This is the second volume of a twovolume work on vectors and tensors. Calculus i 45 lectures 102,126 views precalculus 10. For more comprehensive overviews on tensor calculus we recommend 54, 96, 123, 191, 199, 311, 334. Foundations of tensor analysis for students of physics and. Second, tensor theory, at the most elementary level, requires only linear algebra and some calculus as prerequisites. Australia the goal of this first paper is to formalise the use of certain diagrams for. Tensor calculus and the calculus of moving surfaces. The focus lies mainly on acquiring an understanding of the principles and ideas underlying the concept of tensor. For a deeper understanding, knowledge of manifolds and some pointset topology is required. The first four deal with the basic concepts of tensors, riemannian spaces.
A scalar field describes a onetoone correspondence between a single scalar number. This booklet contains an explanation about tensor calculus for students of physics. Contrasted with the infinitesimal calculus, tensor. Preface this is an introductory text which presents fundamental concepts from the subject areas of tensor calculus, differential geometry and continuum mechanics. Understanding the role of the metric in linking the various forms of tensors1 and, more importantly, in di. Closely associated with tensor calculus is the indicial or index notation. Once the differences and similarities between vectors, oneforms and tensors are clear, we will adopt a unified notation that makes computations easy.
The first four deal with the basic concepts of tensors, riemannian spaces, riemannian curvature, and spaces of constant curvature. Proceeding a small step further, tensor theory requires background in multivariate calculus. If a result is stated as an equation between tensors or vectors or scalars, if it can be proven or interpreted in any coordinate system, it is true for all. Introduction to tensor calculus and continuum mechanics.
Heinbockel department of mathematics and statistics old dominion university. The poor mans introduction to tensors physics the university of. The first part deals with an introduction to tensor calculus and differential geometry which covers such things as the indicial notation,tensor algebra, covariant differentiation, dual tensors,bilinear and multilinear forms,specialtensors,the riemann christoffel tensor,space curves, surface curves, curvature and fundamental quadratic forms. Introduction to tensor analysis and the calculus of moving surfaces hardcover, pavel grinfeld alex j. Ebook pdf physics introduction to tensor calculus and. It follows at once that scalars are tensors of rank 0,0, vectors are tensors of rank 1,0 and oneforms are tensors of. The rst generation of gr textbooks at least partially targeted toward u. It provides an accessible account of most of the current, important mathematical tools required in physics these days.
Tensors for beginners albert tarantola september 15, 2004 1 tensor notations the velocity of the wind at the top of eiffels tower, at a given moment, can be represented by a vector v with components, in some local, given, basis, vi i 1,2,3. Depending on the circumstance, we will represent the partial derivative of a tensor in the following way 3. Heinbockel department of mathematics and statistics old dominion university 2. Cartesian coords, then tensor is called a cartesian tensor. Tips on teaching general relativity with tensors to. The calculus of matrices is presented in 40, 111, 340, for example. What is tensor calculus and why did einstein need it for. The geometry of tensor calculus, i andre joyal cfniuersith du quihec ir montrkal, case postalr 8888, succ. Massachusettsinstituteoftechnology departmentofphysics physics8. If a road slopes upward, how much does its elevation chance for each metre along. The components of this tensor, which can be in covariant g ij or contravariant gij forms, are in general continuous variable functions of coordinates, i. Tensor analysis is useful because of its great generality, computational power, and compact, easy to use notation.
Mathematical methods for physicists a concise introduction this text is designed for an intermediatelevel, twosemester undergraduate course in mathematical physics. Mar 07, 2012 this elementary introduction pays special attention to aspects of tensor calculus and relativity that students tend to find most difficult. An ndimensional vector eld is described by a onetoone correspondence between nnumbers and a point. The central principle of tensor analysis lies in the simple, almost trivial fact that scalars are unaffected by. Tensor analysis is useful because of its great generality, computational power, and compact, easytouse notation. This volume begins with a discussion of euclidean manifolds. This is a general method of argument in tensor calculus. The early chapters have many words and few equations. Volume 1 is concerned with the algebra of vectors and tensors, while this volume is concerned with the geometrical aspects of vectors and tensors. This booklet contains an explanation about tensor calculus for students of physics and engineering with a basic knowledge of linear algebra. Its use of relatively unsophisticated mathematics in the early chapters allows readers to develop their confidence within the framework of cartesian coordinates before undertaking the theory of tensors in curved spaces and its application to general. We now show explicitly that the partial derivative of a.
