1.6 Summation Convention, Kronecker Delta and Permutation Symbols
55
a × b = a i e i × b j e j = a i b j ε i jk e k = a i b j ε jki e k = a i b j ε ki j e k
(1.6.19)
Similarly, the rotation of the vector function a can be written as
∇ × a = e i
∂
∂ x i
× a j e j = ε i jk
∂a j
∂ x i
e k
(1.6.20)
There is the following relations between the permutation symbol and the
Kronecker symbol
ε i jk ε ist =
3 δ is δ it
δ ji δ js δ jt
δ ki δ ks δ kt
= 3δ js δ kt + δ ji δ ks δ it + δ ki δ is δ jt − δ it δ js δ ki − 3δ jt δ ks − δ kt δ is δ ji
= 3δ js δ kt + δ jt δ ks + δ ks δ jt − δ js δ kt − 3δ jt δ ks − δ kt δ js = δ js δ kt − δ ks δ jt =
δ js δ jt
δ ks δ kt
(1.6.21)
Equation (1.6.21) is called the ε −δ identity or epsilon-delta identity. Its general
form is
ε i jk ε rst =
δ ir δ is δ it
δ jr δ js δ jt
δ kr δ ks δ kt
(1.6.22)
Proof Let the expanded form of the third order determinant
A i j
(i, j = 1, 2, 3) be
A i j
=
A 11 A 12 A 13
A 21 A 22 A 23
A 31 A 32 A 33
Exchange the rows or columns of the determinant, then its plus or minus changes,
namely
A 21 A 22 A 23
A 11 A 12 A 13
A 31 A 32 A 33
=
A 12 A 11 A 13
A 22 A 21 A 23
A 32 A 31 A 33
= −
A i j
Exchanging the rows of the determinant, which can be expressed by the
permutation symbol
A i1 A i2 A i3
A j1 A j2 A j3
A k1 A k2 A k3
= ε i jk
A i j