Timeline for Is "Cartan's magic formula" due to Élie or Henri?
Current License: CC BY-SA 3.0
22 events
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| Sep 29, 2022 at 8:22 | comment | added | Wisperwind | Late to the party; but this proof can nowadays also be found in Loring Tu's "An Introduction to Manifolds", Theorem 20.10. (@MartinGisser) | |
| S Jan 31, 2018 at 13:28 | history | suggested | Mizar | CC BY-SA 3.0 |
forgot to change also the last sentence
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| Jan 31, 2018 at 12:45 | review | Suggested edits | |||
| S Jan 31, 2018 at 13:28 | |||||
| S Jan 31, 2018 at 11:33 | history | suggested | Mizar | CC BY-SA 3.0 |
corrected some "anti-"s
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| Jan 31, 2018 at 9:22 | review | Suggested edits | |||
| S Jan 31, 2018 at 11:33 | |||||
| May 13, 2011 at 22:30 | comment | added | Robert Bryant | This formula already appears in É. Cartan's 1922 classic, Leçons sur les invariant intégraux. I believe it's in Chapter IX, where he discusses the effect of `infinitesimal transformations' (i.e., vector fields) on differential forms. Of course, Henri was alive then, but he was only 18. | |
| Sep 22, 2010 at 1:14 | comment | added | Deane Yang | Dick, thanks for the clarification. Now I agree with you. | |
| Sep 22, 2010 at 1:05 | comment | added | Per Vognsen | Martin: Regarding coordinate-based proofs, there's a nice middle ground where you use semi-canonical coordinates: rectify the flow of the vector field so that it looks like $V = \partial/\partial{x_1}$. Then there isn't any crunchy and unenlightening coordinate juggling. | |
| Sep 22, 2010 at 0:00 | comment | added | Martin Gisser | Oh, the book looks extremely yummy. Just had a short look. I'd love to learn more numerics (which I hated at university), esp. the Runge-Kutta/renormalization/Butcher/Hopf algebra connection... | |
| Sep 21, 2010 at 23:10 | comment | added | Dick Palais | I am THE Palais father. :-) BTW, do you know about our co-authored book? See: ode-math.com. | |
| Sep 21, 2010 at 22:38 | comment | added | Martin Gisser | @Dick. Umm, are you THE Palais - or his son? :-) | |
| Sep 21, 2010 at 22:37 | comment | added | Dick Palais | @Maxime. You are right. Henri was born in 1904, so he was already in his thirties when Elie gave the lectures on which the book was based in 1936-37. | |
| Sep 21, 2010 at 22:36 | comment | added | Martin Gisser | ... Henri lived 1904-2008. One of my favorite centennials, after Vietoris (1891-2002). | |
| Sep 21, 2010 at 22:32 | comment | added | Dick Palais | @Martin Gisser. I love Chern's proof too, but my original reaction was total embarrassment! I proudly showed Chern an ugly several page long computational proof, and he said "Yes, Dick, BUT..." and then showed me the above argument in a couple of lines on a blackboard. :-( | |
| Sep 21, 2010 at 22:27 | comment | added | Maxime Bourrigan | /Les systèmes différentiels extérieurs et leurs applications géométriques/ seems to have been written in 1945. Henri was certainly born at this date. | |
| Sep 21, 2010 at 22:23 | comment | added | Dick Palais | Sorry, Deane, you are correct---I screwed up. I should have said that it is obvious for functions $f$ and the differentials of functions, $df$, and that THESE generate the exterior algebra. The reason it is clear for $\omega = df$ is that the RHS becomes $di_X (df) = d(Xf)$. On the other hand, since $d$ is "natural", i.e., commutes with diffeos, it also commutes with Lie derivatives, so the LHS becomes $d L_X f) $ which is also $d(Xf)$. | |
| Sep 21, 2010 at 22:21 | vote | accept | Martin Gisser | ||
| Sep 21, 2010 at 22:21 | comment | added | Martin Gisser | Phantastic! Answered in nanoseconds, plus question already listed by google. Dick, I love Chern's proof. Many, if not most books suck on such matters. Similarly on could prove formula for differential in terms of covariant derivative. Can't tell of any book that doesn't introduce coordinates or bases/indices when treating such things. Deane, the formula is quite trivial on differentials of functions (Lie derivative product rule) and is then extended to general 1-forms by product rule, since these are sums of fdg. | |
| Sep 21, 2010 at 22:02 | comment | added | Deane Yang | I'm not sure I agree that the formula is trivial for $1$-forms. To me that's the first (and only?) nontrivial case, and Chern's argument is a nice way to explain how to extend it to higher degree forms. But I'm just quibbling. | |
| Sep 21, 2010 at 21:54 | history | edited | Dick Palais | CC BY-SA 2.5 |
added 632 characters in body; added 1 characters in body
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| Sep 21, 2010 at 21:44 | history | edited | Dick Palais | CC BY-SA 2.5 |
added 24 characters in body
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| Sep 21, 2010 at 21:34 | history | answered | Dick Palais | CC BY-SA 2.5 |