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Table of Contents
Preface; Acknowledgements; Contents; Selected Symbols; 1 Introduction; 1.1 Space, Geometry, and Linear Algebra; 1.2 Vectors as Geometrical Objects; 1.3 Differentiable Manifolds: First Contact; 1.4 Digression on Notation and Mappings; References; 2 Notes on Point Set Topology; 2.1 Preliminary Remarks and Basic Concepts; 2.2 Topology in Metric Spaces; 2.3 Topological Space: Definition and Basic Notions; 2.4 Connectedness, Compactness, and Separability; 2.5 Product Spaces and Product Topologies; 2.6 Further Reading; References; 3 The Finite-Dimensional Real Vector Space; 3.1 Definitions.
3.2 Linear Independence and Basis3.3 Some Common Examples for Vector Spaces; 3.4 Change of Basis; 3.5 Linear Mappings Between Vector Spaces; 3.6 Linear Forms and the Dual Vector Space; 3.7 The Inner Product, Norm, and Metric; 3.8 The Reciprocal Basis and Its Relations with the Dual Basis; References; 4 Tensor Algebra; 4.1 Tensors and Multi-linear Forms; 4.2 Dyadic Product and Tensor Product Spaces; 4.3 The Dual of a Linear Mapping; 4.4 Remarks on Notation and Inner Product Operations; 4.5 The Exterior Product and Alternating Multi-linear Forms; 4.6 Symmetric and Skew-Symmetric Tensors.
6.3 Gradient of a Scalar Field and Related Concepts in mathbbRN6.4 Differentiability in Euclidean Space Supposing Affine Relations; 6.5 Characteristic Features of Nonlinear Chart Relations; 6.6 Partial Derivatives as Vectors and Tangent Space at a Point; 6.7 Curvilinear Coordinates and Covariant Derivative; 6.8 Differential Forms in mathbbRN and Integration; 6.9 Exterior Derivative and Stokes' Theorem in Form Language; References; 7 A Primer on Smooth Manifolds; 7.1 Introduction; 7.2 Basic Concepts Regarding Analysis on Surfaces in mathbbR3; 7.3 Transition to Smooth Manifolds.
7.4 Tangent Bundle and Vector Fields7.5 Flow of Vector Fields and the Lie Derivative; 7.6 Outlook and Further Reading; References; Appendix Solutions for Selected Problems; Index.
3.2 Linear Independence and Basis3.3 Some Common Examples for Vector Spaces; 3.4 Change of Basis; 3.5 Linear Mappings Between Vector Spaces; 3.6 Linear Forms and the Dual Vector Space; 3.7 The Inner Product, Norm, and Metric; 3.8 The Reciprocal Basis and Its Relations with the Dual Basis; References; 4 Tensor Algebra; 4.1 Tensors and Multi-linear Forms; 4.2 Dyadic Product and Tensor Product Spaces; 4.3 The Dual of a Linear Mapping; 4.4 Remarks on Notation and Inner Product Operations; 4.5 The Exterior Product and Alternating Multi-linear Forms; 4.6 Symmetric and Skew-Symmetric Tensors.
6.3 Gradient of a Scalar Field and Related Concepts in mathbbRN6.4 Differentiability in Euclidean Space Supposing Affine Relations; 6.5 Characteristic Features of Nonlinear Chart Relations; 6.6 Partial Derivatives as Vectors and Tangent Space at a Point; 6.7 Curvilinear Coordinates and Covariant Derivative; 6.8 Differential Forms in mathbbRN and Integration; 6.9 Exterior Derivative and Stokes' Theorem in Form Language; References; 7 A Primer on Smooth Manifolds; 7.1 Introduction; 7.2 Basic Concepts Regarding Analysis on Surfaces in mathbbR3; 7.3 Transition to Smooth Manifolds.
7.4 Tangent Bundle and Vector Fields7.5 Flow of Vector Fields and the Lie Derivative; 7.6 Outlook and Further Reading; References; Appendix Solutions for Selected Problems; Index.