Levi-Civita symbol
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In mathematics, and in particular in tensor calculus, the Levi-Civita symbol, also called the permutation symbol, is defined as follows:
- <math>\epsilon_{ijk} =
\left\{ \begin{matrix} +1 & \mbox{if } (i,j,k) \mbox{ is } (1,2,3), (2,3,1) \mbox{ or } (3,1,2)\\ -1 & \mbox{if } (i,j,k) \mbox{ is } (3,2,1), (1,3,2) \mbox{ or } (2,1,3)\\ 0 & \mbox{otherwise: }i=j \mbox{ or } j=k \mbox{ or } k=i \end{matrix} \right. <math>
It is named after Tullio Levi-Civita. It is used in many areas of mathematics and physics. For example, in linear algebra, the cross product of two vectors can be written as a determinant:
- <math>
\mathbf{a \times b} =
\begin{vmatrix}
\mathbf{e_1} & \mathbf{e_2} & \mathbf{e_3} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
\end{vmatrix}
= \sum_{i,j,k=1}^3 \epsilon_{ijk} \mathbf{e_i} a_j b_k <math> or more simply:
- <math>
\mathbf{a \times b} = \mathbf{c},\ c_i = \sum_{j,k=1}^3 \epsilon_{ijk} a_j b_k <math>
This can be further simplified by using Einstein notation.
The Levi-Civita symbol can be generalized to higher dimensions:
- <math>\epsilon_{ijkl\dots} =
\left\{ \begin{matrix} +1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an even permutation of } (1,2,3,4,\dots) \\ -1 & \mbox{if }(i,j,k,l,\dots) \mbox{ is an odd permutation of } (1,2,3,4,\dots) \\ 0 & \mbox{if any two labels are the same} \end{matrix} \right. <math>
See even permutation or symmetric group for a definition of 'even permutation' and 'odd permutation'
The tensor whose components are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called the permutation tensor. It is actually a pseudotensor because it get a minus sign under orthogonal transformation of jacobian determinant -1 (i.e. a rotation composed with a reflection).
The Levi-Civita symbol is related to the Kronecker delta. In three dimensions, the relationship is given by the following equations:
- <math>
\sum_{i=1}^3 \epsilon_{ijk}\epsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km} <math>
- <math>
\sum_{i,j=1}^3 \epsilon_{ijk}\epsilon_{ijn} = 2\delta_{kn} <math> Furthermore, it can be shown that
- <math>
\sum_{i,j,k,\dots=1}^n \epsilon_{ijk\dots}\epsilon_{ijk\dots} = n! <math> is always fulfilled in n dimensions. In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual.
References
- Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (See section 3.5 for a review of tensors in general relativity).de:Levi-Civita-Symbol
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