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Concept of Tensors
Published in Bhaben Chandra Kalita, Tensor Calculus and Applications, 2019
When the relative position of two indices contravariant or covariant in the components is interchangeable in a tensor, it is called a symmetric tensor with respect to these indices, i.e., Ajki (or Aij) is symmetric if Ajki=Akji (or Aij = Aji).
Preliminary Concepts
Published in Hillel Rubin, Joseph Atkinson, Environmental Fluid Mechanics, 2001
We can identify the principal components of a symmetric tensor, and its principal axes. The symmetric tensor has only diagonal components in a coordinate system comprising its principal axes. These components are called principal components. Basically, the principal components are eigenvalues of the matrix representing the symmetric tensor. The principal axes are represented by a set of unit mutually orthogonal vectors called eigenvectors. The principal components λi of the symmetric tensor Bij satisfy the equation
Tensors
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
which maps T to its anti-symmetric part with components (Tab−Tba)/2. By insertion of the symmetry relations, the symmetric part of an anti-symmetric tensor vanishes and vice versa. It also holds that the symmetric part of a symmetric tensor is equal to the tensor itself, with the equivalent statement also being true for anti-symmetric tensors. The verification of this is the task in Problem 2.10.
Feasible Newton methods for symmetric tensor Z-eigenvalue problems
Published in Optimization Methods and Software, 2023
Jiefeng Xu, Dong-Hui Li, Xueli Bai
Let be the real field. is called an mth order n-dimensional real tensor if it takes the form We use to denote the set of all real tensors of order m and dimension n. If the elements of are invariant under arbitrary permutation of their indices, then is called a symmetric tensor. The set of all mth order n-dimensional symmetric tensors is denoted by . For a tensor and a vector , denotes a n-dimensional vector whose ith element is defined as And the notation denotes a homogeneous polynomial of degree m: It is well known that if is symmetric, then the gradient of the homogenous function is , i.e. and the Hessian is , where the th element of is given by The concept of tensor eigenvalues was introduced by Qi and Lim, independently.
Quantification of turbulent flow anisotropy in an alluvial channel mining pit
Published in Marine Georesources & Geotechnology, 2023
Anisotropic invariant map (AIM) can be used for quantifying anisotropy in turbulent flow. Mostly Lumley triangle technique is used for describing turbulent anisotropy. The method is proposed by Lumley and Newman (Lumley and Newman 1977). It is based on the anisotropy stress tensor represent the non-dimensional form of Reynolds stress tensor. It is a symmetric tensor and has zero trace.