Tao on coordinate vs coordinate-free math reasoning

[dmm]

> On when to use coordinates and other concrete constructions in mathematics, and when to use coordinate-free formulations and abstractions:

> 1. If your priority is to perform computations in mathematics, use coordinates and concrete constructions.
> 2. If your priority is to generalize to as broad a range of use cases as possible, use coordinate-free formulations and abstractions.
> 3. If your priority is to actually understand what is going on behind the mathematical formalism, learn how the coordinate-based and coordinate-free approaches are equivalent.

mathstodon.xyz/@tao/114456756661540097

Comments

[dmm]

Plenty of other interesting things at https://mathstodon.xyz/@tao

[juan_gandhi]

Good link to his profile.
And good idea, of course. I believe Ventzel, ну или И.Грекова, в повести "Кафедра" эту тему затрагивает. Матрицы vs операторы.

[chaource]

At some point I got interested in coordinate-free calculations for differential geometry and general relativity. There are quite a few cases where coordinate-free calculations are actually simpler and faster, just because you have to write fewer symbols. But there are also a few cases where index-based calculations are easier, in particular when dealing with traces of high-rank tensors.

Another detail is that tensor calculations with indices don't have to mean a choice of a coordinate system, as long as we don't introduce it explicitly. Such calculations are equivalent to using abstract index notation in coordinate-free calculations.