Vector and Tensor Analysis with Applications

A. I. Borisenko, I. E. Tarapov, I. E. Tarapov, Richard A. Silverman

• Publisher: Dover Publications
• Year: 2012
• eBook ISBN: 9780486131900
• Topic: Mathematics
• Subtopic: Vector Analysis
• Index: Mathematics

" … the exposition is clear and the choice of topics is excellent  …" Prof. R. E. Williamson, Univ. of Sussex, Brighton, England
This concise introduction to a basic branch of applied mathematics is indispensable to mathematicians, physicists and engineers. Eminently readable, it covers the elements of vector and tensor analysis, with applications of the theory to specific physics and engineering problems. It lays particular stress on the applications of the theory to fluid dynamics.
The authors begin with a definition of vectors and a discussion of algebraic operations on vectors. The vector concept is then generalized in a natural way, leading to the concept of a tensor. Chapter Three considers algebraic operations on tensors. Next, the authors turn to a systematic study of the differential and integral calculus of vector and tensor functions of space and time. Finally, vector and tensor analysis is considered from both a rudimentary standpoint, and in its fuller ramifications, concluding the volume.
The strength of the book lies in the completely worked out problems and solutions at the end of each chapter. In addition, each chapter incorporates abundant exercise material. Intended primarily for advanced undergraduates and graduate students of math, physics and engineering, the work is self-contained and accessible to any student with a good background in calculus.
Vector and Tensor Analysis With Applications is one of a series of SELECTED RUSSIAN PUBLICATIONS IN THE MATHEMATICAL SCIENCES, several of which have already been published by Dover. The authors are distinguished Russian mathematicians and specialists in gas dynamics and numerical analysis. Richard A. Silverman, editor of the series as well as editor and translator of this volume, has revised and improved the original edition and added a bibliography.
" … a concise, clear and comprehensive treatment  … " Prof. Henry G. Booker, University of California, San Diego

1

VECTOR ALGEBRA

In this chapter we define vectors and then discuss algebraic operations on , vectors. The vector concept will be generalized in a natural way in Chapter 2, leading to the concept of a tensor. Then in Chapter 3 we will consider algebraic operations on tensors.

1.1. Vectors and Scalars

Quantities which can be specified by giving just one number (positive, negative or zero) are called scalars. For example, temperature, density, mass and work are all scalars. Scalars can be compared only if they have the same physical dimensions. Two such scalars measured in the same system of units are said to be equal if they have the same magnitude (absolute value) and sign.

However, one must often deal with quantities, called vectors, whose specification requires a direction as well as a numerical value. For example, displacement, velocity, acceleration, force, the moment of a force, electric field strength and dielectric polarization are all vectors. Operations on vectors obey the rules of vector algebra, to be considered below.

Scalars and vectors hardly exhaust the class of quantities of interest in applied mathematics and physics. In fact, there are quantities of a more complicated structure than scalars or vectors, called tensors (of order 2 or higher), whose specification requires more than knowledge of a magnitude and a direction. For example, examination of the set of all stress vectors acting on all elements of area which can be drawn through a given point of an elastic body leads to the concept of the stress tensor (of order 2), while examination of the deformation of an arbitrary elementary volume of an elastic body leads to the concept of the deformation (or strain) tensor. We will defer further discussion of tensors until Chapter 2, concentrating our attention for now on vectors.

A vector A is represented by a directed line segment, whose direction and length coincide with the direction and magnitude (measured in the chosen system of units) of the quantity under consideration. Vector...

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• EDITOR’S PREFACE
• Table of Contents
• 1 - VECTOR ALGEBRA
• 2 - THE TENSOR CONCEPT
• 3 - TENSOR ALGEBRA
• 4 - VECTOR AND TENSOR ANALYSIS: RUDIMENTS
• 5 - VECTOR AND TENSOR ANALYSIS: RAMIFICATIONS
• BIBLIOGRAPHY
• INDEX