Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSRotate by π/2 about the Ox 3 -axis: L 12 = −1, L 21 =1,L 33 = 1, the other L ij =0.(d) T 111 =(−1) × (−1) × (−1) × T 222 = −T 222 ,(e) T 112 =(−1) × (−1) × 1 × T 221 ,(f) T 221 =1× 1 × (−1) × T 112 ,(g) T 123 =(−1) × 1 × 1 × T 213 .Relations (a) and (d) show that elements with all subscripts the same are zero. Relations(e), (f) and (b) show that all elements with repeated subscripts are zero. Relations (g) and(c) show that T 123 = T 231 = T 312 = −T 213 = −T 321 = −T 132 .In total, T ijk differs from ɛ ijk by at most a scalar factor, but since ɛ ijk (and hence λɛ ijk )has already been shown to be an isotropic tensor, T ijk must be the most general third-orderisotropic Cartesian tensor. ◭Using exactly the same procedures as those employed for δ ij and ɛ ijk ,itmaybeshown that the only isotropic first-order tensor is the trivial one with all elementszero.26.10 Improper rotations and pseudotensorsSo far we have considered rigid rotations of the coordinate axes described byan orthogonal matrix L with |L| = +1, (26.4). Strictly speaking such transformationsare called proper rotations. We now broaden our discussion to includetransformations that are still described by an orthogonal matrix L but for which|L| = −1; these are called improper rotations.This kind of transformation can always be considered as an inversion of thecoordinate axes through the origin represented by the equationx ′ i = −x i , (26.38)combined with a proper rotation. The transformation may be looked uponalternatively as one that changes an initially right-handed coordinate system intoa left-handed one; any prior or subsequent proper rotation will not change thisstate of affairs. The most obvious example of a transformation with |L| = −1 isthe matrix corresponding to (26.38) itself; in this case L ij = −δ ij .As we have emphasised in earlier chapters, any real physical vector v may beconsidered as a geometrical object (i.e. an arrow in space), which can be referredto independently of any coordinate system and whose direction and magnitudecannot be altered merely by describing it in terms of a different coordinate system.Thus the components of v transform as v ′ i = L ijv j under all rotations (proper andimproper).We can define another type of object, however, whose components may alsobe labelled by a single subscript but which transforms as v ′ i = L ijv j under properrotations and as v ′ i = −L ij v j (note the minus sign) under improper rotations. Inthis case, the v i are not strictly the components of a true first-order Cartesiantensor but instead are said to form the components of a first-order Cartesianpseudotensor or pseudovector.946