linguistic:paraphrasing, linguistic:translation, linguistic:voice_change
Generalizations

If it is considered as a type (1, 1) {\displaystyle (1,1)} tensor, the Kronecker tensor can be written δ j i {\displaystyle \delta _{j}^{i}} with a covariant index j {\displaystyle j} and contravariant index i {\displaystyle i}:

This tensor represents:

The generalized Kronecker delta or multi-index Kronecker delta of order 2 p {\displaystyle 2p} is a type (p, p) {\displaystyle (p,p)} tensor that is completely antisymmetric in its p {\displaystyle p} upper indices, and also in its p {\displaystyle p} lower indices.

Two definitions that differ by a factor of p ! {\displaystyle p!} are in use. Below, the version is presented has nonzero components scaled to be ± 1 {\displaystyle \pm 1}. The second version has nonzero components that are ± 1 / p ! {\displaystyle \pm 1/p!}, with consequent changes scaling factors in formulae, such as the scaling factors of 1 / p ! {\displaystyle 1/p!} in § Properties of the generalized Kronecker delta below disappearing.