Buch, Englisch, 512 Seiten, Format (B × H): 10 mm x 10 mm, Gewicht: 454 g
ISBN: 978-1-119-79562-9
Verlag: Wiley
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
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
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
1.4 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
2.3 Transformation of Coordinates in S n 27
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
2.7 Algebra of Tensors 43
2.8 Symmetric and Skew-Symmetric Tensors 45
2.9 Outer Multiplication and Contraction 51
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
2.13 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
3.6 Angle Between Two Vectors 86
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
4.3 Transformation of Christoffel Symbols 110
4.4 Covariant Differentiation of Tensor 113
4.5 Gradient, Divergence, and Curl 129
4.6 Exercises 141
5 Riemannian Geometry 143
5.1 Introduction 143
5.2 Riemannian-Christoffel Tensor 143
5.3 Properties of Riemann-Christoffel Tensors 150
5.4 Ricci Tensor, Bianchi Identities, Einstein Tensors 159
5.5 Einstein Space 170
5.6 R
