Talk:Levi-Civita symbol

message: Please add the following equation to the properties section: \hat{e}_k=\varepsilon_{kij}\hat{e}_i\times\hat{e}_j.  — Preceding unsigned comment added by GeraldMeyers (talk • contribs) 00:06, 13 May 2022 (UTC)
thread_title: Add equation to properties

message: I think that for signature (+,–,–,–) the formulas:
:
\begin{align}
        E_{\alpha \beta \gamma \delta} E_{\rho \sigma \mu \nu} &\equiv g_{\alpha \zeta} g_{\beta \eta} g_{\gamma \theta} g_{\delta \iota} \delta^{\zeta \eta \theta \iota}_{\rho \sigma \mu \nu} \\
        E^{\alpha \beta \gamma \delta} E^{\rho \sigma \mu \nu} &\equiv g^{\alpha \zeta} g^{\beta \eta} g^{\gamma \theta} g^{\delta \iota} \delta^{\rho \sigma \mu \nu}_{\zeta \eta \theta \iota} \\
   E^{\alpha \beta \gamma \delta} E_{\rho \beta \gamma \delta} &\equiv -6 \delta^{\alpha}_{\rho} \\
   E^{\alpha \beta \gamma \delta} E_{\rho \sigma \gamma \delta} &\equiv -2 \delta^{\alpha \beta}_{\rho \sigma} \\
   E^{\alpha \beta \gamma \delta} E_{\rho \sigma \theta \delta} &\equiv -\delta^{\alpha \beta \gamma}_{\rho \sigma \theta} \,.
\end{align}
Is't true. At least in three last formulas are need  plus rather than minus. --Mrilluminates (talk) 16:27, 14 April 2017 (UTC)
replies: text: The way it is defined, with E^{\alpha \beta \gamma \delta} = g^{\alpha \zeta} g^{\beta \eta} g^{\gamma \theta} g^{\delta \iota} E_{\zeta \eta \theta \iota} (i.e. normal raising and lowering of indices as with any tensor), then regardless of signature (i.e. either (+−−−) or (−+++)), the minus signs are correct.  This is because the number of negative signs in the signature is odd, and hence the determinant of the matrix of the metric tensor components is negative.  It is almost as though the negative determinant "leaks out" from the square roots and absolute signs.  —Quondum 02:51, 24 August 2017 (UTC)
thread_title:  Minkowski Space 

message: Shortly after I added an "In projective space" section,  requested citations. In the high school where I was a pupil 52 years ago, this was part of the analytic geometry curriculum for the last two years of high school (in the latin-math section) and it was repeated in the geometry course for the following year of university (the first bachelor year about math science or civil engineering). It's been a long time and I don't have my schoolbooks at hand anymore, but equivalent course syllabi ought to be easy to find. The Duality (projective geometry) page might give a hint of what to search where. Also the French-language article parallel to this one mentions in great detail the possibility of having indices starting at either one or zero. — Tonymec (talk) 03:04, 23 April 2020 (UTC)
replies: text: Unfortunately, I did not take the courses which you took on projective geometry. In any case, the burden is on the one seeking to include certain material to justify it (explain what it means and how you know it is true), not on others to figure it out or disprove it. See WP:Burden. JRSpriggs (talk) 22:55, 23 April 2020 (UTC)
replies: text:  Well, in my country all bookshops (other than the ones in supermarkets, which don't sell projective geometry courses) are closed because of the COVID-19 pandemic. When business goes back to normal I'll go to the University bookshop, or somewhere, and try to find an appropriate treatise. In the meantime, you'll have to wait, unless someone living in a country with less restrictive lockdown rules, or someone who has relevant secondary sources at hand, can add the necessary references. — Tonymec (talk) 14:32, 7 May 2020 (UTC)
thread_title:  "Citations needed" for "In projective space" 

message: In my opinion, chapter "Generalization to n dimensions" contains a more expressive definition of the symbol than the introductory chapter.  — Preceding unsigned comment added by 81.39.92.255 (talk) 19:00, 13 September 2020 (UTC)
thread_title:  reorganization proposal 

message: It's erroneous to restrict the index range to positive integers, particularly given the convention, common in treatements of Relativity with the +,-,-,- metric, that time be assigned the index 0.  I see elsewhere here that the French page does not make this mistake.  Taabagg (talk) 19:05, 30 May 2021 (UTC)
thread_title:  Index Range 

message: Certainly, "Most authors" assign the value +1.  Can anyone cite an instance where an author has chosen another value?  If not, we should probably state that the value is always +1.  Taabagg (talk) 19:10, 30 May 2021 (UTC)
thread_title:  Value Assigned to Identity Permutation 

message: I have noticed, on the first formula under the heading Levi-Civita symbol#Generalization to n dimensions, in the +1 branch, one of the as is bizarrely, for a lack of a better term, transposed, or reflected across its own diagonal. Looking at the source of the corresponding formula, the a is a simple ASCII 97 a, without any formatting. I've copied the formula below, painting the a in red to draw attention to it, and it is still transposed. None of the other 11 as seem to suffer this transformation and there is no readily apparent explanation for why this one does.
: \varepsilon_{a_1 a_2 a_3 \ldots a_n} =
  \begin{cases}
         +1 & \text{if }(a_1, {\color{red}a}_2, a_3, \ldots, a_n) \text{ is an even permutation of } (1, 2, 3, \dots, n) \\
         -1 & \text{if }(a_1, a_2, a_3, \ldots, a_n) \text{ is an odd permutation of } (1, 2, 3, \dots, n) \\
    \;\;\,0 & \text{otherwise}
  \end{cases}
Even more bafflingly, when I logged in in order to write this comment, the a became correct. Logging out causes it to become incorrect again. It really makes no sense.
Is this working as intended? Is it some kind of intentional error inserted into the TeX engine so the authors can detect plagiarised code, like the 16th Century mapmakers did? Should we file a bug report with Wikimedia? What do you think? --Wtrmute (talk) 19:53, 30 September 2021 (UTC)
thread_title:  Odd behaviour of the LaTeX renderer 

message: In the "Three dimensions" secion, there  is an image of the 3 × 3 × 3 array with three different colours. But it is very difficult to see the numbers. Should it be replaced with three different 2d matrices? Vpab15 (talk) 15:42, 4 July 2022 (UTC)
thread_title:  Confusing 3d visualisation 

message: There are equations with the Levi-Civita tensor with covariant and contravariant indices, these two coincide only when the metric has determinant 1. 83.52.38.67 (talk) 11:22, 20 June 2023 (UTC)
thread_title:  The article should specify when a Euclidean metric is assumed