Talk:Levi-Civita symbol

Add equation to properties

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)

Minkowski Space

I think that for signature (+,–,–,–) the formulas:

${\begin{aligned}E_{\alpha \beta \gamma \delta }E_{\rho \sigma \mu \nu }&\equiv g_{\alpha \zeta }g_{\beta \eta }g_{\gamma \theta }g_{\delta \iota }\delta _{\rho \sigma \mu \nu }^{\zeta \eta \theta \iota }\\E^{\alpha \beta \gamma \delta }E^{\rho \sigma \mu \nu }&\equiv g^{\alpha \zeta }g^{\beta \eta }g^{\gamma \theta }g^{\delta \iota }\delta _{\zeta \eta \theta \iota }^{\rho \sigma \mu \nu }\\E^{\alpha \beta \gamma \delta }E_{\rho \beta \gamma \delta }&\equiv -6\delta _{\rho }^{\alpha }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \gamma \delta }&\equiv -2\delta _{\rho \sigma }^{\alpha \beta }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \theta \delta }&\equiv -\delta _{\rho \sigma \theta }^{\alpha \beta \gamma }\,.\end{aligned}}$

Is't true. At least in three last formulas are need plus rather than minus. --Mrilluminates (talk) 16:27, 14 April 2017 (UTC)

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)

Ill-formed equation

The equation demonstrating the Levi-Civita symbol's application to denoting the vector cross-product

$(\mathbf {a} \times \mathbf {b} )^{i}=\varepsilon _{ijk}a^{j}b^{k}$

seems ill-formed (I don't have the page open right now so I may have renamed the vectors but I think I got the spirit of what was written in the article). It is equating a vector with a covector, after all. Actually I believe that the cross-product is pseudovector but I guess if improper rotations are ignored it is adequate to consider it a vector. Correct me if I'm wrong, but I believe this equation should use the fully-raised version of the Levi-Civita tensor and the lowered versions of the two vectors:

$(\mathbf {a} \times \mathbf {b} )^{i}=\varepsilon ^{ijk}a_{j}b_{k}$.

-- — Preceding unsigned comment added by 2620:cc:8000:1484:392c:f25e:aab:945f (talk • contribs) 03:46, 16 September 2017‎ (UTC)

The cross product has always been a problem for this article because it cannot be meaningfully defined without a metric (to raise the index), but most of this article is trying to avoid the use of the metric. JRSpriggs (talk) 08:12, 16 September 2017 (UTC)

The first cross-product seen by young pupils is IIRC the moment of a force. They learn it first with no algebra by saying that the moment of a vector is equal to the area of the parallelogram whose sides are the vector and its lever-arm, its action line is perpendicular to both, and its sense is obtained by the rule of the corkscrew (if the force acts with the given lever-arm on a corkscrew, the latter will pull or push as appropriate). Then when a year or two later they learn algebraic geometry, you could just as well write

$({\vec {a}}\times {\vec {f}})_{i}=\varepsilon _{ijk}a^{j}f^{k}$

but if they aren't mature enough by then to grasp the difference between top indices and exponents, just write all the indices on the downside:

$({\vec {a}}\times {\vec {f}})_{i}=\varepsilon _{ijk}a_{j}f_{k}$

or even (without Einstein's summation convention and with everything written explicitly as in lower high school)

$\forall i\in \{1,2,3\}:({\vec {a}}\times {\vec {f}})_{i}=\sum _{j,k=1}^{3}\varepsilon _{ijk}a_{j}f_{k}$

Writing covariant indices below and contravariant ones above is useful and it's a good habit — it helps one avoid some trivial but annoying mistakes — but it is not always absolutely necessary and sometimes there are pedagogical reasons to avoid it. — Tonymec (talk) 04:08, 23 April 2020 (UTC)

"Citations needed" for "In projective space"

Shortly after I added an "In projective space" section, JRSpriggs 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)

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)

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)

reorganization proposal

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)

Index Range

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)

Value Assigned to Identity Permutation

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)

Odd behaviour of the LaTeX renderer

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)

Confusing 3d visualisation

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)

The article should specify when a Euclidean metric is assumed

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)