Tensors and their Applications

APL quote quad, 1986

International Journal of Computer Mathematics, 2013

The algebra of the Kronecker products of matrices is recapitulated using a notation that reveals the tensor structures of the matrices. It is claimed that many of the difficulties that are encountered in working with the algebra can be alleviated by paying close attention to the indices that are concealed beneath the conventional matrix notation. The vectorisation operations and the commutation transformations that are common in multivariate statistical analysis alter the positional relationship of the matrix elements. These elements correspond to numbers that are liable to be stored in contiguous memory cells of a computer, which should remain undisturbed. It is suggested that, in the absence of an adequate index notation that enables the manipulations to be performed without disturbing the data, even the most clear-headed of computer programmers is liable to perform wholly unnecessary and time-wasting operations that shift data between memory cells.

2013

In general, a tensor is a multilinear transformation defined over an underlying finite dimensional vector space. In this brief introduction, tensor spaces of all integral orders will defined inductively. Initially the underlying vector space, V, will be assumed to be an inner product space in order to simplify the discussion. Subsequently, the presentation will be generalized to vector spaces without inner product. Usually, bold-face letters, a, will denote vectors in V and upper case letters, A, will denote tensors. The inner product on V will be denoted by a · b.

A beginner's notes on tensor analysis

American Journal of Physics, 2013

A guide on tensors is proposed for undergraduate students in physics or engineering that ties directly to vector calculus in orthonormal coordinate systems. We show that once orthonormality is relaxed, a dual basis, together with the contravariant and covariant components, naturally emerges. Manipulating these components requires some skill that can be acquired more easily and quickly once a new notation is adopted. This notation distinguishes multi-component quantities in different coordinate systems by a differentiating sign on the index labelling the component rather than on the label of the quantity itself. This tiny stratagem, together with simple rules openly stated at the beginning of this guide, allows an almost automatic, easy-to-pursue procedure for what is otherwise a cumbersome algebra. By the end of the paper, the reader will be skillful enough to tackle many applications involving tensors of any rank in any coordinate system, without indexmanipulation obstacles standing in the way. V

Universitext, 2005

Part I Basic Tensor Algebra Tensor Spaces 3 X Contents 3.5 Einstein's contraction of the tensor product 3.6 Matrix representation of tensors 3.6.1 First-order tensors 3.6.2 Second-order tensors 3.7 Exercises 4 Change-of-basis in Tensor Spaces 65 4.1 Introduction 65 4.2 Change of basis in a third-order tensor product space ....... 65 4.3 Matrix representation of a change-of-basis in tensor spaces ... 4.4 General criteria for tensor character 4.5 Extension to homogeneous tensors 4.6 Matrix operation rules for tensor expressions 74 4.6.1 Second-order tensors (matrices) 4.6.2 Third-order tensors 77 4.6.3 Fourth-order tensors 78 4.7 Change-of-basis invariant tensors: Isotropic tensors 4.8 Main isotropic tensors 4.8.1 The null tensor 4.8.2 Zero-order tensor (scalar invariant) 4.8.3 Kronecker's delta 4.9 Exercises

Tensor Analysis, 2018

In Sect. 2.2.4 the Cauchy stress tensor T was defined. The stress tensor is the "original" tensor as the word tensor means stress. We shall use the definition of the stress tensor as an introduction to the general concept of tensors. We consider a body of continuous material and a material surface A in the body. At a place r a positive side of the surface is defined by a unit vector n as a normal pointing out from the surface. In a Cartesian coordinate system Ox with base vectors e k the normal vector n has the components: n k ; i.e. n ¼ n k e k : The contact force on the positive side of the surface is represented by the stress vector t with Cartesian components: t i ; i.e. t ¼ t i e i : The contact forces on positive coordinate surfaces through the place r are the stress vectors t k with Cartesian components T ik ; i.e. t k ¼ T ik e i : The components T ik are called the coordinate stresses. The Cauchy stress theorem by Eq. (2.2.27

This booklet contains an explanation about tensor calculus for students of physics and engineering with a basic knowledge of linear algebra. The focus lies mainly on acquiring an understanding of the principles and ideas underlying the concept of 'tensor'. We have not pursued mathematical strictness and pureness, but instead emphasise practical use (for a more mathematically pure resumé, please see the bibliography). Although tensors are applied in a very broad range of physics and mathematics, this booklet focuses on the application in special and general relativity.

arXiv: General Relativity and Quantum Cosmology, 1998

We are presenting an algorithm capable of simplifying tensor polynomials with indices when the building tensors have index symmetry properties. These properties include simple symmetry, cyclicity and those due to the presence of partial and covariant derivatives. We are also including some examples using the Riemann tensor as a paradigm. The algorithm is part of a Mathematica package called Tools of Tensor Calculus (TTC) [web address: this http URL]

Journal of Mathematical Physics, 1993

Let s>2 and let a= (ul,az,...,us), where s, ai E Z and a,>u,)~~~>u,>l. Let 1 <t<s. The method of undetermined coefficients was used to investigate natural transfommtions fl:Aa + Aal 0 Aa Q * * * 0 Aas and fi,:A." + A(al*...~'*...~a~) 0 A"'. Necessary and sufficient conditions were found on the coefficients defining Cl and sufficient conditions on the coefficients defining R, Explicit formulas for R1 and R, are given herein.

Springer eBooks, 2019

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Tensor Network Contractions

This chapter is to introduce some basic definitions and concepts of TN. We will show that the TN can be used to represent quantum many-body states, where we explain MPS in 1D and PEPS in 2D systems, as well as the generalizations to thermal states and operators. The quantum entanglement properties of the TN states including the area law of entanglement entropy will also be discussed. Finally, we will present several special TNs that can be exactly contracted, and demonstrate the difficulty of contracting TNs in general cases. 2.1 Scalar, Vector, Matrix, and Tensor Generally speaking, a tensor is defined as a series of numbers labeled by N indexes, with N called the order of the tensor. 1 In this context, a scalar, which is one number and labeled by zero index, is a zeroth-order tensor. Many physical quantities are scalars, including energy, free energy, magnetization, and so on. Graphically, we use a dot to represent a scalar (Fig. 2.1). A D-component vector consists of D numbers labeled by one index, and thus is a first-order tensor. For example, one can write the state vector of a spin-1/2 in a chosen basis (say the eigenstates of the spin operatorŜ [z]) as |ψ = C 1 |0 + C 2 |1 = s=0,1 C s |s , (2.1) with the coefficients C a two-component vector. Here, we use |0 and |1 to represent spin up and down states. Graphically, we use a dot with one open bond to represent a vector (Fig. 2.1). 1 Note that in some references, N is called the tensor rank. Here, the word rank is used in another meaning, which will be explained later.

We are presenting an algorithm capable of simplifying tensor polynomials with indices when the building tensors have index symmetry properties. These properties include simple symmetry, cyclicity and those due to the presence of covariant derivatives. The algorithm is part of a Mathematica package called Tools of Tensor Calculus (TTC)[web address: