38 3 Symmetry of Second Rank Tensors, Cross Product
ab:=⎛
⎝
a 1 b 1 a 1 b 2 a 1 b 3
a 2 b 1 a 2 b 2 a 2 b 3
a 3 b 1 a 3 b 2 a 3 b 3⎞
⎠ (3.21)
is a tensor. It must not be confused with the scalar producta·b=a 1 b 1 +a 2 b 2 +a 3 b 3.
WithAμν=aμbν, the decomposition (3.4) reads:
aμbν=1
3
aλbλδμν+1
2
(
aμbν−aνbμ)
+aμbν. (3.22)The symmetric traceless part, in accord with (3.3), given by
aμbν =1
2
(
aμbν+aνbμ)
−
1
3
aλbλδμν. (3.23)The trace of the dyadicabis the scalar producta·b. In 3D, the antisymmetric part
of the dyadic is linked with the cross producta×b, details later.
The product of dyadic tensors, with total contraction, can be inferred from (3.10).
The case where both dyadics are symmetric traceless is discussed in Sect.3.2.2.
3.1 Exercise:
Symmetric and Antisymmetric Parts of a Dyadic in Matrix Notation
Write the symmetric traceless and the antisymmetric parts of the dyadic tensorAμν=
aμbνinmatrixformforthevectorsa:{ 1 , 0 , 0 }andb:{ 0 , 1 , 0 }.Computethenormof
the symmetric and the antisymmetric parts and compare withAμνAμνandAμνAνμ.
3.2.2 Products of Symmetric Traceless Dyadics
Consider two dyadics, formed by the pairs of vectorsa,bandc,d, respectively. As
discussed above for the tensor multiplication with contraction, the expressionab·cd
stands for the dyadicad, multiplied by the scalar productb·c. The double dot product
then yieldsa·db·c.
Of particular interest is the case, where the symmetric traceless parts of these
dyadics are multiplied. Here one has
(ab·cd)μν=aμbλcλdν=1
4
(b·caμdν+b·daμcν+a·cbμdν+a·dbμcν)−
1
6
c·d(aμbν+aνbμ)−1
6
a·b(cμdν+cνdμ)+1
9
a·bc·dδμν. (3.24)