(p.267) Appendix A Cartesian Tensors
(p.267) Appendix A Cartesian Tensors
(p.267) Appendix A
The essential feature of tensor algebra is that it expresses physical ideas in a form which is independent of coordinate system. It is then often unnecessary to specify a coordinate system at all, and when one is needed, it can be chosen to simplify the treatment of the problem. The description of a physical problem in terms of tensors has two further virtues:
• It is very compact, making the essential features of the mathematics more apparent;
• It is immediately obvious from the form of an expression whether it has the correct (e.g. scalar or vector) behaviour.
This is a condensed summary of the basic ideas. For a fuller account see, for example, Jeffreys (1931).
A.1 Basic definitions
A scalar is a quantity described by a single number, whose value is the same in all coordinate systems. Example: energy.
A vector is a quantity described by three numbers (labelled by a subscript taking values 1, 2 or 3 (or x, y, z)) whose values in one (primed) coordinate system are related to those in a rotated (unprimed) system by
Examples are velocity; angular momentum; dipole moment. The quantities aα′α form an orthogonal matrix A describing the rotation. Consequently
The coefficient aα′α is the cosine of the angle between the α′ and α axes. In the equivalent matrix notation, T denotes the transpose.
It is customary to use the repeated-suffix summation convention, due originally to Einstein, according to which any subscript appearing twice in one term is automatically summed over. Thus (A.1.2) can be written
(p.268) No subscript may occur more than twice in a term. Thus the expression (r · s)2 may not be written asrαsαrαsα (because of the possibility of ambiguity with (r · r)(s · s)) but must be written as, e.g., rαsαrβsβ. α and β here are dummy suffixes; since they are summed over, the actual symbol used is irrelevant. α and β in (A.1.3b), on the other hand, are free suffixes; each such suffix must occur precisely once in each term of the equation, and the equation holds for each value (1, 2 or 3) that they can take.
A tensor of rank n is a quantity described by 3n numbers (labelled by n subscripts each taking the value 1, 2 or 3). Tensors of rank 2 often describe the relationship between two vectors; e.g. the polarizability αξη describes the dipole μξ induced by an electric field Fη:
In order that such equations remain valid in any coordinate system, it is necessary that the values of tensor components in the primed coordinate system are related to those in the unprimed system by
It follows from (A.1.2) that the converse relationship holds:
[Proof: substitute (A.1.5) into the r.h.s. of (A.1.6) and use (A.1.2).] We see that scalars and vectors are tensors of rank 0 and 1 respectively. Note that the order of subscripts on aαα′, is immaterial: aα′α, is the same as aα′α, being the direction cosine between the two axes concerned. Thus a 1′2 = a 21′, but neither of course is the same as a 12′ = a 2′1.
However, the order of the subscripts on the tensor itself is usually significant; that is, Tαβγ…ν ≠ Tβαγ…ν in general. If in a particular case Tαβγ…ν = Tβαγ…ν for any choice of coordinate system, and for all values of the remaining suffixes γ… ν, then the tensor T is said to be symmetric with respect to the first two subscripts.
Addition and subtraction of tensors is straightforward: the quantity
is a component of a tensor W which is the sum of the tensors T and U.
The outer product of two tensors is obtained by multiplying components without summing:
and is readily shown to be a tensor (use (A.1.5)), with a rank which is the sum of the ranks of the factors.
Contraction involves setting two subscripts equal and performing the sum implied by the notation; e.g.
It is easy to show that if T is a tensor of rank n, then Y is a tensor of rank n – 2. (Use (A.1.5) and (A.1.2).) Applying this procedure to an outer product, taking the contracted suffices one from each factor, yields an inner product, e.g.
Clearly Z is a tensor: it is a contraction of X which is a tensor, as stated above. It is also true that if (A.1.10) holds in all coordinate systems, and Z and T are known to be tensors, then it follows that U is a tensor (quotient rule).
The simplest case of an inner product is the scalar product of two vectors: for example s = uαvα = u · v. It is a quantity of rank 0, i.e., a scalar—it is independent of coordinate system. Another common case is the vector formed by taking the inner product of a second-rank tensor with a vector, as in (A.1.4) above.
A.1.1 Isotropic tensors
Normally the application of (A.1.5) yields a set of numbers, describing the tensor in the new axes, which are different from those describing it in the old axes—e.g. T 1′,1′,…1′ ≠ T 11…1. However, some tensors retain the same numerical values in all axis systems, and are called isotropic. Apart from scalar multiplicative factors (scalars being always isotropic by definition) the important isotropic tensors are the Kronecker tensor or Kronecker delta:
and the Levi-Civita tensor:
There are no other isotropic tensors of rank less than 4; moreover all isotropic tensors of any rank can be expressed in terms of outer products of δ’s and ϵ’s.
Notice that δαβ behaves like a subscript substitution operator: in δαβtβγ, for example, there is only one term in the implied sum over β that does not vanish, namely the one for which β = α. Consequently δαβtβγ = tαγ. Notice also that δαα = 3.
An important use of the Levi-Civita tensor occurs in the vector product of two vectors; in tensor notation v = r ∧ s is v α = ϵαβγrβsγ.
A.1.2 Polar and axial tensors
The transformation rule in (A.1.1) or (A.1.5) is appropriate only for proper rotations of the axes. For improper rotations, which involve a reflection and change the handedness of the coordinate system, two possibilities arise. Either the rule in (A.1.5) may apply as it stands, in which case we have a polar tensor, or there may be an additional change of sign for improper rotations, in which case we have an axial tensor. (A.1.5) may be written in a more general form to take account of these possibilities:
where det(a) is the determinant of the matrix whose elements are the direction cosines aα′α, and p is 0 for a polar tensor and 1 for an axial one. Since det(a) is +1 for a proper rotation and (p.270) –1 for an improper one, this gives the extra change of sign for an axial tensor under improper rotation.
It follows that any product, outer or inner, of two polar tensors or two axial tensors is polar, while the product of a polar tensor with an axial tensor is axial. The fundamental axial tensor is the Levi-Civita tensor, which is axial by definition in order that ϵ 123 = +1 in left-handed coordinate systems as well as right-handed ones. Therefore the vector product of two polar tensors is axial. The prime example of this is angular momentum: l = r ∧ p is the vector product of the polar vectors r and p describing position and momentum, expressed in tensor notation as l α = ϵαβγrβpγ.