Cartesian tensors

TENSORS
Scalars behave differently under transformations, however, since they remain
unchanged. For example, the value of the scalar product of two vectors x · y
(which is just a number) is unaffected by the transformation from the unprimed
to the primed basis. Different again is the behaviour of linear operators. If a
linear operator A is represented by some matrix A in a given coordinate system
then in the new (primed) coordinate system it is represented by a new matrix,
A = S−1 AS.
In this chapter we develop a general formulation to describe and classify these
different types of behaviour under a change of basis (or coordinate transformation). In the development, the generic name tensor is introduced, and certain
scalars, vectors and linear operators are described respectively as tensors of zeroth, first and second order (the order – or rank – corresponds to the number of
subscripts needed to specify a particular element of the tensor). Tensors of third
and fourth order will also occupy some of our attention.
26.3 Cartesian tensors
We begin our discussion of tensors by considering a particular class of coordinate
transformation – namely rotations – and we shall confine our attention strictly
to the rotation of Cartesian coordinate systems. Our object is to study the properties of various types of mathematical quantities, and their associated physical
interpretations, when they are described in terms of Cartesian coordinates and
the axes of the coordinate system are rigidly rotated from a basis e1 , e2 , e3 (lying
along the Ox1 , Ox2 and Ox3 axes) to a new one e1 , e2 , e3 (lying along the Ox1 ,
Ox2 and Ox3 axes).
Since we shall be more interested in how the components of a vector or linear
operator are changed by a rotation of the axes than in the relationship between
the two sets of basis vectors ei and ei , let us define the transformation matrix L
as the inverse of the matrix S in (26.2). Thus, from (26.2), the components of a
position vector x, in the old and new bases respectively, are related by
xi = Lij xj .
(26.4)
Because we are considering only rigid rotations of the coordinate axes, the
transformation matrix L will be orthogonal, i.e. such that L−1 = LT . Therefore
the inverse transformation is given by
xi = Lji xj .
(26.5)
The orthogonality of L also implies relations among the elements of L that
express the fact that LLT = LT L = I. In subscript notation they are given by
Lik Ljk = δij
and
Lki Lkj = δij .
(26.6)
Furthermore, in terms of the basis vectors of the primed and unprimed Cartesian
930
26.3 CARTESIAN TENSORS
x2
x2
x1
θ
θ
θ
x1
O
Figure 26.1 Rotation of Cartesian axes by an angle θ about the x3 -axis. The
three angles marked θ and the parallels (broken lines) to the primed axes
show how the first two equations of (26.7) are constructed.
coordinate systems, the transformation matrix is given by
Lij = ei · ej .
We note that the product of two rotations is also a rotation. For example,
suppose that xi = Lij xj and xi = Mij xj ; then the composite rotation is described
by
xi = Mij xj = Mij Ljk xk = (ML)ik xk ,
corresponding to the matrix ML.
Find the transformation matrix L corresponding to a rotation of the coordinate axes
through an angle θ about the e3 -axis (or x3 -axis), as shown in figure 26.1.
Taking x as a position vector – the most obvious choice – we see from the figure that
the components of x with respect to the new (primed) basis are given in terms of the
components in the old (unprimed) basis by
x1 = x1 cos θ + x2 sin θ,
x2 = −x1 sin θ + x2 cos θ,
x3 = x3 .
The (orthogonal) transformation matrix is thus

cos θ
sin θ
L =  − sin θ cos θ
0
0
(26.7)

0
0 .
1
The inverse equations are
x1 = x1 cos θ − x2 sin θ,
x2 = x1 sin θ + x2 cos θ,
x3 = x3 ,
in line with (26.5). 931
(26.8)