Kronecker Delta - Generalizations of The Kronecker Delta

Generalizations of The Kronecker Delta

Tensors
Glossary of tensor theory
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Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric tensor
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Mathematicians Leonhard Euler Carl Friedrich Gauss Augustin-Louis Cauchy Hermann Grassmann Gregorio Ricci-Curbastro Tullio Levi-Civita Jan Arnoldus Schouten Bernhard Riemann Elwin Bruno Christoffel Woldemar Voigt Élie Cartan Hermann Weyl Albert Einstein

If it is considered as a type (1,1) tensor, the Kronecker tensor, it can be written with a covariant index j and contravariant index i:

$\delta^{i}_{j} = \begin{cases} 0 & (i \ne j), \\ 1 & (i = j). \end{cases}$

This (1,1) tensor represents:

The identity mapping (or identity matrix), considered as a linear mapping or
The trace or tensor contraction, considered as a mapping
The map, representing scalar multiplication as a sum of outer products.

The generalized Kronecker delta of order 2p is a type (p,p) tensor that is a completely antisymmetric in its p upper indices, and also in its p lower indices.

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