Timeline for Functions with a Jacobian whose columns are orthogonal
Current License: CC BY-SA 4.0
8 events
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| Sep 30, 2020 at 0:25 | comment | added | Robert Bryant | @DenisSerre: A good remark, but, in fact, the two problems are equivalent up to computing an inverse function. If one regards $x$ and $f$ as columns of height $n$, the Jacobian $J$ satisfies $df = J\,dx$, and the stated condition is that $J^T\,J$ be diagonal. If $K = J^{-1}$, then $dx = K\,df$, and we see that $$K^T\,K = (J^{-1})^T\,(J^{-1})=( J\,J^T)^{-1,},$$ hence $K^TK$ is diagonal if and only if $J\,J^T$ is diagonal. Thus, $f:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the column condition if and only if the inverse function $f^{-1}:\mathbb{R}^n\to\mathbb{R}^n$ satisfies the row condition. | |
| Sep 29, 2020 at 20:52 | vote | accept | oasis93 | ||
| Jul 13, 2020 at 12:15 | answer | added | Robert Bryant | timeline score: 1 | |
| Jul 13, 2020 at 10:37 | comment | added | Denis Serre | Just one remark. Your definition does not imply (is not equivalent to) that the rows be orthogonal. Thus there would be a symmetric question concerning those fields whose Jacobian has orthogonal rows. | |
| Jul 13, 2020 at 10:34 | history | edited | Denis Serre | CC BY-SA 4.0 |
added 4 characters in body
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| Jul 8, 2020 at 6:48 | comment | added | Mateusz Kwaśnicki | Clearly, there are more examples here than in the case of Liouville's theorem: if $g$ is a conformal map and $f_i(x) = h_i(g_i(x))$ for an arbitrary collection of $h_i : \mathbb{R} \to \mathbb{R}$, then the gradients of $f_i$ are orthogonal. | |
| Jul 8, 2020 at 3:49 | review | First posts | |||
| Jul 8, 2020 at 5:53 | |||||
| Jul 8, 2020 at 3:41 | history | asked | oasis93 | CC BY-SA 4.0 |