Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds (eBook)

Preface 6
Acknowledgements 8
Contents 9
Selected Symbols 11
1 Introduction 13
1.1 Space, Geometry, and Linear Algebra 13
1.2 Vectors as Geometrical Objects 14
1.3 Differentiable Manifolds: First Contact 15
1.4 Digression on Notation and Mappings 18
References 20
2 Notes on Point Set Topology 21
2.1 Preliminary Remarks and Basic Concepts 21
2.2 Topology in Metric Spaces 22
2.3 Topological Space: Definition and Basic Notions 27
2.4 Connectedness, Compactness, and Separability 29
2.5 Product Spaces and Product Topologies 31
2.6 Further Reading 33
References 34
3 The Finite-Dimensional Real Vector Space 35
3.1 Definitions 35
3.2 Linear Independence and Basis 37
3.3 Some Common Examples for Vector Spaces 40
3.4 Change of Basis 41
3.5 Linear Mappings Between Vector Spaces 42
3.6 Linear Forms and the Dual Vector Space 44
3.7 The Inner Product, Norm, and Metric 47
3.8 The Reciprocal Basis and Its Relations with the Dual Basis 49
References 52
4 Tensor Algebra 53
4.1 Tensors and Multi-linear Forms 53
4.2 Dyadic Product and Tensor Product Spaces 55
4.3 The Dual of a Linear Mapping 58
4.4 Remarks on Notation and Inner Product Operations 58
4.5 The Exterior Product and Alternating Multi-linear Forms 60
4.6 Symmetric and Skew-Symmetric Tensors 62
4.7 Generalized Kronecker Symbol 63
4.8 The Spaces ?k mathcalV and ?k mathcalV* 64
4.9 Properties of the Exterior Product and the Star-Operator 65
4.10 Relation with Classical Linear Algebra 67
References 69
5 Affine Space and Euclidean Space 70
5.1 Definitions and Basic Notions 70
5.2 Alternative Definition of an Affine Space by Hybrid Addition 72
5.3 Affine Mappings, Coordinate Charts and Topological Aspects 73
References 78
6 Tensor Analysis in Euclidean Space 79
6.1 Differentiability in mathbbR and Related Concepts Briefly Revised 79
6.2 Generalization of the Concept of Differentiability 82
6.3 Gradient of a Scalar Field and Related Concepts in mathbbRN 83
6.4 Differentiability in Euclidean Space Supposing Affine Relations 87
6.5 Characteristic Features of Nonlinear Chart Relations 94
6.6 Partial Derivatives as Vectors and Tangent Space at a Point 96
6.7 Curvilinear Coordinates and Covariant Derivative 99
6.8 Differential Forms in mathbbRN and Integration 103
6.9 Exterior Derivative and Stokes' Theorem in Form Language 105
References 108
7 A Primer on Smooth Manifolds 109
7.1 Introduction 109
7.2 Basic Concepts Regarding Analysis on Surfaces in mathbbR3 113
7.3 Transition to Smooth Manifolds 118
7.4 Tangent Bundle and Vector Fields 119
7.5 Flow of Vector Fields and the Lie Derivative 124
7.6 Outlook and Further Reading 128
References 129
Appendix Solutions for Selected Problems 130
Index 132