3.1 Symmetry 37
3.1.6 Preliminary Remarks on “Antisymmetric Part
and Vector”
The three independent components of the antisymmetric part, in 3D, can be linked
with a vector (tensor of rank=1). This property is specific for 3D, whereas most
relations formulated here, apply also to Cartesian components in 2, 4 and higher
dimensions. This is seen as follows. Inndimensions, the number of elements of a
second rank tensor isn^2. There arenelements in the diagonal, consequently one has
n^2 −noff-diagonal elements. The number of independent elements of the symmetric
part is
Nsym=n+1
2
n(n− 1 )=1
2
n(n+ 1 ), (3.19)that of the antisymmetric part is
Nasy=1
2
n(n− 1 ). (3.20)The number of elements of a vector isn.Forn>0, the relationn=Nasyhas just
the solutionn=3.
3.1.7 Preliminary Remarks on the Symmetric Traceless Part
The symmetric traceless part cannot be expressed in terms of lower rank tensors. For
this reason, it is also referred to as theirreduciblepart of the tensor. In 3D, it has 5
independent components.
The symbol ···is also used for tensors of rank≥2 in order to indicate the
symmetric traceless (irreducible) part which, in general, has 2+1 independent
components, details later.
Different notations for the symmetric traceless part of tensor are found in the
literature. Sometimes the double arrow←→..., in same cases the brackets[...] 0 or
double brackets[[...]] 0 , where the subscript 0 indicates that the trace is zero, are
used instead of···.
3.2 Dyadics
3.2.1 Definition of a Dyadic Tensor
A second rank tensor constructed from the components of two vectors, e.g.aandb
is called adyadic tensor, sometimes also justdyadicordyad. The quantity