Mathematical Methods for Physics and Engineering - Matematica.NET

26.10 IMPROPER ROTATIONS AND PSEUDOTENSORSx 3x ′ 1vpv ′Ox 2Ox 1x ′ 2p ′x ′ 3Figure 26.2 The behaviour of a vector v and a pseudovector p under areflection through the origin of the coordinate system x 1 ,x 2 ,x 3 giving the newsystem x ′ 1 ,x′ 2 ,x′ 3 .It is important to realise that a pseudovector (as its name suggests) is not ageometrical object in the usual sense. In particular, it should not be consideredas a real physical arrow in space, since its direction is reversed by an impropertransformation of the coordinate axes (such as an inversion through the origin).This is illustrated in figure 26.2, in which the pseudovector p is shown as a brokenline to indicate that it is not a real physical vector.Corresponding to vectors and pseudovectors, zeroth-order objects may bedivided into scalars and pseudoscalars – the latter being invariant under rotationbut changing sign on reflection.We may also extend the notion of scalars and pseudoscalars, vectors and pseudovectors,to objects with two or more subscripts. For two subcripts, as definedpreviously, any quantity with components that transform as T ij ′ = L ikL jl T kl underall rotations (proper and improper) is called a second-order Cartesian tensor.If, however, T ij ′ = L ikL jl T kl under proper rotations but T ij ′ = −L ikL jl T kl underimproper ones (which include reflections), then the T ij are the components ofa second-order Cartesian pseudotensor. In general the components of Cartesianpseudotensors of arbitary order transform asT ′ij···k = |L|L il L jm ···L kn T lm···n , (26.39)where |L| is the determinant of the transformation matrix.For example, from (26.29) we have that|L|ɛ ijk = L il L jm L kn ɛ lmn ,947