Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSPhysical examples involving second-order tensors will be discussed in the latersections of this chapter, but we might note here that, for example, magneticsusceptibility and electrical conductivity are described by second-order tensors.26.6 The algebra of tensorsBecause of the similarity of first- and second-order tensors to column vectors andmatrices, it would be expected that similar types of algebraic operation can becarried out with them and so provide ways of constructing new tensors from oldones. In the remainder of this chapter, instead of referring to the T ij (say) as thecomponents of a second-order tensor T, we may sometimes simply refer to T ijas the tensor. It should always be remembered, however, that the T ij are in factjust the components of T in a given coordinate system and that T ij ′ refers to thecomponents of the same tensor T in a different coordinate system.The addition and subtraction of tensors follows an obvious definition; namelythat if V ij···k and W ij···k are (the components of) tensors of the same order, thentheir sum and difference, S ij···k and D ij···k respectively, are given byS ij···k = V ij···k + W ij···k ,D ij···k = V ij···k − W ij···k ,for each set of values i,j,...,k.ThatS ij···k and D ij···k are the components oftensors follows immediately from the linearity of a rotation of coordinates.It is equally straightforward to show that if the T ij···k are the components ofa tensor, then so is the set of quantities formed by interchanging the order of (apair of) indices, e.g. T ji···k .If T ji···k is found to be identical with T ij···k then T ij···k is said to be symmetricwith respect to its first two subscripts (or simply ‘symmetric’, for second-ordertensors). If, however, T ji···k = −T ij···k for every element then it is an antisymmetrictensor. An arbitrary tensor is neither symmetric nor antisymmetric but can alwaysbe written as the sum of a symmetric tensor S ij···k and an antisymmetric tensorA ij···k :T ij···k = 1 2 (T ij···k + T ji···k )+ 1 2 (T ij···k − T ji···k )= S ij···k + A ij···k .Of course these properties are valid for any pair of subscripts.In (26.20) in the previous section we had an example of a kind of ‘multiplication’of two tensors, thereby producing a tensor of higher order – in that case twofirst-order tensors were multiplied to give a second-order tensor. Inspection of(26.21) shows that there is nothing particular about the orders of the tensorsinvolved and it follows as a general result that the outer product of an Nth-ordertensor with an Mth-order tensor will produce an (M + N)th-order tensor.938