Tensors for Physics

394 Appendix: Exercises...

(i) Tetrahedron: the position vectors of the corners are

u^1 =ex+ey+ez, u^2 =−ex−ey+ez,

u^3 =ex−ey−ez, u^4 =−ex+ey−ez.

In the productsuiμuiνthe mixed terms involvingexμeyν,exμezν,eyμezνhave the signs
(+,+,+),(+,−,−),(−,−,+),(−,+,−)fori = 1 , 2 , 3 ,4, respectively. The
sum of these mixed terms vanishes and one finds

∑^4

i= 1

uμiuiν= 4 (eμxexν+eμyeνy+ezμezν)= 4 δμν,

thus the moment of inertia tensor

Θμν= 8 mδμν

is isotropic.
(ii) Octahedron: here the sum

∑ 6

i= 1 u

i

μu

i

νyields 2(e

x

μe

x

ν+e

y

μe

y
ν+ezμezν)and conse-
quently

Θμν= 4 mδμν.

5.2 Verify the Relation(5.51)for the Triple Product of a Symmetric Traceless
Tensor(p.72)
Hint: use the matrix notation

⎛

⎝

a 00

0 b 0

00 c

⎞

⎠,

with c=−(a+b),for the symmetric traceless tensor in its principal axis system.
Compute the expressions on both sides of(5.51)and compare.
In matrix notation, the left hand side of

a·a·a=

1

2

a(a:a)

is

⎛

⎝

2 a^3 / 3 −b^3 / 3 −c^3 / 30 0

02 b^3 / 3 −c^3 / 3 −c^3 / 30

002 c^3 / 3 −a^3 / 3 −b^3 / 3

⎞

⎠. (A.10)