هندسه دیفرانسیل منحنی‌های صفحه

This book features plane curves―the simplest objects in differential geometry―to illustrate many deep and inspiring results in the field in an elementary and accessible way. After an introduction to the basic properties of plane curves, the authors introduce a number of complex and beautiful topics, including the rotation number (with a proof of the fundamental theorem of algebra), rotation index, Jordan curve theorem, isoperimetric inequality, convex curves, curves of constant width, and the four-vertex theorem. The last chapter connects the classical with the modern by giving an introduction to the curve-shortening flow that is based on original articles but requires a minimum of previous knowledge. Over 200 figures and more than 100 exercises illustrate the beauty of plane curves and test the reader's skills. Prerequisites are courses in standard one variable calculus and analytic geometry on the plane. Cover 1 Contents 8 Foreword 12 Preface 14 Background information 15 Structure of the book 15 Acknowledgments 16 Chapter 1. Plane Curves 18 1.1. Continuous curves 20 1.2. Smooth curves, tangent vectors, and tangent lines 33 1.3. Reparametrization and arc length 39 1.4. Normal and tangent vector fields 45 1.5. Curvature and Frenet equations 50 1.6. Geometric interpretation of the curvature 58 1.7. Curves in the complex plane 63 1.8. The fundamental theorem of plane curves 69 1.9. Local canonical form 72 1.10. Parallel curves 75 1.11. Evolutes and involutes 82 1.12. Exercises 90 Chapter 2. Winding Number 114 2.1. The angle function 114 2.2. Winding number of a closed curve 123 2.3. Properties of the winding number 125 2.4. Winding number of deformable curves 134 2.5. Calculation of winding and intersection numbers 139 2.6. Applications 149 2.7. Exercises 153 Chapter 3. Rotation Index 160 3.1. Rotation index 160 3.2. The total curvature 164 3.3. Rotation indices of simple closed curves 167 3.4. The total absolute curvature 172 3.5. Exercises 177 Chapter 4. Jordan Curve Theorem 182 4.1. Jordan curve theorem 183 4.2. Exercises 196 Chapter 5. Isoperimetric Inequality 198 5.1. The classical isoperimetric inequality 198 5.2. Bonnesen isoperimetric inequality 207 5.3. Exercises 208 Chapter 6. Convex Curves 212 6.1. Closed and convex curves 214 6.2. Schur theorem 231 6.3. Curves of constant width 235 6.4. Length and area of convex curves 247 6.5. Gage isoperimetric inequality 257 6.6. Exercises 260 Chapter 7. The Four-Vertex Theorem 268 7.1. The four-vertex theorem for convex curves 271 7.2. A generalization of the four-vertex theorem 277 7.3. The converse of the four-vertex theorem 281 7.4. Exercises 283 Chapter 8. Curve-Shortening Flow 286 8.1. Introduction and basic properties of the flow 286 8.2. Convex curves under the curve-shortening flow 301 8.3. The theorems of Gage and Hamilton 333 8.4. Convergence of the curvature functions of convex curves 344 8.5. Evolution of simple curves: Grayson theorem 352 8.6. Exercises 376 Appendix A. The Class C^{∞} Convergence of the Curvature Function Under the Curve-Shortening Flow 380 Appendix B. Answers to Selected Exercises 402 B.1. Chapter 1 - Page 73 402 B.2. Chapter 2 - Page 136 411 B.3. Chapter 3 - Page 160 413 B.4. Chapter 4 - Page 179 414 B.5. Chapter 5 - Page 191 416 B.6. Chapter 6 - Page 243 418 B.7. Chapter 7 - Page 266 421 B.8. Chapter 8 - Page 359 422 Bibliography 426 Index 430 Back Cover 435 Features plane curves to illustrate many deep and inspiring results in the field in an elementary and accessible way. After an introduction to the basic properties of plane curves, the authors introduce a number of complex and beautiful topics, including the rotation number, rotation index, Jordan curve theorem, and isoperimetric inequality.