Levi-Civita tensor: Difference between revisions

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#REDIRECT [[Levi-Civita symbol]]
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The [[Levi-Civita symbol]] is used to denote the components of the '''Levi-Civita tensor''', sometimes called the '''Levi-Civita form''', and in ''n'' dimensions this tensor is an invariant of the special unitary group [[SU(n)]].<ref name=Vaughn>
 
{{cite book |title=Introduction to mathematical physics |author=Michael T. Vaughn |pages=p. 484 |url=http://books.google.com/books?id=E6_DiJDIptoC&pg=PA484 |isbn=3527406271 |publisher=Wiley-VCH |year=2007}}
 
</ref> It flips sign under reflections, and physicists call it a ''pseudo''-tensor.<ref name=Felsager>
 
{{cite book |title=Geometry, particles, and fields |author=Bjørn Felsager |pages=p. 358 |url=http://books.google.com/books?id=R1XkarKY7AwC&pg=PA358 |year=1998 |isbn=0387982671 |publisher=Springer}}
 
</ref>  It also is called the ''alternating tensor''<ref name=Sharma>
 
{{cite book |title=Matrix Methods and Vector Spaces in Physics |author=Vinod K. Sharma |url=http://books.google.com/books?id=Kg2ZjUmOB9EC&pg=PT386 |pages=p. 370|chapter=§9.2 Alternating tensor (or Levi-Civita symbol) |isbn=8120338669 |publisher=Prentice-Hall of India Pvt.Ltd |year=2009}}
 
 
</ref>  or the ''completely antisymmetric tensor'' with ''n'' indices in ''n'' dimensions. The completely antisymmetric tensor with ''n'' indices in ''n''-dimensions has only one independent component, and is denoted in two, three and four dimensions as &epsilon;<sub>ij</sub>, &epsilon;<sub>ijk</sub>, &epsilon;<sub>ijkl</sub>.<ref name=Padmanabhan>
 
{{cite book |title=Gravitation: Foundations and Frontiers |author=T. Padmanabhan |url=http://books.google.com/books?id=BSfe2MjbQ3gC&pg=PA22 |pages=p. 22 |isbn=0521882230 |publisher=Cambridge University Press |year=2010}}
 
</ref> Consequently, in three dimensions the completely antisymmetric tensor with three indices is entirely specified by stating &epsilon;<sub>123</sub>&nbsp;=&nbsp;&epsilon;<sub>xyz</sub>&nbsp;=&nbsp;1 in [[Cartesian coordinates]].
 
==Notes==
<references/>

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The Levi-Civita symbol is used to denote the components of the Levi-Civita tensor, sometimes called the Levi-Civita form, and in n dimensions this tensor is an invariant of the special unitary group SU(n).[1] It flips sign under reflections, and physicists call it a pseudo-tensor.[2] It also is called the alternating tensor[3] or the completely antisymmetric tensor with n indices in n dimensions. The completely antisymmetric tensor with n indices in n-dimensions has only one independent component, and is denoted in two, three and four dimensions as εij, εijk, εijkl.[4] Consequently, in three dimensions the completely antisymmetric tensor with three indices is entirely specified by stating ε123 = εxyz = 1 in Cartesian coordinates.

Notes

  1. ↑ Michael T. Vaughn (2007). Introduction to mathematical physics. Wiley-VCH, p. 484. ISBN 3527406271. 
  2. ↑ Bjørn Felsager (1998). Geometry, particles, and fields. Springer, p. 358. ISBN 0387982671. 
  3. ↑ Vinod K. Sharma (2009). “§9.2 Alternating tensor (or Levi-Civita symbol)”, Matrix Methods and Vector Spaces in Physics. Prentice-Hall of India Pvt.Ltd, p. 370. ISBN 8120338669. 
  4. ↑ T. Padmanabhan (2010). Gravitation: Foundations and Frontiers. Cambridge University Press, p. 22. ISBN 0521882230. 
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