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!
Best Answer
You have asked a good number of questions. I'll answer the one in the title.
The point is that differential forms are "the things you can integrate on manifolds". Manifolds are more general objects than open subsets of $\Bbb R^n$, and in some sense one of the reasons one wants to introduce forms.
Suppose you have a 1-form $\alpha$ on a manifold $M$, and a smooth curve $\gamma: [0,1] \to M$. Then I can define the integral of $\alpha$ over $\gamma$ - and this does not depend on the parameterization of $\gamma$. That is, if I precompose with a(n orientation preserving) diffeomorphism $[0,1] \to [0,1]$, the integral will still be the same.
You might object "But I already know how to do this for functions. I can just take a line integral." Let's write down the standard formula for the line integral in $\Bbb R^n$: $\int_\gamma f := \int_0^1 f(\gamma(t))\|\gamma'(t)\|$. This is indeed independent of reparameterizations. But the key thing here is that $\|\gamma'(t)\|$ term that we had to introduce. 1) When you integrate a 1-form, such a term doesn't really show up. This is desirable, because... 2) When you're on a manifold that's not a subset of $\Bbb R^n$, we no longer have a way to write down $\|\gamma'(t)\|$ without introducing extra structure (being able to measure how big a tangent vector is is almost precisely the same as a Riemannian metric). It is not so desirable to introduce this structure if all you're interested is the manifold.
There are higher things called $k$-forms (which exist on any manifold of dimension at least $k$). These are also defined almost precisely so that "$k$-forms are the things that you can integrate over $k$-dimensional submanifolds". Again, this is independent of any number of things, eg local parameterizations of the submanifold. When you see the construction, you'll see that they're defined as "sections of the $k$th exterior power $\Lambda^k T^*M$". I'll justify this for $k$ the dimension of the manifold. The point of this is that the way an $n$-form changes under a coordinate transformation is precisely by the determinant of the Jacobian of the coordinate transformation. Now, if you write down the formula for integrating a function after doing a coordinate change there is an $|\det J(\varphi)|$ term in the integral. So we set things up precisely so that the integral of a differential form is defined independent of (local) choices of coordinates, which I hope justifies my claim that differential forms are built to be the things you integrate.