Preface 1 Normed Vector Spaces 1.1 First Notions 1.2 Limits and Continuity 1.3 Open and Closed Sets 1.4 Compactness 1.5 Banach Spaces 1.6 Linear and Polynomial Mappings 1.7 Normed Algebras 1.8 The Exponential Mapping Appendix： The Fundamental Theorem of Algebra 2 Differentiation 2.1 Directional Derivatives 2.2 The Differential 2.3 Differentials of Compositions 2.4 Mappings of Class C1 2.5 Extrema 2.6 Differentiability of the Norm Appendix： Gateaux Differentiability 3 Mean Value Theorems 3.1 Generalizing the Mean Value Theorem 3.2 Partial Differentials 3.3 Integration 3.4 Differentiation under the Integral Sign 4 Higher Derivatives and Differentials 4.1 Schwarz's Theorem 4.2 Operationson Ck-Mappings 4.3 Multilinear Mappings 4.4 Higher Differentials 4.5 Higher Differentials and Higher Derivatives 4.6 Cartesian Product Image Spaces 4.7 Higher Partial Differentials 4.8 Generalizing Ck to Normed Vector Spaces 4.9 Leibniz's Rule 5 Taylor Theorems and Applicahons 5.1 Taylor Formulas 5.2 Asymptotic Developments 5.3 Extrema： Second-Order Conditions Appendix： Homogeneous Polynomials 6 Hilbert Spaces 6.1 Basic Notions 6.2 Projections 6.3 The Distance Mapping 6.4 The Riesz Representation Theorem 7 Convex Functions 7.1 Preliminary Results 7.2 Continuity of Convex Functions 7.3 Differentiable Convex Functions 7.4 Extrema of Convex Functions Appendix： Convex Polyhedra
8 The Inverse and Implicit Mapping Theorems 8.1 The Inverse Mapping Theorem 8.2 The Implicit Mapping Theorem 8.3 The Rank Theorem 8.4 Constrained Extrema Appendix 1： Bijective Continuous Linear Mappings Appendix 2： Contractions 9 Vector Fields 9.1 Existence of Integral Curves 9.2 Initial Conditions 9.3 Geometrical Properties of Integral Curves 9.4 Complete Vector Fields Appendix： A Useful Result on Smooth Functions 10 The Flow of a Vector Field 10.1 Continuity of the Flow 10.2 Differentiability of the Flow 10.3 Higher Differentiability of the Flow 10.4 The Reduced Flow 10.5 One-Parameter Subgroups 11 The Calculus of Variations： An Introduction 11.1 The Space C1（I，E） 11.2 Lagrangian Mappings 11.3 Fixed Endpoint Problems 11.4 Euler-LagrangeEquations 11.5 Convexity 11.6 The Class of an Extremal References Index
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