Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. So, for example, delta_{12} = 0, but delta_{33} = 1. It is written as the symbol δij, and treated as a notational shorthand rather than as a function.

Extensions of the delta function

In the same fashion, we may define an analogous, multi-dimensional function of many variables

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:delta^{j_1 j_2 ... j_n}_{i_1 i_2 ...i_n}:= prod_{k=1}^n delta_{i_k j_k}.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This function takes the value 1 if and only if all the upper indices match the corresponding lower one, and the value zero otherwise.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~