390 Appendix: Exercises...
respectively. The corresponding angles are exactly 45◦for the angle betweenaand
b, and≈ 70. 5 ◦and≈ 35. 3 ◦, for the other two angles.
Exercises Chapter 3
3.1 Symmetric and Antisymmetric Parts of a Dyadic in Matrix Notation(p.38)
Write the symmetric traceless and the antisymmetric parts of the dyadic tensor Aμν=
aμbνin matrix form for the vectorsa:{ 1 , 0 , 0 }andb:{ 0 , 1 , 0 }.Compute the norm
squared of the symmetric and the antisymmetric parts and compare with AμνAμν
and AμνAνμ.
In matrix notation, the tensorAμνis equal to
A=
⎛
⎝
010
000
000
⎞
⎠. (A.1)
The trace of this matrix is zero. So its symmetric part coincides with its symmetric
traceless part
A =1
2
⎛
⎝
010
100
000
⎞
⎠. (A.2)
The antisymmetric part of this tensor is
Aasy=1
2
⎛
⎝
010
− 100
000
⎞
⎠. (A.3)
The tensor product A·Ayields
Aμλ Aλν =1
4
⎛
⎝
100
010
000
⎞
⎠, (A.4)
and consequently
Aμλ Aλμ =1
2
.
Similarly, the product of the antisymmetric part with its transposed, viz.
AasyμλAasyνλ