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

List of contents

Preface xv
 
About the Book xvii
 
Introduction 1
 
Part I: Tensor Theory 7
 
1 Preliminaries 9
 
1.1 Introduction 9
 
1.2 Systems of Different Orders 9
 
1.3 Summation Convention Certain Index 10
 
1.3.1 Dummy Index 11
 
1.3.2 Free Index 11
 
1.4 Kronecker Symbols 11
 
1.5 Linear Equations 14
 
1.6 Results on Matrices and Determinants of Systems 15
 
1.7 Differentiation of a Determinant 18
 
1.8 Examples 19
 
1.9 Exercises 23
 
2 Tensor Algebra 25
 
2.1 Introduction 25
 
2.2 Scope of Tensor Analysis 25
 
2.2.1 n-Dimensional Space 26
 
2.3 Transformation of Coordinates in S n 27
 
2.3.1 Properties of Admissible Transformation of Coordinates 30
 
2.4 Transformation by Invariance 31
 
2.5 Transformation by Covariant Tensor and Contravariant Tensor 32
 
2.6 The Tensor Concept: Contravariant and Covariant Tensors 34
 
2.6.1 Covariant Tensors 34
 
2.6.2 Contravariant Vectors 35
 
2.6.3 Tensor of Higher Order 40
 
2.6.3.1 Contravariant Tensors of Order Two 40
 
2.6.3.2 Covariant Tensor of Order Two 41
 
2.6.3.3 Mixed Tensors of Order Two 42
 
2.7 Algebra of Tensors 43
 
2.7.1 Equality of Two Tensors of Same Type 45
 
2.8 Symmetric and Skew-Symmetric Tensors 45
 
2.8.1 Symmetric Tensors 45
 
2.8.2 Skew-Symmetric Tensors 46
 
2.9 Outer Multiplication and Contraction 51
 
2.9.1 Outer Multiplication 51
 
2.9.2 Contraction of a Tensor 53
 
2.9.3 Inner Product of Two Tensors 54
 
2.10 Quotient Law of Tensors 56
 
2.11 Reciprocal Tensor of a Tensor 58
 
2.12 Relative Tensor, Cartesian Tensor, Affine Tensor, and Isotropic Tensors 60
 
2.12.1 Relative Tensors 60
 
2.12.2 Cartesian Tensors 63
 
2.12.3 Affine Tensors 63
 
2.12.4 Isotropic Tensor 64
 
2.12.5 Pseudo-Tensors 64
 
2.13 Examples 65
 
2.14 Exercises 71
 
3 Riemannian Metric 73
 
3.1 Introduction 73
 
3.2 The Metric Tensor 74
 
3.3 Conjugate Tensor 75
 
3.4 Associated Tensors 77
 
3.5 Length of a Vector 84
 
3.5.1 Length of Vector 84
 
3.5.2 Unit Vector 85
 
3.5.3 Null Vector 86
 
3.6 Angle Between Two Vectors 86
 
3.6.1 Orthogonality of Two Vectors 87
 
3.7 Hypersurface 88
 
3.8 Angle Between Two Coordinate Hypersurfaces 89
 
3.9 Exercises 95
 
4 Tensor Calculus 97
 
4.1 Introduction 97
 
4.2 Christoffel Symbols 97
 
4.2.1 Properties of Christoffel Symbols 98
 
4.3 Transformation of Christoffel Symbols 110
 
4.3.1 Law of Transformation of Christoffel Symbols of 1st Kind 110
 
4.3.2 Law of Transformation of Christoffel Symbols of 2nd Kind 111
 
4.4 Covariant Differentiation of Tensor 113
 
4.4.1 Covariant Derivative of Covariant Tensor 114
 
4.4.2 Covariant Derivative of Contravariant Tensor 115
 
4.4.3 Covariant Derivative of Tensors of Type (0,2) 116
 
4.4.4 Covariant Derivative of Tensors of Type (2,0) 118
 
4.4.5 Covariant Derivative of Mixed Tensor of Type (s, r) 120
 
4.4.6 Covariant Derivatives of Fundamental Tensors and the Kronecker Delta 120
 
4.4.7 Formulas for Covariant Differentiation 122
 
4.4.8 Covariant Differentiation of Relative Tensors 123
 
4.5 Gradient, Divergence, and Curl 129
 
4.5.1 Gradient 130
 
4.5.2 Divergence 130
 
4.5.2.1 Divergence of a Mixed Tensor (1,1) 13

About the author

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.

Summary

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

Product details

• Authors: De, D De, Dipankar De, Dipankar (University of Calcutta De, Tripu
• Assisted by: Dipanka De (Editor), Dipankar De (Editor)
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 31.07.2021
• EAN: 9781119795629
• ISBN: 978-1-119-79562-9
• No. of pages: 512
• Series: Modern Mathematics in Computer Science; Modern Mathematics in Computer
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Geometry; Statistik, Mathematik, Informatik, Geometrie, Visualisierung, Statistics, Tensor, Mathematics, computer science, Differentialgeometrie, Rechnergestützte u. graphische Statistik, Computational & Graphical Statistics, Geometrie u. Topologie, Geometry & Topology, Visualization & Computer Graphics, Visualisierung u. Computergraphik