Tensor Analysis and Its Applications

By Quddus Khan

This book is intended to serve as a textbook for undergraduate and postgraduate students of mathematics. It will be useful to the researchers working in the field of differential geometry and its applications to general theory of relativity and other applied areas. It will also be helpful in preparing for the competitive examinations like IAS, IES, NET, PCS, and other higher education tests. The text starts with the basic concepts and results, which shall refer throughout this book and is followed by the study of the tensor algebra and its calculus, consisting the notion of tensor, its operations, and its different types; Christoffels symbols and its properties, the concept of covariant differentiation of tensors and its properties, tensor form of gradient, divergence, laplacian and curl, divergence of a tensor, intrinsic derivatives, and parallel displacement of vectors, Riemanns symbols and its properties, and application of tensor in different areas.

• Language: English
• Publisher: Partridge Publishing India
• Release date: Aug 27, 2015
• ISBN: 9781482850680

Quddus Khan

Dr. Quddus Khan, PhD, is a senior assistant professor in the Department of Maths., Shibli National P. G. College, Azamgarh (UP), India. He has been teaching undergraduate and postgraduate classes for the last sixteen years. Dr. Khan has to his credit sixteen research papers published in various national and international journals and has coauthored three books, authored three books, and produced two PhDs.

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Copyright © 2015 by Khan.

All rights reserved. No part of this book may be used or reproduced by any means, graphic, electronic, or mechanical, including photocopying, recording, taping or by any information storage retrieval system without the written permission of the author except in the case of brief quotations embodied in critical articles and reviews.

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Contents

Preface

1 Preliminaries

1.1 n - Dimensional Space

1.2 Superscript and Subscript

1.3 Einstein’s Summation Convention

1.4 Dummy Suffix and Real Suffix

1.5 Transformation of Coordinates

1.6 Kronecker Delta

1.7 Matrices and Determinants

2 Tensor Algebra and its Calculus

2.1 Tensors

2.2 Operations on Tensors

2.2.1 Addition and Subtraction of Tensors

2.2.2 Multiplication of Tensors

2.2.3 Contraction

2.2.4 Inner Product or Scalar Product

2.2.5 Quotient Law

2.2.6 Extension of Rank of a Tensor

2.3 Types of Tensors

2.3.1 Invariant Tensors and Equality of Tensors

2.3.2 Fundamental Tensor

2.3.3 Symmetric Tensor and Anti - Symmetric (Skew - Symmetric) Tensor

2.3.4 Reciprocal Tensor

2.3.5 Reducible Tensor and Irreducible Tensor

2.3.6 Relative Tensor

2.3.7 Raising and Lowering Suffix: Associated Tensors

3 Christoffel’s Symbols and their Properties

3.1 Christoffel’s Symbols or Christoffel’s Brackets

3.2 Symmetric Properties of Christoffel’s Symbols

3.3 Tensor Laws of Transformation of Christoffel’s Symbols

3.4 Transitive Property for Laws of Transformation of Christoffel’s Symbols

4 Covariant Differentiation of Tensors and their Properties

4.1 Comma Notation

4.2 Covariant Differentiation of Vectors (Tensors of Rank one)

4.3 Covariant Derivative of a Covariant Tensor of Rank two

4.4 Covariant Derivative of a Contravariant Tensor of Rank two

4.5 Covariant Derivative of a Mixed Tensor of Rank two

4.6 Covariant Derivative of the Fundamental Tensors

4.7 Laws of Covariant Differentiation

4.8 Covariant Derivative of an Invariant (or a Scalar)

4.9 Tensor Form of Gradient, Divergence, Laplacian and Curl

4.10 Divergence of a Tensor

4.11 Intrinsic Derivatives

4.12 Parallel Displacement of Vectors

5 Riemannian Symbols and its Properties

5.1 Riemannian Symbols of Second Kind

5.2 Properties of the Riemann Curvature Tensor 35559.png

5.3 Contraction of the Riemann Curvature Tensor 35559.png

5.4 Riemannian Symbols of First Kind

5.5 Properties of Covariant Curvature Tensor 36621.png

5.6 Contraction of Bianchi’s Identities (Einstein tensor)

5.7 Riemannian Curvature

5.8 Flat Space

5.9 Uniform Vector Field

5.10 Conditions for Flat Space - time

5.11 Space of Constant Curvature

5.12 Einstein space

6 Application of Tensor Calculus

6.1 Introduction

6.2 Application of Tensor Calculus

6.2.1 Tensor Calculus and Differential Geometry

6.2.2 Tensor Calculus and Riemannian Geometry

6.2.3 Tensor Calculus and Theory of Relativity

6.2.4 Tensor Calculus and Elasticity

6.2.5 Tensor Calculus and Physics

6.2.6 Tensor Calculus and Other Fields like Economics, Probability and Engineering

Bibliography

Preface

T he notion of a tensor was introduced and studied by Professor Gregorio Ricci of the University of Padua (Italy) in 1887 primarily as the extension of vectors. The classical definitions of scalars (having magnitude only) and vectors (with magnitude and direction as well) do not cover completely many physical and geometrical quantities. For instance, stress in an elastic body and curl of a vector supposedly misunderstood as vectors are much more than vectors. Indeed, they are tensors.

