Timeline for Exterior derivative independence from coordinate systems
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 11, 2019 at 13:23 | comment | added | Lo Scrondo | I deeply thank you! | |
| Jul 11, 2019 at 13:12 | comment | added | Willie Wong | Because the derivatives agree @LoScrondo: if two functions have the same derivative at one point then there Taylor expansion agrees to first order. | |
| Jul 11, 2019 at 11:38 | comment | added | Lo Scrondo | Thank you @WillieWong , I think yours is the only possible explanation. Just to verify if I've understood correctly...why your Taylor expansion lacks the linear term, i.e. why it is $\xi t \to \xi t + O(\xi^2t^2)$ and not $\xi t \to \xi t + O(\xi t)$? | |
| Jul 9, 2019 at 16:08 | answer | added | Bazin | timeline score: 2 | |
| Jul 9, 2019 at 13:22 | comment | added | Willie Wong | I am pretty sure implicitly the coordinate systems defining $V$ and $V'$ are the "same" at $x$ (the transition map should have derivative that equals the identity there) (this is given the figure illustrating the situation on pg 191). Then it really should just be a change of variables and application of Taylor's theorem. Following Arnold's notation, for $\partial V'(\epsilon)$, instead of integrating over the four segments $\xi t, \xi t + \eta, \eta t, \eta t + \xi$, you would be integrating over $\xi t + O(\xi^2t^2)$ etc. | |
| Jul 9, 2019 at 12:49 | comment | added | Lo Scrondo | I just followed the description given by Arnold, but in fact he treats $V$ as an ordinary parallepiped, and $V'$ as a curvilinear one | |
| Jul 9, 2019 at 11:53 | comment | added | Ben McKay | I think you will have to define the term "curvilinear parallelipiped". If $V$ is just a parallelipiped, then $V'$ isn't. But if you allow arbitrary curvilinearity, it is not clear how it behaves with $\varepsilon$. | |
| Jul 8, 2019 at 23:15 | review | First posts | |||
| Jul 9, 2019 at 3:05 | |||||
| Jul 8, 2019 at 23:13 | history | asked | Lo Scrondo | CC BY-SA 4.0 |