Tensor Calculus

J. L. Synge, A. Schild, A. Schild

• Publisher: Dover Publications
• Year: 2012
• eBook ISBN: 9780486141398
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

Mathematicians, theoretical physicists, and engineers unacquainted with tensor calculus are at a serious disadvantage in several fields of pure and applied mathematics. They are cut off from the study of Reimannian geometry and the general theory of relativity. Even in Euclidean geometry and Newtonian mechanics (particularly the mechanics of continua), they are compelled to work in notations which lack the compactness of tensor calculus. 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.
Tensor Calculus contains eight chapters. The first four deal with the basic concepts of tensors, Riemannian spaces, Riemannian curvature, and spaces of constant curvature. The next three chapters are concerned with applications to classical dynamics, hydrodynamics, elasticity, electromagnetic radiation, and the theorems of Stokes and Green. In the final chapter, an introduction is given to non-Riemannian spaces including such subjects as affine, Weyl, and projective spaces. There are two appendixes which discuss the reduction of a quadratic form and multiple integration. At the conclusion of each chapter a summary of the most important formulas and a set of exercises are given. More exercises are scattered throughout the text. The special and general theory of relativity is briefly discussed where applicable.

CHAPTER I

SPACES AND TENSORS

1.1. The generalized idea of a space. In dealing with two real variables (the pressure and volume of a gas, for example), it is a common practice to use a geometrical representation. The variables are represented by the Cartesian coordinates of a point in a plane. If we have to deal with three variables, a point in ordinary Euclidean space of three dimensions may be used. The advantages of such geometrical representation are too well known to require emphasis. The analytic aspect of the problem assists us with the geometry and vice versa.

When the number of variables exceeds three, the geometrical representation presents some difficulty, for we require a space of more than three dimensions. Although such a space need not be regarded as having an actual physical existence, it is an extremely valuable concept, because the language of geometry may be employed with reference to it. With due caution, we may even draw diagrams in this “space,” or rather we may imagine multidimensional diagrams projected on to a two-dimensional sheet of paper; after all, this is what we do in the case of a diagram of a three-dimensional figure.

Suppose we are dealing with N real variables x¹, x², ... , x^N. For reasons which will appear later, it is best to write the numerical labels as superscripts rather than as subscripts. This may seem to be a dangerous notation on account of possible confusion with powers, but this danger does not turn out to be serious.

We call a set of values of x¹, x², .... x^N a point. The variables x¹, x², ... , x^N are called coordinates. The totality of points corresponding to all values of the coordinates within certain ranges constitute a space of N dimensions. Other words, such as hyperspace, manifold, or variety are also used to avoid confusion with the familiar meaning of the word “space.” The ranges of the coordinates may be from − ∞ to + ∞, or they may be restricted. A space of N dimensions is referred to by a symbol such as V_N.

Excellent examples of generalized spaces are given by dynamical systems consisting of particles and rigid bodies. Suppose we have a bar which can slide on a plane. Its position (or configuration) may be fixed by assigning the Cartesian coordinates x, y of one end and the angle θ which the bar makes with a fixed direction. Here the space of configurations is of three dimensions and the ranges of the coordinates are

− ∞ < x < + ∞, − ∞ < y < + ∞, 0 ≤ θ < 2π.

Exercise. How many dimensions has the configuration-space of a rigid body free to move in ordinary space? Assign coordinates and give their ranges.

It will be most convenient in our general developments to discuss a space with an unspecified number of dimensions N, where N ≥ 2. It is a remarkable feature of the tensor calculus that no essential simplifi...

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• PREFACE
• Table of Contents
• CHAPTER I - SPACES AND TENSORS
• CHAPTER II - BASIC OPERATIONS IN RIEMANNIAN SPACE
• CHAPTER III - CURVATURE OF SPACE
• CHAPTER IV - SPECIAL TYPES OF SPACE
• CHAPTER V - APPLICATIONS TO CLASSICAL DYNAMICS
• CHAPTER VI - APPLICATIONS TO HYDRODYNAMICS, ELASTICITY, AND ELECTROMAGNETIC RADIATION
• CHAPTER VII - RELATIVE TENSORS, IDEAS OF VOLUME, GREEN-STOKES THEOREMS
• CHAPTER VIII - NON-RIEMANNIAN SPACES
• APPENDIX A - REDUCTION OF A QUADRATIC FORM
• APPENDIX B - MULTIPLE INTEGRATION
• BIBLIOGRAPHY
• INDEX