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7 Quadrupole Contributions from Interface and Bulk
Using the Levi-Civita symbols thus defined, the vector product is represented in
the following form,
[B × C] i = ε ij k B j C k .
In this section of Appendix we employ the Einstein’s convention of contraction,
and omit the summation symbol
over i, j , or k. 5 The rotation of a vector is
represented in a similar way, [∇ × C] i = ε ij k ∂ j C k . The scalar triple product is
given by A · (B × C) = ε ij k A i B j C k .
The Levi-Civita tensor satisfies the following contraction formulas,
ε ij k ε pqk = δ ip δ jq − δ iq δ jp ,
(7.133)
ε ij k ε pj k = 2δ ip ,
(7.134)
ε ij k ε ij k = 6.
(7.135)
Proof ε ij k in Eq. (7.132) is expressed using a set of orthonormal vectors
{e x , e y , e z } as
ε ij k =
(e x · e i ) (e x · e j ) (e x · e k )
(e y · e i ) (e y · e j ) (e y · e k )
(e z · e i ) (e z · e j ) (e z · e k )
≡ | U (xyz, ij k) | .
(7.136)
Therefore,
ε ij k ε pqr = | U (xyz, ij k) | · | U (xyz, pqr) | =
U (xyz, ij k)
T
· | U (xyz, pqr) |
= | U (ij k, pqr) |
=
δ ip δ iq δ ir
δ jp δ jq δ jr
δ kp δ kq δ kr
= δ ir (δ jp δ kq − δ jq δ kp ) − δ jr (δ ip δ kq − δ iq δ kp ) + δ kr (δ ip δ jq − δ iq δ jp ).
Equations (7.133), (7.134), (7.135) are proved by taking the contraction.
ε ij k ε pqk = δ ik (δ jp δ kq − δ jq δ kp ) − δ jk (δ ip δ kq − δ iq δ kp ) + δ kk (δ ip δ jq − δ iq δ jp )
= δ iq δ jp − δ ip δ jq − δ jq δ ip + δ jp δ iq + 3(δ ip δ jq − δ iq δ jp )
= δ ip δ jq − δ iq δ jp ,
(7.133)
5 Therefore, ε ij k B j C k ≡
j,k
ε ij k B j C k .