Timeline for Why is the standard definition of a $(p, q)$-tensor so bizarre?
Current License: CC BY-SA 4.0
36 events
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| Nov 17, 2020 at 12:33 | comment | added | Student | coordinate-free v.s coordinate-full | |
| S Aug 25, 2020 at 17:44 | history | suggested | RobPratt |
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| Aug 25, 2020 at 16:54 | review | Suggested edits | |||
| S Aug 25, 2020 at 17:44 | |||||
| Aug 25, 2020 at 16:50 | comment | added | Qfwfq | @liuyao: the first definition of orientation you provide seems wrong to me: it seems you are first taking a global frame, and then modding out by global $GL^+_n$-valued functions. But a manifold, to admit a global frame at all, needs to be parallelizable (en.wikipedia.org/wiki/Parallelizable_manifold). The correct definition of orientation $\mathfrak{o}$, as you suggest, is a global volume form $\eta$ modulo the multiplication of strictly positive global functions: $\mathfrak{o}=[\eta]=\eta\cdot C^\infty (M,\mathbb{R}_{>0})$. | |
| Aug 25, 2020 at 16:34 | answer | added | Tim Campion | timeline score: 1 | |
| Aug 25, 2020 at 2:06 | answer | added | Mozibur Ullah | timeline score: 2 | |
| Apr 11, 2019 at 13:26 | comment | added | user44143 | You can see online how Einstein would have learned tensors, following Ricci and Levi-Civita's paper in the Annalen from 1900: eudml.org/doc/157997. It has applications to analysis (quadratic forms), geometry (embedded surfaces), mechanics (integrable vector fields), and physics (electrodynamics, thermodynamics, elasticity). | |
| Apr 2, 2019 at 16:16 | comment | added | liuyao | Here’s a similar situation: an orientation of a manifold is often defined as an assignment of ordered basis at each point, smoothly varying, modulo GL(n) with positive determinant. Very cumbersome. The short definition would be a volume form, i.e., a non-vanishing n-form. I’d say the long definition should be kept informal, or not branded as definition at all. | |
| Mar 31, 2019 at 4:18 | comment | added | Steven Landsburg | @NateEldredge: That cabbage/goat mnemonic only works if you remember that a cabbage can also eat a goat. | |
| Mar 31, 2019 at 4:03 | comment | added | liuyao | I think the first definition is already beyond what physicists and computer scientists are comfortable with (and should not be branded as standard). But being an article on wikipedia, it makes sense to have a definition that satisfies mathematician's demand of precision, while also making it clear to the reader how in the world it has to do with a multi-dimensional array. Even in a purely math context, one may also want to put out a definition that is easy to work with (such as a line bundle as a collection of "functions" over open sets that satisfy similar transformation rules). | |
| Mar 30, 2019 at 18:37 | comment | added | David Roberts♦ | @MichaelBächtold apparently he knew enough to grasp the concept of fibre bundle, which is more than the old approach, though it might have been implicit. That's why i included Ehresmann :-) | |
| Mar 30, 2019 at 11:15 | vote | accept | Arthur | ||
| Mar 30, 2019 at 11:06 | comment | added | Michael Bächtold | @DavidRoberts do you know if Élie used the modern definition of tensor? I had the impression he didn't, at least not for 1-forms in his works around 1900. I still don't know when the modern definition arose and hope someone answers this question. | |
| Mar 30, 2019 at 10:42 | comment | added | David Roberts♦ | @MichaelBächtold I learned about tensors for the purposes of GR from an old physics book in my first year of university. It was a long time before I realised that tensors are tuples of functions of the coordinates, not just symbols, which is why the coordinate transformation rules are so important. It just wasn't said that they were not just arrays of numbers! | |
