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969:{\displaystyle {\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}}}
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1342:. 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
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1104:(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. —
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2219:{\displaystyle \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}}}
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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
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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
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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
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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
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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.
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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. —
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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?
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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:
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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.
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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. —
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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.
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s 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
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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
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There are equations with the Levi-Civita tensor with covariant and contravariant indices, these two coincide only when the metric has determinant 1.
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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:
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In my opinion, chapter "Generalization to n dimensions" contains a more expressive definition of the symbol than the introductory chapter.
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1701:{\displaystyle \forall i\in \{1,2,3\}:({\vec {a}}\times {\vec {f}})_{i}=\sum _{j,k=1}^{3}\varepsilon _{ijk}a_{j}f_{k}}
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s seem to suffer this transformation and there is no readily apparent explanation for why this one does.
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The equation demonstrating the Levi-Civita symbol's application to denoting the vector cross-product
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Knowledge. If you would like to participate, please visit the project page, where you can join
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mentions in great detail the possibility of having indices starting at either one or zero. —
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became correct. Logging out causes it to become incorrect again. It really makes no sense.
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Is't true. At least in three last formulas are need plus rather than minus. --
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1433:{\displaystyle ({\vec {a}}\times {\vec {f}})_{i}=\varepsilon _{ijk}a^{j}f^{k}}
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and with everything written explicitly as in lower high school)
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The article should specify when a
Euclidean metric is assumed
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Please add the following equation to the properties section:
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The first cross-product seen by young pupils is IIRC the
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Shortly after I added an "In projective space" section,
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I have noticed, on the first formula under the heading
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page might give a hint of what to search where. Also
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729:
726:
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693:
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394:
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137:Mid-importance
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90:Physics portal
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65:
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62:Mid‑importance
56:
44:
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37:
26:
13:
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2199:
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2159:
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2103:
2098:
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2080:
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2067:
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2055:
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2049:
2046:
2043:
2027:
2023:
2019:
2016:
2013:
2008:
2004:
2000:
1995:
1989:
1983:
1978:
1974:
1960:
1957:
1951:
1946:
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1015:
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1002:
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959:
952:
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938:
935:
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927:
924:
922:
915:
912:
909:
906:
902:
896:
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890:
887:
883:
873:
870:
865:
862:
858:
854:
851:
848:
846:
839:
836:
833:
830:
826:
820:
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814:
811:
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797:
792:
788:
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781:
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776:
769:
766:
763:
760:
756:
750:
747:
744:
741:
737:
727:
724:
721:
718:
713:
710:
707:
704:
700:
694:
691:
687:
681:
678:
674:
668:
665:
661:
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652:
648:
644:
642:
635:
632:
629:
626:
622:
616:
613:
610:
607:
603:
593:
590:
587:
584:
579:
576:
573:
570:
566:
560:
557:
553:
547:
544:
540:
534:
531:
527:
521:
518:
514:
510:
508:
501:
498:
495:
492:
488:
482:
479:
476:
473:
469:
457:
456:
455:
449:
447:
445:
441:
437:
433:
429:
409:
399:
392:
387:
377:
368:
365:
362:
358:
354:
349:
339:
323:
315:
310:
305:
304:
290:
289:
286:
285:
281:
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196:
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185:
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162:
159:
155:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
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2278:
2255:
2235:
2230:
2228:
1883:
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1850:
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1813:81.39.92.255
1807:— Preceding
1804:
1734:
1299:— Preceding
1296:
1206:
1118:
978:
453:
432:GeraldMeyers
426:— Preceding
327:
308:
282:
274:
261:
242:Mid-priority
241:
201:
167:Mid‑priority
136:
96:
40:WikiProjects
2281:83.52.38.67
1827:Index Range
217:Mathematics
208:mathematics
164:Mathematics
2297:Categories
2206:otherwise
1770:JRSpriggs
1766:WP:Burden
1737:JRSpriggs
1326:JRSpriggs
1809:unsigned
1313:contribs
1301:unsigned
440:contribs
428:unsigned
309:365 days
275:Archives
2256:In the
2239:Wtrmute
1853:Taabagg
1834:Taabagg
1785:Tonymec
1750:Tonymec
1715:Tonymec
1106:Quondum
244:on the
139:on the
112:Physics
103:Physics
59:Physics
30:C-class
2262:Vpab15
36:scale.
2285:talk
2266:talk
2243:talk
1857:talk
1838:talk
1817:talk
1789:talk
1774:talk
1754:talk
1719:talk
1330:talk
1309:talk
985:talk
436:talk
2087:if
1967:if
1297:--
424:.
236:Mid
131:Mid
2299::
2287:)
2268:)
2245:)
2237:--
2178:…
2133:…
2078:−
2062:…
2017:…
1932:…
1898:ε
1859:)
1840:)
1819:)
1791:)
1776:)
1768:.
1756:)
1721:)
1664:ε
1637:∑
1617:→
1608:×
1602:→
1566:∈
1560:∀
1498:ε
1478:→
1469:×
1463:→
1396:ε
1376:→
1367:×
1361:→
1332:)
1315:)
1311:•
1249:ε
1227:×
1160:ε
1138:×
1090:ι
1087:θ
1084:η
1081:ζ
1071:ι
1068:δ
1058:θ
1055:γ
1045:η
1042:β
1032:ζ
1029:α
1016:δ
1013:γ
1010:β
1007:α
987:)
953:γ
950:β
947:α
942:θ
939:σ
936:ρ
932:δ
928:−
925:≡
916:δ
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910:σ
907:ρ
897:δ
894:γ
891:β
888:α
874:β
871:α
866:σ
863:ρ
859:δ
852:−
849:≡
840:δ
837:γ
834:σ
831:ρ
821:δ
818:γ
815:β
812:α
798:α
793:ρ
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782:−
779:≡
770:δ
767:γ
764:β
761:ρ
751:δ
748:γ
745:β
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728:ν
725:μ
722:σ
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714:ι
711:θ
708:η
705:ζ
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692:δ
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679:γ
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666:β
656:ζ
653:α
645:≡
636:ν
633:μ
630:σ
627:ρ
617:δ
614:γ
611:β
608:α
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591:θ
588:η
585:ζ
580:ν
577:μ
574:σ
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567:δ
561:ι
558:δ
548:θ
545:γ
535:η
532:β
522:ζ
519:α
511:≡
502:ν
499:μ
496:σ
493:ρ
483:δ
480:γ
477:β
474:α
442:)
438:•
403:^
393:×
381:^
359:ε
343:^
2283:(
2264:(
2241:(
2231:a
2200:0
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2184:n
2181:,
2175:,
2172:3
2169:,
2166:2
2163:,
2160:1
2157:(
2149:)
2144:n
2140:a
2136:,
2130:,
2125:3
2121:a
2117:,
2112:2
2108:a
2104:,
2099:1
2095:a
2091:(
2081:1
2071:)
2068:n
2065:,
2059:,
2056:3
2053:,
2050:2
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2044:1
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2033:)
2028:n
2024:a
2020:,
2014:,
2009:3
2005:a
2001:,
1996:2
1990:a
1984:,
1979:1
1975:a
1971:(
1961:1
1958:+
1952:{
1947:=
1940:n
1936:a
1927:3
1923:a
1917:2
1913:a
1907:1
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1884:a
1880:a
1876:a
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42::
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