Kronecker Delta

Author

Eric W. Weisstein
June 9, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/KroneckerDelta.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/KroneckerDelta.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Identities

(∑_(i=1)^3)(∑_(j=1)^3)(Subscript[δ, i,j])

Sum[KroneckerDelta[i,j],{i,3},{j,3}]

(∑_(i=1)^3)(∑_(j=1)^3)(∑_(k=1)^3)(Subscript[δ, i,j])(Subscript[ϵ, i,j,k])

Sum[KroneckerDelta[i,j]Signature[{i,j,k}],{i,3},{j,3},{k,3}]

(∑_(i=1)^3)(∑_(j=1)^3)(∑_(k=1)^3)(Subscript[ϵ, i,p,q])(Subscript[ϵ, j,p,q])=2(Subscript[δ, i,j])

Table[Sum[Signature[{i,p,q}]Signature[{j,p,q}],{p,3},{q,3}],{i,3},{j,3}]

Table[2KroneckerDelta[i,j],{i,3},{j,3}]

(∑_(k=1)^3)(Subscript[ϵ, i,j,k]) (Subscript[ϵ, p,q,k])=(Subscript[δ, i,p]) (Subscript[δ, j,q])-(Subscript[δ, i,q]) (Subscript[δ, j,p])

Table[Sum[Signature[{i,j,k}]Signature[{p,q,k}],{k,3}],{i,3},{j,3},{p,3},{q,3}]

Table[KroneckerDelta[i,p]KroneckerDelta[j,q]-KroneckerDelta[i,q]KroneckerDelta[j,p],{i,3},{j,3},{p,3},{q,3}]