Introduction to tensor calculus and continuum mechanics vtk gent. The last part of the textbook is devoted to the calculus of moving surfaces. Developed by gregorio riccicurbastro and his student tullio levicivita, 1 it was used by albert einstein to develop his theory of general relativity. This textbook is distinguished from other texts on the subject by the depth of the presentation and the discussion of the calculus of moving surfaces, which is an extension of tensor calculus to deforming manifolds. Introduction to tensor calculus a scalar eld describes a onetoone correspondence between a single scalar number and a point. Introduction these notes were written for a broad audiencei wrote these notes to be accessible to anyone with a basic knowledge of linear algebra and vector calculus. An introduction to tensor calculus, relativity, and cosmology. Ap calculus bc sample test questions 6 lectures 20,846 views math 210. The definition of a tensor comes only in chapter 6 when the reader is ready for it.
The coordinates a i can be found by ei a a ei amem ei am. Its use of relatively unsophisticated mathematics in the early chapters allows readers to develop their confidence within the framework of cartesian coordinates before undertaking the theory of tensors in curved spaces. The tensor approach can be of benefit to members of all technical sciences including mathematics and all engineering disciplines. Vectors from r3 are denoted by bold lower case letters, e. Pdf these are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. An introduction to tensors for students of physics and. While this text maintains a reasonable level of rigor, it takes great care to avoid formalizing the subject. Volume 1 is concerned with the algebra of vectors and tensors, while this volume is concerned with the geometrical. Introduction to tensor calculus for general relativity mit.
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. Introduction to tensor analysis and the calculus of. Let us generalize these concepts by assigning nsquared numbers to a single point or ncubed numbers to a single. Tensor calculus is a powerful tool that combines the geometric and analytical perspectives and enables us to take full advantage of the computational utility of coordinate systems. An introduction to tensors for students of physics and engineering joseph c. It is the output of the function iafter the independent variable. The metric tensor and arc lengths flat space by eigenchris.
Introduction to tensor calculus and continuum mechanics by j. A gentle introduction to tensors washington university in. Introduction to tensor analysis and the calculus of moving surfaces. Zero tensor 0 has all its components zero in all coord systems. For an analogy, retreat to the case of a real valued function. The next three chapters are concerned with applications to classical dynamics, hydrodynamics, elasticity, electromagnetic radiation, and the theorems of stokes and green. This classic text is a fundamental introduction to the subject for the beginning student of absolute differential calculus, and for those interested in the applications of tensor calculus to mathematical physics and engineering. 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. Or how fast does the position of a sports car change for each second that ticks by. The purpose of this book is to empower the reader with a magnificent new perspective on a wide range of fundamental topics in mathematics.
Kolecki national aeronautics and space administration glenn research center cleveland, ohio 445 tensor analysis is the type of subject that can make even the best of students shudder. Partial differentiation of a tensor is in general not a tensor. Uno cial errata for introduction to tensor analysis and the calculus of moving surfaces hardcover, pavel grinfeld alex j. In both of these examples, the functions involved were vector valued functions of vector variables. An introduction to riemannian geometry and the tensor calculus, cambridge university press, cambridge, 1957. They have the further property that they are linear functions, i. Description of the book introduction to tensor analysis and the calculus of moving surfaces. 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.
This elementary introduction pays special attention to aspects of tensor calculus and relativity that students tend to find most difficult. Dec 12, 2014 introduction to tensor calculus and continuum mechanics by j. It seems abstruse because of the intellectual gap that exists between where most physics and engineering mathematics end and where tensor. That is the power of tensor calculus and its general properties of transformation between coordinates. Hzc3pn and ross street macquarie university, new south wales 2109. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is presumed. A readable introduction to tensor calculos from profesor heinbockel, from old dominion university. Although tensors are applied in a very broad range of physics and math ematics, this booklet focuses on the application in special and general relativity. Introduction to tensor calculus for general relativity. Introduction to vector and tensor analysis jesper ferkingho borg september 6, 2007. Introduction to tensor analysis and the calculus of moving.