96 7 Fields, Spatial Differential Operators
Fig. 7.7 Uniaxial tensor
fields for the director in a
nematic liquid crystal
Fig. 7.8 Bricks indicating
the alignment tensor of a
nematic liquid crystal
Examples are shown in Fig.7.7, adapted from [23]. The ‘defect’ structure in the lower
left corner of the right figure is typical for a nematic substance, cf. Chap. 15.
In the general biaxial case, ellipsoids associated the tensor, as discussed in
Sect.5.4, could be used to to visualize a second rank tensor field. However, “bricks”
with their sides proportional to the principal semi-axes can more easily convey the
information about the different principal values of the tensor at different space points.
As an example, the alignment tensor in the vicinity of a “defect” in a nematic liquid
crystal is shown in Fig.7.8, adapted from [24, 83].
7.4.2 Spatial Derivatives of Tensor Fields
LetTμν(r)be a tensor field. Application of the nabla operator∇λyields the third
rank tensor∇λTμν. By analogy to (7.31), the tensor of rank three can be decomposed
into parts associated with a vector, with a second rank tensor, and with the pertaining
irreducible symmetric traceless third rank tensor. The first one of these parts involves
thetensor divergence
∇λTλν,
which is a vector. Notice that this expression has to be distinguished from∇λTνλ,
when one hasTμν =Tνμ.
An application of the tensor divergence used for the pressure tensor, occurs in the
local conservation equation for the linear momentum.