|
|
|
|
Levi-Civita permutation symbol
|
(Definition)
|
|
Definition 1 Let
for all
. The Levi-Civita permutation symbols
and
are defined as
The Levi-Civita permutation symbol is a special case of the generalized Kronecker delta symbol. Using this fact one can write the Levi-Civita permutation symbol as the determinant of an matrix consisting of traditional delta symbols. See the entry on the generalized Kronecker symbol for details.
When using the Levi-Civita permutation symbol and the generalized Kronecker delta symbol, the Einstein summation convention is usually employed. In the below, we shall also use this convention.
- When
, we have for all in ,
- When
, we have for all in ,
Let us prove these properties. The proofs are instructional since they demonstrate typical argumentation methods for manipulating the permutation symbols.
Proof. For equation 1, let us first note that both sides are antisymmetric with respect of and . We therefore only need to consider the case and . By substitution, we see that the equation holds for
, i.e., for and . (Both sides are then one). Since the equation is anti-symmetric in and , any set of values for these can be reduced the above case (which holds). The equation thus holds for all values of and . Using equation 1, we have for equation 2
Here we used the Einstein summation convention with going from to . Equation 3 follows similarly from equation 2. To establish equation 4, let us first observe that both sides vanish when . Indeed, if , then one can not choose and such that both permutation symbols on the left are nonzero. Then, with fixed, there are only two ways to choose and from the remaining two indices. For any such indices, we have
(no summation), and the result follows. The last property follows since and for any distinct indices in , we have
(no summation).
|
"Levi-Civita permutation symbol" is owned by matte. [ full author list (2) ]
|
|
(view preamble)
Cross-references: curl, Cartesian coordinates, open set, vector field, arguments, adjacent, triple scalar product, clear, expression, cross product, component, orthonormal basis, oriented, right, vectors, summation, indices, fixed, vanish, reduced, anti-symmetric, antisymmetric, sides, equation, permutation, proofs, properties, Einstein summation convention, Kronecker symbol, matrix, determinant, generalized Kronecker delta symbol
There are 6 references to this entry.
This is version 10 of Levi-Civita permutation symbol, born on 2003-03-17, modified 2006-01-12.
Object id is 4116, canonical name is LeviCivitaPermutationSymbol3.
Accessed 18972 times total.
Classification:
| AMS MSC: | 05A10 (Combinatorics :: Enumerative combinatorics :: Factorials, binomial coefficients, combinatorial functions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|