Kronecker delta

English

Etymology

Named after German mathematician Leopold Kronecker (1823–1891)

Noun

Kronecker delta (plural Kronecker deltas)

1. (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 [see Problem 2.5(d)],

     $e_{ijk}e_{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}$.      (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.

2. (mathematics) A unary function, written as δ with a single index, which evaluates to 1 at zero, and 0 elsewhere.

Usage notes

• 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.

Synonyms

• (binary function): Kronecker tensor, substitution tensor

Derived terms

• generalized Kronecker delta

Translations

Translations

• Chinese:

  Mandarin: 克羅內克爾δ／克罗内克尔δ (Kèluónèikè'ěr δ)

See also

• Dirac delta
• Dirac delta function
• Dirac measure
• Iverson bracket
• Kronecker product
• Kronecker symbol
• Levi-Civita symbol

Further reading

• Unit function on Wikipedia.Wikipedia
• Iverson bracket on Wikipedia.Wikipedia