Introduction to Differential Geometry with Tensor Applications

INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH TENSOR APPLICATIONS This is the only volume of its kind to explain, in precise and easy-to-understand language, the fundamentals of tensors and their applications in differential geometry and analytical mechanics with examples for practical applications and questions for use in a course setting.

Introduction to Differential Geometry with Tensor Applications discusses the theory of tensors, curves and surfaces and their applications in Newtonian mechanics. Since tensor analysis deals with entities and properties that are independent of the choice of reference frames, it forms an ideal tool for the study of differential geometry and also of classical and celestial mechanics. This book provides a profound introduction to the basic theory of differential geometry: curves and surfaces and analytical mechanics with tensor applications. The author has tried to keep the treatment of the advanced material as lucid and comprehensive as possible, mainly by including utmost detailed calculations, numerous illustrative examples, and a wealth of complementing exercises with complete solutions making the book easily accessible even to beginners in the field.

Groundbreaking and thought-provoking, this volume is an outstanding primer for modern differential geometry and is a basic source for a profound introductory course or as a valuable reference. It can even be used for self-study, by students or by practicing engineers interested in the subject.

Whether for the student or the veteran engineer or scientist, Introduction to Differential Geometry with Tensor Applications is a must-have for any library.

This outstanding new volume:

• Presents a unique perspective on the theories in the field not available anywhere else
• Explains the basic concepts of tensors and matrices and their applications in differential geometry and analytical mechanics
• Is filled with hundreds of examples and unworked problems, useful not just for the student, but also for the engineer in the field
• Is a valuable reference for the professional engineer or a textbook for the engineering student

Dipankar De, PhD, received his BSc and MSc in mathematics from the University of Calcutta, India and his PhD in mathematics from Tripura University, India. He has over 40 years of teaching experience and is an associate professor and guest lecturer in India. He has published many research papers in various reputed journal in the field of fuzzy mathematics and differential geometry.

INTRODUCTION TO DIFFERENTIAL GEOMETRY WITH TENSOR APPLICATIONS

This is the only volume of its kind to explain, in precise and easy-to-understand language, the fundamentals of tensors and their applications in differential geometry and analytical mechanics with examples for practical applications and questions for use in a course setting.

Introduction to Differential Geometry with Tensor Applications discusses the theory of tensors, curves and surfaces and their applications in Newtonian mechanics. Since tensor analysis deals with entities and properties that are independent of the choice of reference frames, it forms an ideal tool for the study of differential geometry and also of classical and celestial mechanics. This book provides a profound introduction to the basic theory of differential geometry: curves and surfaces and analytical mechanics with tensor applications. The author has tried to keep the treatment of the advanced material as lucid and comprehensive as possible, mainly by including utmost detailed calculations, numerous illustrative examples, and a wealth of complementing exercises with complete solutions making the book easily accessible even to beginners in the field.

Groundbreaking and thought-provoking, this volume is an outstanding primer for modern differential geometry and is a basic source for a profound introductory course or as a valuable reference. It can even be used for self-study, by students or by practicing engineers interested in the subject.

Whether for the student or the veteran engineer or scientist, Introduction to Differential Geometry with Tensor Applications is a must-have for any library.

This outstanding new volume:

• Presents a unique perspective on the theories in the field not available anywhere else

• Explains the basic concepts of tensors and matrices and their applications in differential geometry and analytical mechanics

• Is filled with hundreds of examples and unworked problems, useful not just for the student, but also for the engineer in the field

• Is a valuable reference for the professional engineer or a textbook for the engineering student

Auteur

Dipankar De, PhD, received his BSc and MSc in mathematics from the University of Calcutta, India and his PhD in mathematics from Tripura University, India. He has over 40 years of teaching experience and is an associate professor and guest lecturer in India. He has published many research papers in various reputed journal in the field of fuzzy mathematics and differential geometry.

Échantillon de lecture
1
Preliminaries
1.1 Introduction

Some quantities are associated with their magnitude and direction, but certain quantities are associated with two or more directions. Such a quantity is called a tensor, e.g., the stress at a point of an elastic solid is an example of a tensor which depends on two directions: one is normal and the other is that of force on the area. Tensor comes from the word tension.

In this chapter, we discuss the notation of systems of different orders, which are applied in the theory of determinants, symbols, and summation conventions. Also, results on some matrices and determinants are discussed because they will be used frequently later on. 1.2 Systems of Different Orders

Let us consider the two quantities, a*1, *a*1 *or a*1, *a*2, which are represented by *ai or ai, respectively, for i = 1, 2. In such cases, the expressions ai, ai, ai j, ai j, and are called systems. In each value of ai and ai are called systems of first order and each value of ai j, ai j, and is called a double system or system of second order, of which a*12, *a*22*a*23, *a*13, *and are called their respective components. Similarly, we have systems of the third order that depend on three indices shown as ai jk, aikl, ai jm, ai jn, and and each number of their respective components are 8.

In a system of order zero, it is shown that the quantity has no index, such as a. The upper and lower indices of a system are called its indices of contravariance and covariance, respectively. For a system of , i and j are indices of a contravariant and k is of covariance. Accordingly, the system Aij is called a contravariant system, Aklm is called a covariant system, and is called a mixed system. 1.3 Summation Convention Certain Index

If in some expressions a certain index occurs twice, this means that this expression is summed with respect to that index for all admissible values of the index.

Thus, the linear form has an index, i, occurring in it twice. We will omit the summation symbol Sigma and write aixi to mean a*1*x*1 + *a*2*x*2 + *a*3*x*3 + *a*4*x*4. In order to avoid Sigma, we shall make use of a convention used by A. Einstein which is accordingly called the *Einstein Summation Convention or Summation Convention.

Of course, the range of admissible values of the index, 1 to 4 in this case, must be specified. If the symbol i has a range of values from 1 to 3 and j ranges from 1 to 4, the expression (1.1) represents three linear forms: (1.2) Here, index i is the identifying (free) index and since index j, occurs twice, it is the summation index.

We shall adopt this convention throughout the chapters and take the sum whenever a letter appears in a t…