266 14 Rotation of Tensors
and
R···(),···(φ)=P···(^0 ),···+∑
m= 1[
(cos(mφ)(
P···(m,)···+P···(−,m···))
+sin(mφ)i(
P(···m,)···−P···(−,m···))]
. (14.42)
The projection operators also allow the solution of tensor equations, as discussed
next.
14.4 Solution of Tensor Equations
14.4.1 Inversion of Linear Equations.
Letaμ 1 ·μandbμ 1 ·μbe irreducible tensors of rankwhich obey the rotation-like
linear relation
aμ 1 μ 2 ···μ+φHμ() 1 μ 2 ···μ(),ν 1 ν 2 ···νaν 1 ν 2 ···ν=c 0 bμ 1 μ 2 ···μ, (14.43)wherec 0 is a given coefficient. With the properties of the projectorsP(m)given
above, this equation is inverted foraμ 1 ·μaccording to
aμ 1 μ 2 ···μ=∑
m=−c(m)Pμ(m 1 μ) 2 ···μ(),ν 1 ν 2 ···νbμ 1 μ 2 ···μ, (14.44)with
c(m)=c 0 ( 1 +miφ)−^1. (14.45)
When a more general linear relation between two tensors is cast into the form
∑
m=−c(m)Pμ(m 1 μ) 2 ···μ(),ν 1 ν 2 ···νaμ 1 μ 2 ···μ=bμ 1 μ 2 ···μ, (14.46)with given coefficientsc(m), the inversion of this equation reads
aμ 1 μ 2 ···μ=∑
m=−(
c(m))− 1
Pμ(m 1 μ) 2 ···μ(),ν 1 ν 2 ···νbμ 1 μ 2 ···μ, (14.47)