Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSScalars behave differently under transformations, however, since they remainunchanged. 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 unprimedto the primed basis. Different again is the behaviour of linear operators. If alinear operator A is represented by some matrix A in a given coordinate systemthen 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 thesedifferent types of behaviour under a change of basis (or coordinate transformation).In the development, the generic name tensor is introduced, and certainscalars, vectors and linear operators are described respectively as tensors of zeroth,first and second order (the order –orrank – corresponds to the number ofsubscripts needed to specify a particular element of the tensor). Tensors of thirdand fourth order will also occupy some of our attention.26.3 Cartesian tensorsWe begin our discussion of tensors by considering a particular class of coordinatetransformation – namely rotations – and we shall confine our attention strictlyto the rotation of Cartesian coordinate systems. Our object is to study the propertiesof various types of mathematical quantities, and their associated physicalinterpretations, when they are described in terms of Cartesian coordinates andthe axes of the coordinate system are rigidly rotated from a basis e 1 , e 2 , e 3 (lyingalong the Ox 1 , Ox 2 and Ox 3 axes) to a new one e ′ 1 , e′ 2 , e′ 3 (lying along the Ox′ 1 ,Ox ′ 2 and Ox′ 3 axes).Since we shall be more interested in how the components of a vector or linearoperator are changed by a rotation of the axes than in the relationship betweenthe two sets of basis vectors e i and e ′ i , let us define the transformation matrix Las the inverse of the matrix S in (26.2). Thus, from (26.2), the components of aposition vector x, in the old and new bases respectively, are related byx ′ i = L ij x j . (26.4)Because we are considering only rigid rotations of the coordinate axes, thetransformation matrix L will be orthogonal, i.e. such that L −1 = L T .Thereforethe inverse transformation is given byx i = L ji x ′ j. (26.5)The orthogonality of L also implies relations among the elements of L thatexpress the fact that LL T = L T L = I. In subscript notation they are given byL ik L jk = δ ij and L ki L kj = δ ij . (26.6)Furthermore, in terms of the basis vectors of the primed and unprimed Cartesian930