Levi-Civita symbol: Difference between revisions
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The symbol can be generalized to ''n''-dimensions, as completely antisymmetric in its indices with ε<sub>123...''n''</sub> = 1. More specifically, the symbol is one for even [[Permutation group|permutations]] of the indices, −1 for odd permutations, and 0 otherwise. | The symbol can be generalized to ''n''-dimensions, as completely antisymmetric in its indices with ε<sub>123...''n''</sub> = 1. More specifically, the symbol is one for even [[Permutation group|permutations]] of the indices, −1 for odd permutations, and 0 otherwise. | ||
The Levi-Civita symbol also is used to denote the components of the ''alternating tensor''<ref name=Sharma> | ===Levi-Civita tensor=== | ||
The Levi-Civita symbol also is used to denote the components of the ''Levi-Civita tensor'', and in ''n'' dimensions it 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 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}} | {{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 three indices in three dimensions. The completely antisymmetric tensor with '' | </ref> or the ''completely antisymmetric tensor'' with three indices in three 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 ε<sub>ij</sub>, ε<sub>ijk</sub>, ε<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}} | {{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}} | ||
Revision as of 12:26, 2 January 2011
The Levi-Civita symbol, usually denoted as εijk equals one if i,j,k = 1,2,3 or any permutation that keeps the same cyclic order,[1] or minus one if the order is different, or zero if any two of the indices are the same. It is named after the Italian mathematician and physicist Tullio Levi-Civita.
The symbol can be generalized to n-dimensions, as completely antisymmetric in its indices with ε123...n = 1. More specifically, the symbol is one for even permutations of the indices, −1 for odd permutations, and 0 otherwise.
Levi-Civita tensor
The Levi-Civita symbol also is used to denote the components of the Levi-Civita tensor, and in n dimensions it is an invariant of the special unitary group SU(n).[2] It also is called the alternating tensor[3] or the completely antisymmetric tensor with three indices in three 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
- ↑ The term "cyclic order" imagines the items in a list, say a, b, c, ... arranged in a circle. Then all sequences that could be encountered by going once around the circle in the direction of the sequence a, b, c, ... are in cyclic order, regardless of the starting point. See Scoby McCurdy (1894). “Cyclic order”, An exercise book in algebra. D. C. Heath & Co., p. 59.
- ↑ Michael T. Vaughn (2007). Introduction to mathematical physics. Wiley-VCH, p. 484. ISBN 3527406271.
- ↑ 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.
- ↑ T. Padmanabhan (2010). Gravitation: Foundations and Frontiers. Cambridge University Press, p. 22. ISBN 0521882230.