The simplest interpretation of the Kronecker delta is as the discrete version of the Delta Function defined by
![\begin{displaymath}
\delta_{ij} \equiv \cases{
0 & for $i \not = j$\cr
1 & for $i = j$.\cr}
\end{displaymath}](k_838.gif) |
(1) |
It has the Complex Generating Function
![\begin{displaymath}
\delta_{mn} = {1\over 2\pi i} \int z^{m-n-1}\,dz,
\end{displaymath}](k_839.gif) |
(2) |
where
and
are Integers. In 3-space, the Kronecker delta satisfies the identities
![\begin{displaymath}
\delta_{ii} = 3
\end{displaymath}](k_840.gif) |
(3) |
![\begin{displaymath}
\delta_{ij}\epsilon_{ijk} = 0
\end{displaymath}](k_841.gif) |
(4) |
![\begin{displaymath}
\epsilon_{ipq}\epsilon_{jpq} = 2\delta_{ij}
\end{displaymath}](k_842.gif) |
(5) |
![\begin{displaymath}
\epsilon_{ijk}\epsilon_{pqk} = \delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp},
\end{displaymath}](k_843.gif) |
(6) |
where Einstein Summation is implicitly assumed,
, and
is the Permutation Symbol.
Technically, the Kronecker delta is a Tensor defined by the relationship
![\begin{displaymath}
\delta_l^k {\partial x_i'\over\partial x_k} {\partial x_l\ov...
...l x_k\over\partial x_j'}
= {\partial x_i'\over\partial x_j'}.
\end{displaymath}](k_846.gif) |
(7) |
Since, by definition, the coordinates
and
are independent for
,
![\begin{displaymath}
{\partial x_i'\over\partial x_j'} = {\delta'}_j^i,
\end{displaymath}](k_850.gif) |
(8) |
so
![\begin{displaymath}
{\delta'}_j^i = {\partial x_i'\over\partial x_k} {\partial x_l\over\partial x_j'}\delta_l^k,
\end{displaymath}](k_851.gif) |
(9) |
and
is really a mixed second Rank Tensor. It satisfies
![\begin{displaymath}
{\delta_{ab}}^{jk} = \epsilon_{abi}\epsilon^{jki} = \delta_a^j\delta_b^k-\delta_a^k\delta_b^j
\end{displaymath}](k_853.gif) |
(10) |
![\begin{displaymath}
\delta_{abjk}=g_{aj}g_{bk}-g_{ak}g_{bj}
\end{displaymath}](k_854.gif) |
(11) |
![\begin{displaymath}
\epsilon_{aij}\epsilon^{bij} = {\delta_{ai}}^{bi}= 2\delta_a^b.
\end{displaymath}](k_855.gif) |
(12) |
See also Delta Function, Permutation Symbol
© 1996-9 Eric W. Weisstein
1999-05-26