Most introductory texts on differential forms praise their property of allowing for a "coordinate free formulation".
What exactly does this mean? What would be a concrete example for which a coordinate free formulation is superior to choosing a coordinate system?
I am aware that expressing calculations in differential geometry in local parametrizations can be a mess, but aren't we also always choosing a basis for differential forms in writing something like
$$\omega = \sum_{k=1}^n a_i dx_i,$$
where $\{dx_1,\dots, dx_n\}$ is a basis of $T_p \mathbb{R}^n = \mathbb{R}^n$ for some point $p \in \mathbb{R}^n$?
Best Answer
In linear algebra it is important to understand the distinction between a linear transformation $T:V\to W$ and a matrix $A$ which represents $T$. Different choices of bases for $V$ and $W$ will give different matrices. The transformation $T$ is "coordinate-free" but the matrix $A$ is not.
Another example is an inner product on a vector space. It exists independently of any choice of basis ("coordinate-free") but if we choose a basis we can represent the inner product using a matrix (which would not be coordinate-free).
Differential forms are coordinate-free in the same way. They exist independently of any choice of basis but if we choose we can express them using a basis.
Why is a coordinate-free formulation superior? Sometimes equations look simpler. For example, compare the coordinate-free geodesic equation to the coordinate-dependent geodesic equation: $$\nabla_{\gamma'}\gamma'=0$$ vs. $$\frac{d^2\gamma^k}{dt^2}+\Gamma_{ij}^k\frac{d\gamma^i}{dt}\frac{d\gamma^j}{dt}=0.$$ Or compare the coordinate-free equation for closedness of a differential form ($d\omega=0$) to what it is written out in coordinates.
Also, from a philosophical perspective, coordinates are rather arbitrary (nature doesn't come with axes), so it is more pleasing if we can write something in a coordinate-free way.