Encyclopedia > Levi-Civita symbol

  Article Content

Levi-Civita symbol

The Levi-Civita[?] symbol, also called the permutation symbol, is defined as follow:
<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 used in many areas of mathematics and physics. For example, in linear algebra, the cross product of two vectors can be written as:

<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 tensor whose components are given by the Levi-Civita symbol (a tensor of covariant rank 3) is sometimes called the permutation tensor.

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'

A related symbol is the Kronecker delta.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Springs, New York

... an average density of 177.0/km² (458.2/mi²). The racial makeup of the town is 89.82% White, 1.47% African American, 0.20% Native American, 1.45% Asian, 0.02% ...

 
 
 
This page was created in 38.2 ms