Note: Supplemental materials are not guaranteed with Rental or Used book purchases.
Purchase Benefits
What is included with this book?
Preface | p. xi |
Basics | p. 1 |
Smooth Functions | p. 1 |
Measures | p. 6 |
Lebesgue Measure | p. 10 |
Integration | p. 12 |
Measurable Functions | p. 12 |
The Integral | p. 14 |
Lebesgue Spaces | p. 20 |
Product Measures and the Fubini-Tonelli Theorem | p. 21 |
The Exterior Algebra | p. 22 |
The Generalized Pythagorean Theorem | p. 25 |
The Hausdorff Distance and Steiner Symmetrization | p. 33 |
Borel and Suslin Sets | p. 42 |
Caratheodory's Construction and Lower-Dimensional Measures | p. 53 |
The Basic Definition | p. 53 |
Hausdorff Measure and Spherical Measure | p. 55 |
A Measure Based on Parallelepipeds | p. 56 |
Projections and Convexity | p. 57 |
Other Geometric Measures | p. 58 |
Summary | p. 59 |
The Densities of a Measure | p. 61 |
A One-Dimensional Example | p. 63 |
Caratheodory's Construction and Mappings | p. 64 |
The Concept of Hausdorff Dimension | p. 67 |
Some Cantor Set Examples | p. 69 |
Basic Examples | p. 70 |
Some Generalized Cantor Sets | p. 72 |
Cantor Sets in Higher Dimensions | p. 73 |
Invariant Measures and the Construction of Haar Measure | p. 77 |
The Fundamental Theorem | p. 78 |
Haar Measure for the Orthogonal Group and the Grassmannian | p. 84 |
Remarks on the Manifold Structure of G(N, M) | p. 88 |
Covering Theorems and the Differentiation of Integrals | p. 91 |
Wiener's Covering Lemma and Its Variants | p. 91 |
The Besicovitch Covering Theorem | p. 99 |
Decomposition and Differentiation of Measures | p. 109 |
The Riesz Representation Theorem | p. 115 |
Maximal Functions Redux | p. 122 |
Analytical Tools: The Area Formula, the Coarea Formula, and Poincare Inequalities | p. 125 |
The Area Formula | p. 125 |
Linear Maps | p. 126 |
C[superscript 1] Functions | p. 132 |
Rademacher's Theorem | p. 134 |
The Coarea Formula | p. 137 |
Measure Theory of Lipschitz Maps | p. 140 |
Proof of the Coarea Formula | p. 142 |
The Area and Coarea Formulas for C[superscript 1] Submanifolds | p. 143 |
Rectifiable Sets | p. 148 |
Poincare Inequalities | p. 151 |
The Calculus of Differential Forms and Stokes's Theorem | p. 159 |
Differential Forms and Exterior Differentiation | p. 159 |
Stokes's Theorem | p. 164 |
Introduction to Currents | p. 173 |
A Few Words about Distributions | p. 174 |
The Definition of a Current | p. 177 |
Constructions Using Currents and the Constancy Theorem | p. 183 |
Further Constructions with Currents | p. 189 |
Products of Currents | p. 189 |
The Pushforward | p. 190 |
The Homotopy Formula | p. 193 |
Applications of the Homotopy Formula | p. 193 |
Rectifiable Currents with Integer Multiplicity | p. 195 |
Slicing | p. 204 |
The Deformation Theorem | p. 211 |
Proof of the Unscaled Deformation Theorem | p. 217 |
Applications of the Deformation Theorem | p. 222 |
Currents and Calculus of Variations | p. 225 |
Proof of the Compactness Theorem | p. 225 |
Integer-Multiplicity 0-Currents | p. 226 |
A Rectifiability Criterion for Currents | p. 231 |
MBV Functions | p. 232 |
The Slicing Lemma | p. 237 |
The Density Lemma | p. 238 |
Completion of the Proof of the Compactness Theorem | p. 240 |
The Flat Metric | p. 241 |
Existence of Currents Minimizing Variational Integrals | p. 244 |
Minimizing Mass | p. 244 |
Other Integrands and Integrals | p. 245 |
Density Estimates for Minimizing Currents | p. 250 |
Regularity of Mass-Minimizing Currents | p. 255 |
Preliminaries | p. 256 |
The Height Bound and Lipschitz Approximation | p. 262 |
Currents Defined by Integrating over Graphs | p. 269 |
Estimates for Harmonic Functions | p. 272 |
The Main Estimate | p. 286 |
The Regularity Theorem | p. 303 |
Epilogue | p. 308 |
Appendix | p. 311 |
Transfinite Induction | p. 311 |
Dual Spaces | p. 313 |
Line Integrals | p. 316 |
Pullbacks and Exterior Derivatives | p. 319 |
References | p. 323 |
Index of Notation | p. 329 |
Index | p. 335 |
Table of Contents provided by Ingram. All Rights Reserved. |
The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.
The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.