Definition:Kronecker Delta/Number

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Definition

Let $\Gamma$ be a set.

Then $\delta_{\alpha \beta}: \Gamma \times \Gamma \to \set {0, 1}$ is the mapping on the cartesian square of $\Gamma$ defined as:

$\quad \forall \tuple {\alpha, \beta} \in \Gamma \times \Gamma: \delta_{\alpha \beta} := \begin {cases} 1 & : \alpha = \beta \\ 0 & : \alpha \ne \beta \end {cases}$


This use of $\delta$ is known as the Kronecker delta notation or Kronecker delta convention.


Also denoted as

When used in the context of tensors, the notation can often be seen as ${\delta^i}_j$.


Also presented as

The Kronecker delta can be expressed using Iverson bracket notation as:

$\delta_{\alpha \beta} := \sqbrk {\alpha = \beta}$


Sources

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