11.4 Isotropic Coupling Tensors 193
aμbλ aλaν ≡Δμν,λκ,σ τ aλbκ aσaτ=1
6
a^2 aμbν+1
6
a·baμaν. (11.46)Puttingb=a, in these equations yields
aμaλ aλaν ≡Δμν,λκ,σ τ aλaκ aσaτ =1
3
a^2 aμaν. (11.47)Multiplication of the last equation byaμaν leads to
aμaν aνaλ aλaμ =Δμν,λκ,σ τ aμaνaλaκ aσaτ =2
9
a^6. (11.48)11.5 Coupling of a Vector with Irreducible Tensors
The product of a vectorbwith an irreducible tensorAof rankyields a tensor of
rank+1 which can be decomposed into an irreducible tensor of rank+1 and
terms involving irreducible tensors of ranksand−1. With the help of isotropic
Δ-tensors and the-tensor, this decomposition reads:
bμAμ 1 μ 2 ···μ=Δ(μμ+ 11 μ) 2 ···μ,νν 1 ν 2 ···νbνAν 1 ν 2 ···ν−( 2 − 1 )
( 2 + 1 )− 1
()μ 1 μ 2 ···μ,μ,ν 1 ν 2 ···ν(b×A)ν 1 ν 2 ···ν+
( 2 − 1 )
( 2 + 1 )
Δ()μ 1 μ 2 ···μ,ν ν 1 ν 2 ···ν− 1 (b·A)ν 1 ν 2 ···ν− 1. (11.49)The first term on the right hand side of (11.49) corresponds to
bμAμ 1 μ 2 ···μ. (11.50)The cross product and the dot product occurring in the second and third term are
given by
(b×A)ν 1 ν 2 ···ν=εν 1 λκbλAκν 2 ···ν, (11.51)and
(b·A)ν 1 ν 2 ···ν− 1 =bλAλν 1 ν 2 ···ν− 1. (11.52)
For=1, i.e. whenAis a vector, these relations reduce to the expressions given in
Chap. 6 for the decomposition of dyadics.