Kronecker delta

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Kronecker delta

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Named after German mathematician Leopold Kronecker (1823–1891)

Kronecker delta (plural Kronecker deltas)

(mathematics) A binary function, written as δ with two subscripts, which evaluates to 1 when its arguments are equal, and 0 otherwise.
- 1998, Robert G. Deissler, Turbulent Fluid Motion, Taylor & Francis, page 24:
  The Kronecker delta $\delta _{ij}$ is an example of an isotropic tensor. That is, its components remain invariant with rotation of coordinate axes.
- 2007, J. N. Reddy, An Introduction to Continuum Mechanics, Cambridge University Press, page 20,
  Further, the Kronecker delta and the permutation symbol are related by the identity, known as the $e$ - $\delta$ identity ,
  $e_{ijk}e_{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}$ . (2.2.43)
  
  The permutation symbol and the Kronecker delta prove to be very useful in proving vector identities.
- 2017, Sadri Hassani, Special Relativity: A Heuristic Approach, Elsevier, page 257:
  The most important property of the Kronecker delta occurs when it shares a common repeated index with another tensor:
  Note 10.1.3. When an index of a tensor $T$ is contracted with one of the indices of the Kronecker delta, the result is an expression in which the Kronecker delta is removed and the contracted index of $T$ is replaced by the other index of the Kronecker delta.
(mathematics) A unary function, written as δ with a single index, which evaluates to 1 at zero, and 0 elsewhere.

The notation $\delta ^{ij}$ and $\delta _{j}^{i}$ are also sometimes used.
In linear algebra, the Kronecker delta can be regarded as a tensor of type (1,1).
The function can also be expressed using Iverson bracket notation, as $[i = j]$ .
The single-argument function is equivalent to setting $j=0$ in the binary function.