The object of this book is to provide a compact exposition of the fundamental results in the theory of tensors and also to illustrate the power of the tensor technique by applications to differential geometry, elasticity, and relativity. This book is intended to be a textbook, suitable for use in a undergraduate (honors) / postgraduate Mathematics course. This book is useful for those who are totally unfamiliar with tensor analysis. Those who are already acquainted with tensor analysis will fell interested in application of tensor analysis in Riemannian geometry, theory of relativity and many other disciplines of science and technology.

This book consists of six chapters. The first chapter deals with brief concepts of n - dimensional space, superscript and subscript, Einstein’s summation convention, dummy suffix and real suffix, transformation of coordinates, Kronecker delta and matrices and determinants etc. which are referred to in subsequent chapters. We begin second chapter by defining tensors and in sequence, we discuss in detail about the operations of the tensors and the different types of the tensors. We introduce the notion of the Christoffel’s symbols and discuss in detail about its properties in chapter third. The concept of covariant differentiation of tensors and its properties, tensor form of gradient, divergence, Laplacian and curl, divergence of a tensor, Intrinsic derivatives and parallel displacement of vectors etc. have been studied in detail in chapter fourth. In chapter five, we have studied about Riemann’s symbols (first kind and second kind) and its properties and have also discussed on a flat space, uniform vector field, conditions for flat space - time, space of constant curvature, Einstein space etc., which are mainly useful for postgraduate students and also for researchers. In last chapter, we have discussed the applications of tensor calculus which are mainly useful for researchers.

I have freely consultanted the books on tensor analysis, Riemannian geometry by the authors given in the bibliography. I am indebted to these authors for providing a good and exhaustive literature on the subject due to which the writing of this book could be made possible.

I wish to express my best thanks to the publisher for bringing out the book in such a nice form. Any comment and suggestions for improving the text will be most welcome.

Quddus Khan

1 Preliminaries

W e begin this chapter by defining n - dimensional space and giving some other fundamental conceptual definitions to which we shall refer throughout this book.

1.1 n - Dimensional Space

Consider an ordered set of n real variables 8222.png These variables will be called the coordinates of a point. The space generated by all points corresponding to distinct values of the coordinates is called an n - dimensional space and is denoted by Vn.

A curve in Vn is defined as the collection of the points which satisfy the n equations

8234.png

where u is a parameter and 8245.png are n functions of u, which obey certain continuity condition.

A subspace Vm of Vn is defined for 8259.png as the collection of points which satisfy the n equations

8270.png

where the variables 8280.png are the coordinates of Vm. Also, 8292.png are n functions of 8303.png satisfying certain conditions of continuity.

In particular, the subspace Vm is called a curve, a surface and a hypersurface according as 8315.png and 8326.png respectively.

1.2 Superscript and Subscript

The suffixes i and j in 8344.png are called superscript and subscript, respetively. The upper position always denotes superscript and lower position denotes subscript. The suffix i in the coordinate 8355.png do not have the character of power indices. Usually powers will be dented by bracket. Thus, 8367.png means square of 8377.png

1.3 Einstein’s Summation Convention

If any index in a term is repeated, then a summation with respect to that index over the range 1, 2, …, n is implied. This convention is known as Einstein’s Summation. The expression 8393.png is represented by 8405.png Summation convention means drop the sigma sign and adopt the convention. In view of summation convention, we have

8417.png

(1.1)

Hence by summation convention we mean that if a suffix occurs twice in a term, once in upper position and once in lower position, then that suffix implied sum over defined range. If the range is not given, then we assume that the range is from 1 to n.

1.4 Dummy Suffix and Real Suffix

If a suffix occurs twice in a term, once in upper position and once in lower position, then that suffix is called a dummy suffix. For example, in 8431.png , i is a dummy suffix. Evedently,

8443.png

(1.2)

and 8457.png

(1.3)

From (1.2) and (1.3), we have

8471.png

(1.4)

which shows that a dummy suffix can be replaced by another dummy suffix not already appearing in the expressin. Also, two or more than two dummy suffixes can be interchanged. For example,

8487.png

(1.5)

Dummy suffix is also known as umbral suffix or dextral index.

A suffix which is not repeated is called a real or free suffix. For example, 8503.png is a real suffix in 8516.png A