Tensors for Physics

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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)
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