Multidimensional real analysis. Differentiation 1

Volume 1 provides a comprehensive review of differential analysis in multidimensional Euclidean space. Half-title......Page 2 Series-title......Page 3 Title......Page 4 Copyright......Page 5 Dedication......Page 6 Contents......Page 8 Preface......Page 12 Acknowledgments......Page 14 Introduction......Page 16 1.1 Inner product and norm......Page 20 1.2 Open and closed sets......Page 25 1.3 Limits and continuous mappings......Page 30 1.4 Composition of mappings......Page 36 1.5 Homeomorphisms......Page 38 1.6 Completeness......Page 39 1.7 Contractions......Page 42 1.8 Compactness and uniform continuity......Page 43 1.9 Connectedness......Page 52 2.1 Linear mappings......Page 56 2.2 Differentiable mappings......Page 61 2.3 Directional and partial derivatives......Page 66 2.4 Chain rule......Page 70 2.5 Mean Value Theorem......Page 75 2.6 Gradient......Page 77 2.7 Higher-order derivatives......Page 80 2.8 Taylor’s formula......Page 85 2.9 Critical points......Page 89 2.10 Commuting limit operations......Page 95 3.1 Diffeomorphisms......Page 106 3.2 Inverse Function Theorems......Page 108 3.3 Applications of Inverse Function Theorems......Page 113 3.4 Implicitly defined mappings......Page 115 (B) An idea about the solution......Page 116 (C) Formula for the derivative of the solution......Page 117 (D) The conditions are necessary......Page 118 3.5 Implicit Function Theorem......Page 119 3.6 Applications of the Implicit Function Theorem......Page 120 3.7 Implicit and Inverse Function Theorems on C......Page 124 4.1 Introductory remarks......Page 126 4.2 Manifolds......Page 128 4.3 Immersion Theorem......Page 133 4.4 Examples of immersions......Page 137 4.5 Submersion Theorem......Page 139 4.6 Examples of submersions......Page 143 4.7 Equivalent definitions of manifold......Page 145 4.8 Morse’s Lemma......Page 147 5.1 Definition of tangent space......Page 152 5.3 Examples of tangent spaces......Page 156 5.4 Method of Lagrange multipliers......Page 168 5.5 Applications of the method of multipliers......Page 170 5.6 Closer investigation of critical points......Page 173 5.7 Gaussian curvature of surface......Page 175 5.8 Curvature and torsion of curve in R3......Page 178 5.9 One-parameter groups and infinitesimal generators......Page 181 5.10 Linear Lie groups and their Lie algebras......Page 185 5.11 Transversality......Page 191 Review Exercises......Page 194 Exercises for Chapter 1......Page 220 Exercises for Chapter 2......Page 236 Exercises for Chapter 3......Page 278 Exercises for Chapter 4......Page 312 Exercises for Chapter 5......Page 336 Notation......Page 430 Index......Page 432 Part one of the authors'comprehensive and innovative work on multidimensional real analysis. This book is based on extensive teaching experience at Utrecht University and gives a thorough account of differential analysis in multidimensional Euclidean space. It is an ideal preparation for students who wish to go on to more advanced study. The notation is carefully organized and all proofs are clean, complete and rigorous. The authors have taken care to pay proper attention to all aspects of the theory. In many respects this book presents an original treatment of the subject and it contains many results and exercises that cannot be found elsewhere. The numerous exercises illustrate a variety of applications in mathematics and physics. This combined with the exhaustive and transparent treatment of subject matter make the book ideal as either the text for a course, a source of problems for a seminar or for self study. Part one of the authors' comprehensive and innovative work on multidimensional real analysis. This book is based on extensive teaching experience at Utrecht University and gives a thorough account of differential analysis in multidimensional Euclidean space. It is an ideal preparation for students who wish to go on to more advanced study. The notation is carefully organized and all proofs are clean, complete and rigorous. The authors have taken care to pay proper attention to all aspects of the theory. In many respects this book presents an original treatment of the subject and it contains man Part one of the authors' comprehensive and innovative work on multidimensional real analysis. Numerous illustrative exercises combined with an exhaustive and transparent treatment of subject matter make the book ideal as either the text for a course, a source of problems for a seminar or for self study