Foundations of tensor analysis for students of physics and. Calculus of tensors, mathematical physics, physique mathe matique, calcul tensoriel, city planning, cities and towns. Earlier physicists from newton to maxwell had to work much harder to solve their problems. An introductory tensor calculus for physics book is a most welcome addition to the libraries of both young stu dents in physics and instructors with teaching duties at the. The reader is expected to be familiar with some notions of vector spaces or matrix algebra. Tensor analysis for physicists jan arnoldus schouten. This is the second volume of a twovolume work on vectors and tensors. For algebra on vectors and tensors, an index must show up twice and only twice.
The base vectors and metric tensors will be introduced, and the local base vectors in curvilinear coordinates and tensor algebra will be presented. A gentle introduction to tensors washington university in. Such visualizations may further develop ones geometric and physical intuition. It follows at once that scalars are tensors of rank 0,0, vectors are tensors of rank 1,0 and oneforms are tensors of. Applications for electrodynamics and relativity are presented. If an index shows up once on the left hand side lhs of sign, it must show up once and only once on the right hand side rhs of sign. This is tensor analysis for physicists, written from the point of view of a mathematician. The tensor analysis in xn is developed in chapter iv. Is free for individual use and educational purposes. Tensor analysis and elementary differential geometry for. Thebehaviourofthetransformationofthecomponentsofavectorunder the transformation of a. Elementary tensor analysis this appendix is intended to provide the mathematical preliminaries needed for a clear and rigorous presentation of the basic principles in continuum mechanics. Tensor analysis for physicists 1989 edition open library.
The reader must be prepared to do some mathematics and to think. According to the geometric picture of gr, the connection as the potential is related with energy and the curvature tensor is related with force strength with the mass to connect the connectioncurvature with energyforce. In our daily life we see large number of physical quantities. However, it wasnt until i was introduced to the abstract index notation see, e. Several times during the preparation of this book we taught a one semester course to students with a very limited background in linear algebra and. Tensor visualizations inspired by schoutens tensor calculus for physicists, the tensor algebra of vectors and differential forms are accurately represented pictorially. Im a fan of bishop and goldbergs tensor analysis on manifolds. For this reason, here we introduce the notion of a.
Tensor analysisintroductionde nitionde nition of di erent rank tensorscontraction and direct productquotient rulepseudo tensorsgeneral tensorsmetric tensors. So, since you have been warned, we shall henceforth abide by the following principle. In rowvector notation, the basis vectors themselves are just i ex 1,0,0 j ey 0,1,0 k ez 0,0,1 1. Buy a cheap copy of tensor analysis for physicists book by jan arnoldus schouten. This brilliant study by a famed mathematical scholar and former professor of mathematics at the university of amsterdam integrates a concise exposition of the mathematical basis of tensor analysis with admirably chosen physical examples of the theory. Im sure there are better books out there, but this one is nice for its niche. Tensor analysis for physicists book by jan arnoldus schouten. With an introduction to tensors and group theory for physicists, physicist nadir jeevanjee has produced a masterly book that will help other physicists understand those subjects tensors and groups as mathematicians understand them from the first pages, jeevanjee shows amazing skill in finding fresh, compelling words to bring forward the insight that animates the modern.
Bewley envisioned that the tensor approach to electrical engineering would become a standard. Foundations of tensor analysis for students of physics and engineering with an introduction to the theory of relativity joseph c. Albert einstein used tensor analysis as the mathematical basis for the general theory of relativity. This chapter will prepare basic knowledge about the tensor analysis in. In my attempt to summarize tensor analysis in a single paragraph, i have left out many important details. On occasions when we have taught a one semester course, we covered material. Cartesian coords, then tensor is called a cartesian tensor. Closely associated with tensor calculus is the indicial or index notation.
Introduction to tensor analysis and the calculus of moving. Any commercial use without written consent from the author is prohibited. Its fairly physically motivated while still keeping the math rigorous. On a related note, walds general relativity is a great book that has a couple good chapters on tensor calculus in the beginning. Chapter 1 introduction the content of these lecture notes covers the second part1 of the lectures of a graduate course in modern mathematical physics at the university of trento. The first five chapters incisively set out the mathematical theory underlying the use of tensors. Developed by gregorio riccicurbastro and his student tullio levicivita, 1 it was used by albert einstein to develop his theory of general relativity. Volume 1 is concerned with the algebra of vectors and tensors, while this volume is concerned with the geometrical. Tensor analysis 1967 missing p74 princeton university. Introduction to tensor calculus and continuum mechanics. Tensor fields do not behave reliably under mappings.
