Elements of Tensor Calculus

A. Lichnerowicz, J.W. Leech, J.W. Leech, D.J. Newman

• Publisher: Dover Publications
• Year: 2016
• Print ISBN: 9780486805177
• eBook ISBN: 9780486811864
• Edition: 1
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

This classic introductory text, geared toward undergraduate students of mathematics, is the work of an internationally renowned authority on tensor calculus. The two-part treatment offers a rigorous presentation of tensor calculus as a development of vector analysis as well as discussions of the most important applications of tensor calculus.
Starting with a chapter on vector spaces, Part I explores affine Euclidean point spaces, tensor algebra, curvilinear coordinates in Euclidean space, and Riemannian spaces. Part II examines the use of tensors in classical analytical dynamics and details the role of tensors in special relativity theory. The book concludes with a brief presentation of the field equations of general relativity theory.

PART I: TENSOR CALCULUS

CHAPTER I

Vector Spaces

I. CONCEPT OF A VECTOR SPACE

1. Definition of a vector space. Consider the set of displacement vectors of elementary vector analysis. These satisfy the following rules:

(i) The result of vector addition of any two vectors, x and y, is their vector sum, or resultant, x + y. Vector addition has the following properties:

(a) x + y = y + x (commutative property);

(b) x + (y + z) = (x + y) + z (associative property);

(c) there exists a zero vector denoted by 0 such that x + 0 = x;

(d) for every vector x there is a corresponding negative vector (–x), such that x + (–x) = 0.

(ii) The result of multiplying a vector x by a real scalar α is a vector denoted by αx. Scalar multiplication has the following properties:

(a′) 1x = x;

(b′) α(βx) = (αβ)x (associative property);

(c′) (α + β)x = αx + βx (distributive property for scalar addition);

(d′) α(x + y) = αx + αy (distributive property for vector addition).

Using the above properties as a guide, we now consider a general set E of arbitrary elements x, y etc., which obey the following rules:

(1) To every pair x, y, there corresponds an element x + y having the properties (a), (b), (c), (d).

(2) To every combination of an element x and a real number α there corresponds an element αx having the properties (a′), (b′), (c′), (d′).

We then say that E is a vector space over the field of real numbers and that the elements x, y, etc., are vectors in E. If the second rule holds for all complex numbers a then E is a vector space over the field of complex numbers. Except when otherwise stated we shall confine ourselves in this book to the study of vector spaces over the field of real numbers.

2. Examples of vector spaces. There are several other simple examples of vector spaces which may be quoted to give an idea of the interest and application of the general concept.

(a) Consider the set of complex numbers a + ib, where a and b are real. The addition of any two complex numbers (a + ib, c + id, etc.) and the multiplication of a complex number by a real number α obviously have the properties listed in §1. It follows that the...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface
• Part I: Tensor Calculus
• Part II: Applications
• Bibliography
• Index