2.5 Definitions of Vectors and Tensors in Physics 23
2.5.2 What is a Tensor?.
Tensors are important “tools” for the characterization of anisotropies; but what is
meant by the notiontensor? Here mainly Cartesian tensors of rank,=0, 1, 2,...
are treated. These are quantities withindices which change in a specific way, when
the coordinate system is rotated. More specifically: a Cartesian tensor of rankis a
quantity withindices, e.g.Aμ 1 μ 2 ...μ, whose Cartesian componentsA′μ 1 μ 2 ...μin a
rotated coordinate system are obtained from the original ones by the application of
rotation matricesUto each one of the indices, viz.:
A′μ 1 μ 2 ...μ=Uμ 1 ν 1 Uμ 2 ν 2 ...UμνAν 1 ν 2 ...ν. (2.45)In this sense,scalarsandvectorsare tensors of rank=0 and=1. Examples for
vectors are the position vectorrof a particle, its velocityv, its linear momentump,
as already mentioned before, but also its orbital angular momentumL, its spins,as
well as an electric fieldEand a magnetic fieldB.
Tensors of rank=2 are frequently referred to astensorswithout indicating their
rank. Examples are the moment of inertia tensor, the pressure tensor or the stress
tensor. Applications will be discussed later.
A second rank tensor can also be written as a matrix. However, it is distinguished
from an arbitrary 3×3-matrix by the transformation properties of its components,
just as not any 3-tuple is a vector in the sense described above. Of course, the matrix
notation does not work for tensors of rank 3 or of higher rank.
2.5.3 Multiplication by Numbers and Addition of Tensors
The multiplication of a tensor by real numberkmeans the multiplication of all its
elements by this number, which is almost trivial in component notation:
k(A)μ 1 μ 2 ...μ=kAμ 1 μ 2 ...μ. (2.46)The addition of two tensors of the same rank implies that the corresponding compo-
nents are added. When a tensorCis said to be the sum of the tensorsAandB,this
means:
Cμ 1 μ 2 ...μ=Aμ 1 μ 2 ...μ+Bμ 1 μ 2 ...μ. (2.47)
The addition of two tensors makes sense only when both have the same rank.Of
course, the rank of the resulting sum is also.
Notice, though it may sound somewhat confusing, tensors of a fixed rank(with
= 0 , 1 , 2 ,...) are elements of a vector space.