Linear invariants of a Cartesian tensor

Isotropic Tensors Under Non-compact Rotation Groups

“…We find that their number is exactly the same as that of corresponding linearly independent isotropic tensors under the compact rotation groups SO(2), SO(3) and SO(4) as given by Naila [11] and Faiz and Rashid [10,12]. showing that is indeed isotropic underSL(3, ℝ ).…”

“…This is the motivation for the present work in which, using the direct method, we explicitly construct a set of isotropic tensors, of rank up to 6, under each of these non-compact groups, select a linearly independent subset of each of these sets, and then show that each of these selected subsets is complete in the sense that every isotropic tensor of the relevant type, is a linear combination of elements of this subset. This means that the current work is an immediate generalization of the work of Ahmad and Rashid [10] for SO (2) and SO(3) and of Naila Amir [11] for SO (4). An interesting fact which appears from this work, is that the number of elements of a complete set of linearly independent isotropic tensor of any particular type, under a non-compact group, is identical with the corresponding number, under the corresponding compact group.…”

“…[8] and Gusev and Lurie [9] have used them in isotropic strain gradient elasticity theory. Next, Ahmad and Rashid [10] first showed that the number of linearly independent linear invariants of tensors of any rank under any rotation group SO(p), is equal to the dimension of the space of isotropic tensors of SO(p), of that rank, and then obtained all the linearly independent linear invariants under SO(2) of tensors up to rank 6 and those under SO(3) also up to rank 6. They also found the number of linearly independent linear invariants of a tensor of general rank, under the group of rotations about a fixed axis, say the 3-axis.…”

“…As in references [10,11], we denote by d the dimensions of the space in which rotations take place, and by r the rank of the tensor. To proceed systematically, we start with the case d = 2, and then move on to the higher values of d = 3, 4, one by one.…”

Linear Invariants of a Cartesian Tensor Under SO(2), SO(3) and SO(4)

“…Norris [5] showed that the set of four invariants of Ahmad, combined with C 2 1 , C 2 2 and C 1 C 2 (which are indeed quadratic invariants) is complete, also the complete set of quadratic invariants, with one axis fixed, contains 35 members. Recently Ahmad and Rashid [6] have discussed linear invariants of a Cartesian tensor, in three dimensions, under SO(3) as well as SO (2). The fundamental result of the theory is Theorem 1 The number of independent linear invariants of a Cartesian tensor of rank r is the same as the dimension of the space of isotropic tensors of rank r.…”