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Kronecker Delta


The simplest interpretation of the Kronecker delta is as the discrete version of the delta function defined by

 delta_(ij)={0   for i!=j; 1   for i=j.
(1)

The Kronecker delta is implemented in the Wolfram Language as KroneckerDelta[i, j], as well as in a generalized form KroneckerDelta[i, j, ...] that returns 1 iff all arguments are equal and 0 otherwise.

It has the contour integral representation

 delta_(mn)=1/(2pii)∮_gammaz^(m-n-1)dz,
(2)

where gamma is a contour corresponding to the unit circle and m and n are integers.

In three-space, the Kronecker delta satisfies the identities

delta_(ii)=3
(3)
delta_(ij)epsilon_(ijk)=0
(4)
epsilon_(ipq)epsilon_(jpq)=2delta_(ij)
(5)
epsilon_(ijk)epsilon_(pqk)=delta_(ip)delta_(jq)-delta_(iq)delta_(jp),
(6)

where Einstein summation is implicitly assumed, i,j=1, 2, 3, and epsilon_(ijk) is the permutation symbol.

Technically, the Kronecker delta is a tensor defined by the relationship

 delta_l^k(partialx_i^')/(partialx_k)(partialx_l)/(partialx_j^')=(partialx_i^')/(partialx_k)(partialx_k)/(partialx_j^')=(partialx_i^')/(partialx_j^').
(7)

Since, by definition, the coordinates x_i and x_j are independent for i!=j,

 (partialx_i^')/(partialx_j^')=delta^'_j^i,
(8)

so

 delta^'_j^i=(partialx_i^')/(partialx_k)(partialx_l)/(partialx_j^')delta_l^k,
(9)

and delta_j^i is really a mixed second-rank tensor. It satisfies

delta_(ab)^(jk)=epsilon_(abi)epsilon^(jki)
(10)
=delta_a^jdelta_b^k-delta_a^kdelta_b^j
(11)
delta_(abjk)=g_(aj)g_(bk)-g_(ak)g_(bj)
(12)
epsilon_(aij)epsilon^(bij)=delta_(ai)^(bi)
(13)
=2delta_a^b.
(14)

See also

Delta Function, Permutation Symbol, Permutation Tensor

Related Wolfram sites

http://functions.wolfram.com/IntegerFunctions/KroneckerDelta/

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Kronecker Delta." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KroneckerDelta.html

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