Now by keeping the vectors r i and r i at their positions in (1. (1.3) It is clear that the basis vector r i occupies all possible positions in (1.3). When we have considered tensors in Euclidean space E3, we have not. To obtain these linearly independent tensors of rank 6, it suffices, for example, to consider the following tensors: r i r i r k r k r m r m, r i r k r i r k r m r m, r i r k r k r i r m r m, r i r k r k r m r i r m, r i r k r k r m r m r i. Chapter 7 Foundations of Tensor Analysis 7.1 Tensor Fields and Their Differentiation 1. Of these tensors, only fifteen are linearly independent (irreducible to each other). Obviously, by rearranging the basis vectors in (1.2), we obtain 6! = 720 permutations (isotropic tensors of rank 6) in the general case. (1.2) All other isotropic tensors of rank 6 can be obtained from (1.2) by permutations of basis vectors. For example, r i r i r k r k r m r m = E ˜ EE˜EE˜E. By contracting the indices pairwise arbitrarily, we obtain some isotropic tensor of rank 6. The multiplicative basis of a tensor of rank 6 is r i r j r k r l r m r n. By way of example, let us also construct all linearly independent isotropic tensors of rank 6. If we pay attention to the structure of isotropic tensors of rank 2 and rank 4 in (1.1), then we easily see that they can be obtained from the corresponding multiplicative bases by pairwise convolution (contraction) of indices of the basis vectors and by exhausting all possible cases of such contraction. The general expression for an arbitrary isotropic tensor of rank 4 is their linear combination C ˜ C ˜ C = 3 k=1 a k C ˜ C ˜ C (k). The tensors C ˜ C ˜ C (1) = E ˜ EE˜E = r i r i r j r j, C ˜ C ˜ C (2) = r i r j r i r j, C ˜ C ˜ C (3) = r i E ˜ Er i = r i r j r j r i (1.1) are three linearly independent (irreducible to each other) tensors of rank 4. Tensors are mathematical objects which have. It is known that E ˜ E = r i r i = g ij r i r j is the only isotropic tensor of rank 2 that can be used to represent any other isotropic tensor a ˜ a of rank 2 in the form a ˜ a = aE˜E, where a is a scalar i.e., an arbitrary isotropic tensor of rank 2 is a spherical tensor. Tensor calculus is, at its most basic, the set of rules and methods for manipulating and calculating with tensors. The components of the tensor may have no symmetry or have symmetries of various types. We find linearly independent above-mentioned tensors up to and including rank six. Vectors, Tensors and the Basic Equations of Fluid Mechanics. We state assertions and theorem that permit one to construct these tensors. An Introduction to Linear Algebra & Tensors. We consider various methods for constructing linearly independent isotropic, gyrotropic, orthotropic, and transversally isotropic tensors.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |