generalized Kronecker delta symbol
Let and be natural numbers such that .
Further, let and be natural numbers in
for all in .
Then the
generalized Kronecker delta symbol, denoted by
,
is zero if
or for some , or if
as sets.
If none of the above conditions are met, then
is defined as the sign of the permutation
that maps
to .
From the definition, it follows that when , the generalized Kronecker delta symbol reduces to the traditional delta symbol . Also, for , we obtain
where is the Levi-Civita permutation symbol.
For any we can write the generalized delta function
as a determinant of traditional delta symbols. Indeed,
if is the permutation group
of elements, then
The first equality follows since the sum one the first line has only one non-zero term; the term for which . The second equality follows from the definition of the determinant.
Title | generalized Kronecker delta symbol |
---|---|
Canonical name | GeneralizedKroneckerDeltaSymbol |
Date of creation | 2013-03-22 13:31:38 |
Last modified on | 2013-03-22 13:31:38 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 5 |
Author | matte (1858) |
Entry type | Definition |
Classification | msc 15A99 |
Related topic | LeviCivitaPermutationSymbol3 |