David Quintero on Nostr: "A Geometrical Introduction to Tensor Calculus," by Jeroen Tromp, teaches you ...

"A Geometrical Introduction to Tensor Calculus," by Jeroen Tromp, teaches you differential geometry with a focus on continuum mechanics and materials, not relativity (although examples for relativity are given). This has been surprising for me in many ways.

For instance, in continuum mechanics, you may need to account for deformations. For instance, dislocations and disclinations are 2-forms (vector valued form and tensor valued form). They are the equivalent of torsion and curvature in relativity, respectively. They obey Bianchi identities, too.

In General Relativity, you can have torsion, then you have the Einstein-Cartan gravitation. But it seems that the universe doesn't need torsion. We're lucky because the field equations with torsion are (even more) horrible to solve!

But in materials science (continuum mechanics), you can have these deformations. So, you need more sophisticated differential geometry than for gravitation. But although you could need all that sophistication to _describe_ your continuum media, I'm not sure if people are actually using it. (I mean, if it's practical).