A Brief Introduction to Tensors ∗

The present article offers an exposition of the axiomatic definition of tensors and their further developments from this very standpoint. Various types of tensor sand their examples have been included.

This chapter is not meant as a replacement for a course in tensor analysis, but it will provide a working background to tensor notation and algebra.

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

Journal of Mathematical Sciences, 2009

Basic definitions of linear algebra and functional analysis are given. In particular, the definitions of a semigroup, group, ring, field, module, and linear space are given [1-3, 6]. A local theorem on the existence of homeomorphisms is stated. Definitions of the inner r-product, local inner product of tensors whose rank is not less than r, and of local norm of a tensor [22] are also given. Definitions are given and basic theorems and propositions are stated and proved concerning the linear dependence and independence of a system of tensors of any rank. Moreover, definitions and proofs of some theorems connected with orthogonal and biorthonormal tensor systems are given. The definition of a multiplicative basis (multibasis) is given and ways of construction bases of modules using bases of modules of smaller dimensions. In this connection, several theorems are stated and proved. Tensor modules of even orders and problems on finding eigenvalues and eigentensors of any even rank are studied in more detail than in [22]. Canonical representations of a tensor of any even rank are given. It is worth while to note that it was studied by the Soviet scientist I. N. Vekua, and an analogous problem for the elasticity modulus tensor was considered by the Polish scientist Ya. Rikhlevskii in 1983-1984.

Tensors As mentioned in the introduction, all laws of continuum mechanics must be formulated in terms of quantities that are independent of coordinates. It is the purpose of this chapter to introduce such mathematical entities. We shall begin by introducing a shorthand notation-the indicial notation-in Part A of this chapter, which will be followed by the concept of tensors introduced as a linear transformation in Part B. The basic field operations needed for continuum formulations are presented in Part C and their representations in curvilinear coordinates in Part D. Part A The Indicia1 Notation 2A1 Summation Convention, Dummy Indices Consider the sum s = a p l + as2 + a3x3 +-* + a,&,, (2A1.1) We can write the above equation in a compact form by using the summation sign: n s = ajxi i = l (2A1.2) It is obvious that the following equations have exactly the same meaning as Eq. (2A1.2) n j=l s = 2 ajxj (2A1.3) n s = c a m x m m = l (2A1.4) etc. 3

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.

Tensors and their Applications, 2006

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.

A beginner's notes on tensor analysis

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.