Tensor Analysis

Tensors and their Applications, 2006

2001

Tensors are mathematical objects that generalize vectors and matrices. They describe geometrical quantities and they are used in various applied settings including mathematical physics. The indicial notation of tensors permits us to write an expression in a compact manner and to use simplifying mathematical operations. In a large number of problems in differential geometry and general relativity, the time consuming and straightforward algebraic manipulation is obviously very important. Thus, tensor computation came into existence and became necessary and desirable at the same time. Over the past 25 years, few algorithms have appeared for simplifying tensor expressions. Among the most important tensor computation systems, we can mention SHEEP, Macsyma !Tensor Package, MathTensor and GRTensorII. Meanwhile, graph theory, which had been lying almost dormant for hundreds of years since the time of Euler, started to explode by the turn of the 20th century. It has now grown into a major discipline in mathematics, which has branched off today in various directions such as coloring problems, Ramsey theory, factorization theory and optimization, with problems permeating into many scientific areas such as physics, chemistry, engineering, psychology, and of course computer science. Investigating some of the tensor computation packages will show that they have some deficiencies. Thus, rather than building a new system and adding more features to it, it was an objective in this thesis to express an efficient algorithms by removing most, if not all, restrictions compared to other packages, using graph theory. A summary of the implementation and the advantages of this system is also included. iii I am thankful to my supervisor, Professor Stephen Watt, who taught me everything I needed to know about tensor expressions, for accepting to supervise me, for his kindness and generous contributions of time and for his careful commentary of my thesis. Without him, this work would never been completed. Also, I would like to thank every person in the SCL lab for their helpful advice and useful comments on this thesis. Furthermore, I am sincerely grateful to my parents who kept supporting me regardless of the consequences and to my family, especially my wife, for their endless support and love. I shall never forget that.

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

K24389 Illustrating the important aspects of tensor calculus, and highlighting its most practical features, Physical Components of Tensors presents an authoritative and complete explanation of tensor calculus that is based on transformations of bases of vector spaces rather than on transformations of coordinates. Written with graduate students, professors, and researchers in the areas of elasticity and shell theories in mind, this text focuses on the physical and nonholonomic components of tensors and applies them to the theories. It establishes a theory of physical and anholonomic components of tensors and applies the theory of dimensional analysis to tensors and (anholonomic) connections. This theory shows the relationship and compatibility among several existing definitions of physical components of tensors when referred to nonorthogonal coordinates. The book assumes a basic knowledge of linear algebra and elementary calculus, but revisits these subjects and introduces the mathematical backgrounds for the theory in the first three chapters. In addition, all field equations are also given in physical components as well. Comprised of five chapters, this noteworthy text: • Deals with the basic concepts of linear algebra, introducing the vector spaces and the further structures imposed on them by the notions of inner products, norms, and metrics • Focuses on the main algebraic operations for vectors and tensors and also on the notions of duality, tensor products, and component representation of tensors • Presents the classical tensor calculus that functions as the advanced prerequisite for the development of subsequent chapters • Provides the theory of physical and anholonomic components of tensors by associating them to the spaces of linear transformations and of tensor products and advances two applications of this theory Physical Components of Tensors contains a comprehensive account of tensor calculus , and is an essential reference for graduate students or engineers concerned with solid and structural mechanics.

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

2023

This booklet would like to whet your appetite to immerse yourself into the world of tensors with small bites of special tensors to take a closer look at tensor calculus. For the novice, tensor notations and operations are somehow clumsy and uncomfortable and accompanied by heavy calculations. Therefore here the focus is on praxis with selected examples - using Eigenmath as your computer algebraic companion to unburden calculation in this field.