Editing Kronecker delta (section)

=== Definitions of the generalized Kronecker delta ===
In terms of the indices, the generalized Kronecker delta is defined as:<ref>{{cite book|first=Theodore|last=Frankel|title=The Geometry of Physics: An Introduction|edition=3rd|date=2012|publisher=Cambridge University Press|isbn=9781107602601}}</ref><ref>{{cite book|first=D. C.|last=Agarwal|title=Tensor Calculus and Riemannian Geometry|edition=22nd|date=2007|publisher=Krishna Prakashan Media}}{{ISBN missing}}</ref>
<math display="block">\delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} = \begin{cases}
\phantom-1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an even permutation of } \mu_1 \dots \mu_p \\
-1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an odd permutation of } \mu_1 \dots \mu_p \\
\phantom-0 & \quad \text{in all other cases}.
\end{cases}</math>

Let <math>\mathrm{S}_p</math> be the [[symmetric group]] of degree <math>p</math>, then:
<math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
= \sum_{\sigma \in \mathrm{S}_p} \sgn(\sigma)\, \delta^{\mu_1}_{\nu_{\sigma(1)}}\cdots\delta^{\mu_p}_{\nu_{\sigma(p)}}
= \sum_{\sigma \in \mathrm{S}_p} \sgn(\sigma)\, \delta^{\mu_{\sigma(1)}}_{\nu_1}\cdots\delta^{\mu_{\sigma(p)}}_{\nu_p}. </math>

Using [[Antisymmetric tensor#Notation|anti-symmetrization]]:
<math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
= p! \delta^{\mu_1}_{[ \nu_1} \dots \delta^{\mu_p}_{\nu_p ]}
= p! \delta^{[ \mu_1}_{\nu_1} \dots \delta^{\mu_p ]}_{\nu_p}.</math>

In terms of a <math>p\times p</math> [[determinant]]:<ref>{{cite book |first1=David |last1=Lovelock |first2=Hanno |last2=Rund |title=Tensors, Differential Forms, and Variational Principles |publisher=Courier Dover Publications |year=1989 |isbn=0-486-65840-6 }}</ref>
<math display="block">\delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} =
\begin{vmatrix}
\delta^{\mu_1}_{\nu_1} & \cdots & \delta^{\mu_1}_{\nu_p} \\
\vdots & \ddots & \vdots \\
\delta^{\mu_p}_{\nu_1} & \cdots & \delta^{\mu_p}_{\nu_p}
\end{vmatrix}.</math>

Using the [[Laplace expansion]] ([[Determinant#Laplace's expansion and the adjugate matrix|Laplace's formula]]) of determinant, it may be defined [[Recursion|recursively]]:<ref>A recursive definition requires a first case, which may be taken as {{math|1=''δ'' = 1}} for {{math|1=''p'' = 0}}, or alternatively {{math|1=''δ''{{su|p=''μ''|b=''ν''|lh=0.9em}} = ''δ''{{su|p=''μ''|b=''ν''|lh=0.9em}}}} for {{math|1=''p'' = 1}} (generalized delta in terms of standard delta).</ref>
<math display="block">\begin{align}
  \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
    &= \sum_{k=1}^p (-1)^{p+k} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k} \dots \check\mu_p}_{\nu_1 \dots \check\nu_k \dots \nu_p} \\
    &= \delta^{\mu_p}_{\nu_p} \delta^{\mu_1 \dots \mu_{p - 1}}_{\nu_1 \dots \nu_{p-1}} - \sum_{k=1}^{p-1} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k-1}\, \mu_k\, \mu_{k+1} \dots \mu_{p-1}}_{\nu_1 \dots \nu_{k-1}\, \nu_p\, \nu_{k+1} \dots \nu_{p-1}},
\end{align}</math>
where the caron, <math>\check{}</math>, indicates an index that is omitted from the sequence.

When <math>p=n</math> (the dimension of the vector space), in terms of the [[Levi-Civita symbol]]:
<math display="block">\delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n} = \varepsilon^{\mu_1 \dots \mu_n}\varepsilon_{\nu_1 \dots \nu_n}\,.</math>
More generally, for <math>m=n-p</math>, using the [[Einstein summation convention]]:
<math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = \tfrac{1}{m!} \varepsilon^{\kappa_1 \dots \kappa_m \mu_1 \dots \mu_p}\varepsilon_{\kappa_1 \dots \kappa_m \nu_1 \dots \nu_p}\,.</math>