Timeline for Functions with a Jacobian whose columns are orthogonal
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Sep 29, 2020 at 23:47 | comment | added | Robert Bryant | @oasis93: Well, if you read French, you can start with Gaston Darboux' 1898 book, Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes, which contains considerable information about the $n$-dimensional case for $n>3$. It covers pretty much everything that was known about the subject before 1900. There is an informative and thorough review of this book (in English) by E. O. Lovett published by the AMS at ams.org/journals/bull/1899-05-04/S0002-9904-1899-00584-6/… | |
| Sep 29, 2020 at 20:52 | vote | accept | oasis93 | ||
| Sep 29, 2020 at 20:50 | comment | added | oasis93 | Can you please introduce some references for the general case of n>3? I tried terms such as "over-determined orthogonal coordinate systems" on Google, but it seems there's no exact result. When searching for "orthogonal coordinate systems" almost all the results are for the case of n=3, which is not of interest to me. I would appreciate if you could introduce a book or paper on the general case of n>3 that characterizes the solutions. Huge thanks. | |
| Jul 18, 2020 at 10:39 | comment | added | Robert Bryant | @oasis93: There are plenty of invertible solutions for all $n$. For example, if $f_j$ is an invertible function of $x_j$ only for all $j$, you have a solution. Of course, this is not all of the solutions. The involutivity property implies that there exist many solutions locally and describes how to construct them using PDE. Overdetermined means that you can't arbitrarily prescribe initial conditions for the PDE on a hypersurface, say, of the form $\mathbb{R}^{n-1}\subset\mathbb{R}^n$. | |
| Jul 18, 2020 at 10:19 | comment | added | oasis93 | Thanks for your response. Actually I'm interested in the case where $n>3$ and the function is invertible. Does the over-determined means that they don't have any solution? | |
| Jul 13, 2020 at 12:15 | history | answered | Robert Bryant | CC BY-SA 4.0 |