Timeline for History of the component-free approach to tensors
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 15, 2015 at 6:30 | comment | added | Alan U. Kennington | @Ben Crowell: I'm not 100% "sure". I'm just going by what I read everywhere. The Cartan formalism is used in physics as a basis-free approach. en.wikipedia.org/wiki/Cartan_connection. Élie Cartan is also well known for referring to tensor calculus as "the debauch of indices". I'm not saying that Cartan never used indices/components. He certainly did. But he eliminated many of them. See Misner/Thorne/Wheeler page 348. Cartan made tensors more geometric, and so forth. So Cartan was in important figure in the "component-free campaign" or the "war on coordinates". | |
| Oct 15, 2015 at 4:55 | comment | added | user466 | @AlanU.Kennington: Are you sure about Cartan? AFAIK, Cartan's contribution was the use of notations like dx and $\partial_x$ for basis vectors in the covector and tangent vector spaces. These are coordinate-based notations, not coordinate-independent ones. | |
| Oct 14, 2015 at 22:38 | answer | added | Martin Gisser | timeline score: 4 | |
| Oct 14, 2015 at 22:02 | comment | added | Martin Gisser | Good question. I have a hate-love relationship with tensor calculus, e.g. hating Christoffel symbols. Yet much of the component-free stuff is also junk: Koszul's notation got rid of the "debauch of indices" but also is a source of new superfluous mess. I found the physicists's "abstract index" notation the best of the whole mess. Philosophically it is component free, the indices just indicating the structure of the tensor. But still oversimpling, generating new mess. Cf. R.Wald, General Relativity, or D.Malament, Topics in the Foundations of General Relativity and Newtonian Gravitation Theory. | |
| Oct 13, 2015 at 3:12 | comment | added | Alan U. Kennington | That question 599 was about intrinsic geometry, not about the component-free approach. Intrinsic geometry was evolving already with Gauss and Christoffel, Bianchi, Ricci, Levi-Civita and others. It really started with the Gaussian curvature discovery. The component-free approach is generally attributed to Élie Cartan, early in the 20th century. | |
| Oct 12, 2015 at 20:40 | comment | added | Conifold | This question is answered here hsm.stackexchange.com/questions/599/… | |
| Oct 12, 2015 at 14:37 | history | edited | HDE 226868♦ | CC BY-SA 3.0 |
added 10 characters in body; edited title
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| Oct 12, 2015 at 14:16 | history | asked | set5 | CC BY-SA 3.0 |