Tensors for Physics

Appendix: Exercises... 403

Exercise Chapter 9

9.1 Verify the Required Properties of the Third and Fourth Rank Irreducible
Tensors(9.5)and(9.6) (p. 157)

The required symmetry of the third and fourth rank tensorsaμaνaλandaμaνaλaκ,
as given explicitly by (9.5) and (9.6) is seen by inspection, note thataμaν=aνaμ
andδμν=δνμ.
Settingλ=ν,in(9.5), leads to

aμaνaν=aμa^2 −

1

5

a^2 (aμ 3 +aνδμν+aνδμν)= 0.

Hereδνν=3 andaνδμν=aμare used.
Likewise, puttingκ=λin (9.6), yields

aμaνaλaκ =aμaνa^2

−

1

7

a^2 (aμaν 3 +aμaλδνλ+aμaλδνλ+aνaλδμλ+aνaλδμλ+a^2 δμν)

+

1

35

a^4 (δμν 3 +δμλδνλ+δμλδνλ)

=aμaνa^2

(

1 −

1

7

7

)

−

1

7

a^4 δμν+

1

35

a^4 δμν 5 = 0 ,

where, e.g.aλδμλ=aμandδμλδνλ=δμνhave been used.

Exercises Chapter 10

10.1 Prove the Product Rule(10.13)for the Laplace Operator(p.166)
The proof of (10.13), viz.

Δ(g(r)Xμ 1 μ 2 ···μ)=(g′′− 2 r−^1 g′)Xμ 1 μ 2 ···μ

starts from the general relation

Δ(fg)=fΔg+ 2 (∇κf)(∇κg)+gΔf,

for any two functionsfandg. Now choose forfthe-th descending multipole tensor
Xμ 1 μ 2 ···μand assume that the scalargdepends onr=|r|only. SinceΔX...=0,
one obtains

Δ(g(r)Xμ 1 μ 2 ···μ)=Xμ 1 μ 2 ···μΔg+ 2 g′r−^1 rκ∇κXμ 1 μ 2 ···μ.