A basic knowledge of vectors, matrices, and physics is assumed. I am curious about the dimensional analysis of the metric tensor. A good exercise is to take the tensorial form of maxwells equations and use the decomposition by an observer with a unittimelike vector to obtain the set of vectorial equations found in. The appearance of jeevanjees an introduction to tensors and group theory for physicists is an example of excellent timing. The third chapter extends tensor theory to spaces other than vector spaces, namely manifolds. The important property of a physical quantity is that it can be measured and expressed in terms of a mathematical quantity like number. An introduction to tensors for students of physics and. Several times during the preparation of this book we taught a one semester course to students with a very limited background in linear algebra and no background in tensor analysis.
However, it is likely that teachers will wish to generate additional exercises. Introduction to tensor calculus for general relativity. Lecture notes on vector and tensor algebra and analysis. Kolecki national aeronautics and space administration glenn research center cleveland, ohio 445 summary although one of the more useful subjects in higher mathematics, tensor analysis has the tendency to be. In both of these examples, the functions involved were vector valued functions of vector variables. This brilliant study by a famed mathematical scholar and former professor of mathematics at the university of amsterdam integrates a concise exposition of the. Schouten, 1989, dover publications edition, in english 2nd ed. Many physicists, at least in the general relativity area, are familiar with the definition of a tensor in terms of multilinear maps. For an analogy, retreat to the case of a real valued function.
Tensor is the mathematical tool used to express these physical quantities. Tensor analysis and curvilinear coordinates phil lucht rimrock digital technology, salt lake city, utah 84103 last update. It is this chapter that provides the foundations for tensor applications in physics. The analytic version regarding applications to linear functional analysis to quantum and quantum relativistic theories is covered by my books morettia, morettib and the chapter khmo15. It is the output of the function iafter the independent variable. In chapters vi through ix, professor schouten presents applications of the theory that are both intrinsically interesting and good examples of the use and advantages of the calculus. In minkowski space and under poincare transformations, tensors are lorentz tensors, or, 4tensors. An introduction to tensors and group theory for physicists. Zero tensor 0 has all its components zero in all coord systems. Contrasted with the infinitesimal calculus, tensor calculus allows. Tensor analysis for physicists, second edition dover books on physics by schouten, j. This book was written as lecture notes for classes that i taught to undergraduate. Volume 1 is concerned with the algebra of vectors and tensors, while this volume is concerned with the geometrical aspects of vectors and tensors.
Tensors home department of physics university of miami. As others have correctly pointed out, tensors are particularly useful because of their transformation properties. The title, the poor mans introduction to tensors, is a reference to gravitation by misner, thorne and wheeler, which characterizes simplified approaches to a problem as the poor mans way to do x. The stress at a point of an elastic solid is an example of a tensor which depends on two directions one normal to the area and other that of the force on it. Tensor analysis for physicists, second edition dover books. Any physical property that can be quanti ed is called a physical quantity. They have the further property that they are linear functions, i. Tensor analysis for physicists, second edition dover. In fact, that is how they are defined in all gr books i have looked at carroll, misnerthornewheeler, hawkingellis, wald. May 22, 2005 however, it wasnt until i was introduced to the abstract index notation see, e.
In this report we will limit our self to describe the. Tensor analysis applied to the equations of continuum. For instance, the definition for tensors in the first sentence is an incomplete one. Dimensional analysis of metric tensor physics stack exchange. The secondorder tensors will be discussed in detail. What is tensor analysis and how is it used in physics.
In tensor analysis, the maze of indices disguises the great differences between various quantities. In mathematics, tensor calculus, tensor analysis, or ricci calculus is an extension of vector calculus to tensor fields tensors that may vary over a manifold, e. The title, the poor mans introduction to tensors, is a reference to gravitation by misner, thorne and wheeler. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity. Bochner isoperimetric inequalities in mathematical by c. This volume begins with a discussion of euclidean manifolds. But for writing vectors, the easiest notation is the underline. Using tensor methods, it can be difficult to discern the invariants in geometric and physical situations even the local ones. I want to elaborate a little bit on the other answers. Nobody actually defines a tensor as something that transforms like a tensor anymore. A semiintuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and similar higherorder vector products. Lecture notes on vector and tensor algebra and analysis ilyal. In nonrelativistic physics once we have chosen a co.
1476 1431 575 1528 1597 304 671 1104 1035 625 713 513 1589 79 145 1092 901 1554 1385 646 107 574 205 515 878 729 205 123 89 176 1418