of calculus is both easily proved and generalised, e.g generalised Stokes Theorem)

An english translation of E. Cartan, "ON CERTAIN DIFFERENTIAL EXPRESSIONS AND THE PFAFF PROBLEM" is available here (and in wikipedia article)
In order to genertalise a particular (coordinate-dependent) representation of these operations, they are related to geometric entities (as spaces) of the space itself (for example the tangent and co-tangent space) and not to specific representations of these entities. Effectively it is a recursive construction of forms, a $k$ form is constructed by an operation $d$ on a $k-1$ form and so on. Nowhere is a specific representation taking place. So one can indeed use these operations and do calculus as needed on any manifold. Moreover like any good recursive scheme should do, $0$ forms can be identified with special entities on the space and the whole differential forms scheme is well defined.