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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} \equiv \cases{
0 & for $i \not = j$\cr
1 & for $i = j$.\cr}
\end{displaymath} (1)

It has the Complex Generating Function
\delta_{mn} = {1\over 2\pi i} \int z^{m-n-1}\,dz,
\end{displaymath} (2)

where $m$ and $n$ are Integers. In 3-space, the Kronecker delta satisfies the identities
\delta_{ii} = 3
\end{displaymath} (3)

\delta_{ij}\epsilon_{ijk} = 0
\end{displaymath} (4)

\epsilon_{ipq}\epsilon_{jpq} = 2\delta_{ij}
\end{displaymath} (5)

\epsilon_{ijk}\epsilon_{pqk} = \delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp},
\end{displaymath} (6)

where Einstein Summation is implicitly assumed, $i,j=1, 2, 3$, and $\epsilon$ is the Permutation Symbol.

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

\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} (7)

Since, by definition, the coordinates $x_i$ and $x_j$ are independent for $i \not = j$,
{\partial x_i'\over\partial x_j'} = {\delta'}_j^i,
\end{displaymath} (8)

{\delta'}_j^i = {\partial x_i'\over\partial x_k} {\partial x_l\over\partial x_j'}\delta_l^k,
\end{displaymath} (9)

and $\delta_j^i$ is really a mixed second Rank Tensor. It satisfies
{\delta_{ab}}^{jk} = \epsilon_{abi}\epsilon^{jki} = \delta_a^j\delta_b^k-\delta_a^k\delta_b^j
\end{displaymath} (10)

\end{displaymath} (11)

\epsilon_{aij}\epsilon^{bij} = {\delta_{ai}}^{bi}= 2\delta_a^b.
\end{displaymath} (12)

See also Delta Function, Permutation Symbol

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© 1996-9 Eric W. Weisstein