Tensors for Physics

Appendix: Exercises... 407

Δ(μνλ,μ^3 ) ′ν′λ′=

1

6

[

(δμμ′δνν′+δμν′δνμ′)δλλ′+(δμμ′δνλ′+δμλ′δνμ′)δλν′

+(δμλ′δνν′+δμν′δνλ′)δλμ′

]

−

1

15

[

(δνλδμμ′+δμλδνμ′+δμνδλμ′)δν′λ′

+(δνλδμν′+δμλδνν′+δμνδλν′)δμ′λ′

+(δνλδμλ′+δμλδνλ′+δμνδλλ′)δμ′ν′

]

. (A.13)

Notice that allδ-tensors in the bracket[...]behind the factor^16 in (A.13) contain
one primed and one unprimed subscript. On the other hand, the triple products of
δ-tensors behind the factor 151 contain one unit tensor with two primed subscripts,
one with two unprimed ones, and one with mixed subscripts.
The contractionλ′=λyieldsΔμνλ,μ(^3 ) ′ν′λ=^75 Δ(μν,μ^2 ) ′ν′. This is in accord with (11.6).

Exercises Chapter 12

12.1 Verify the Numerical Factor in(12.7)for the Integral over a Triple Product
of Tensors(p.202)
Hint: Putν=λ, κ=σ, τ=μand use the relevant formulae given in Sect.11.4.
The recommended contraction, on the left hand side of (12.7)involves

̂rμ̂rν̂rν̂rκ̂rκ̂rμ =

1

3

̂rμ̂rκ̂rκ̂rμ =

2

9

,

and the subsequent integration still yields^29.

On the other hand, due toΔμν,νκ,κμ=^3512 ,cf.(11.38), the right hand side becomes

8

105

35

12 =

2

9 , just as expected.

12.2 Prove that the Fokker-Planck Equation Implies an Increase of the
Orientational Entropy with Increasing Time(p.212)
Hint: The time change of an orientational average isd〈ψ〉/dt=

∫

∂(ψf)/∂td^2 u.
The time change offln(f/f 0 )is ln(f/f 0 )∂f/∂t+ff−^1 ∂f/∂t. Thus the time change
of the orientational entropysa=−kB

∫

fln(f/f 0 )d^2 uis

d

dt

sa=−kB

∫

ln(f/f 0 )

∂

∂t

fd^2 u=−kBν 0

∫

ln(f/f 0 )LμLμfd^2 u,

where

∫ ∂

∂tfd

(^2) u=0 and (12.41) have been used. HereLμ=εμνλuν∂
∂uλis the
relevant differential operator. An integration by part leads to the expression
d
dt
sa=kBν 0

∫

f−^1 (Lμf)Lμfd^2 u,