What exactly does "differential forms are coordinate free" mean?

Solution 1:

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.