| Mar 30, 2019 at 10:39 | comment | added | David Roberts♦ | @Michael Élie (see en.wikipedia.org/wiki/%C3%89lie_Cartan#Differential_Geometry). All the amazing new geometry he did that people at the time had a hard time with. | |
| Mar 30, 2019 at 10:20 | comment | added | Michael Bächtold | @DavidRoberts do you mean Elie or Henri Cartan? | |
| Mar 30, 2019 at 9:53 | comment | added | Michael Bächtold | @EinfacherSchreiberling "Obviously the "standard definition" is easier to understand..." The fact that the OP posed this question and is getting up-votes seems to contradict this assumption. | |
| Mar 30, 2019 at 9:49 | comment | added | Michael Bächtold | "this is a bad definition from a pedagogical point of view". Why do you think so? | |
| Mar 30, 2019 at 5:06 | history | edited | Konstantinos Kanakoglou |
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| Mar 30, 2019 at 3:23 | comment | added | Nate Eldredge | I still like the definition I learned from my undergraduate differential geometry / linear algebra professor. "A vector is like a cabbage. A linear functional is like a goat, it eats a cabbage and spits out a number. [A goat with an unusual digestive system, I guess]. A $(p,q)$ tensor is like a dragon that eats $p$ goats and $q$ cabbages, and spits out a number." | |
| Mar 30, 2019 at 0:31 | history | edited | Konstantinos Kanakoglou |
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| Mar 29, 2019 at 22:09 | comment | added | Konstantinos Kanakoglou | @ David Roberts, you are right. However, they have some arguments in doing so. (this is what i am trying to explain in my answer). | |
| Mar 29, 2019 at 21:53 | comment | added | David Roberts♦ | Physicists use Riemannian geometry as it stood around 1900-1920, pre-Cartan, Ehresmann etc etc. They haven't updated the definition to keep pace with mathematical perceptions. | |
| Mar 29, 2019 at 21:46 | history | edited | Sergei Akbarov | CC BY-SA 4.0 |
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| Mar 29, 2019 at 21:41 | history | edited | YCor | CC BY-SA 4.0 |
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| Mar 29, 2019 at 21:21 | answer | added | Konstantinos Kanakoglou | timeline score: 21 | |
| Mar 29, 2019 at 19:45 | review | Close votes | |||
| Mar 31, 2019 at 10:14 | |||||
| Mar 29, 2019 at 19:32 | comment | added | Arthur | But what confuses me is that the alternative definition doesn't have to be 'coordinate-free' (if that's what puts people off). The tensor product can be defined with respect to bases! | |
| Mar 29, 2019 at 19:28 | comment | added | Francesco Polizzi | @EinfacherSchreiberling: right, of course. | |
| Mar 29, 2019 at 19:27 | comment | added | ssx | @FrancescoPolizzi and Ricci you do not mention? | |
| Mar 29, 2019 at 19:23 | comment | added | Francesco Polizzi | "Einstein was using that definition in the early 1900". Let us recall that, without the deep work of T. Levi-Civita, no mathematical formulation of General Relativity could have been possible. | |
| Mar 29, 2019 at 19:21 | comment | added | ssx | Einstein was using that definition in the early 1900. Whitney came up with the modern definition in 1940 or so. Obviously the "standard definition" (coordinatewise) is easier to understand, but less nice to think about than abstract tensor products. | |
| Mar 29, 2019 at 19:21 | comment | added | Francesco Polizzi | More seriously, the notion of "dual vector space" is often a delicate one, at least for non-mathematicians. Computations in coordinates can be less transparent than coordinate-free ones, but they can be grasped more easily by people whose background in abstract linear algebra is not strong. | |
| Mar 29, 2019 at 19:16 | comment | added | Francesco Polizzi | When you do long computations in coordinates, operations like contractions and similar become automatic by using upper and lower indices. Or, at least, I have been told so :) | |
| Mar 29, 2019 at 19:14 | comment | added | JJJ | Physicists and Computer Scientists are why | |
| Mar 29, 2019 at 19:13 | history | asked | Arthur | CC BY-SA 4.0 |