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1 The Genesis of Differential Methods 1

1.1 The Static Approach to Curves 2

1.2 The Dynamic Approach to Curves 4

1.3 Cartesian Versus Parametric 10

1.4 Singularities and Multiplicities 15

1.5 Chasing the Tangents 19

1.6 Tangent：The Differential Approach 24

1.7 Rectification of a Curve 27

1.8 Length Versus Curve Integral 31

1.9 Clocks,Cycloids and Envelopes 33

1.10 Radius of Curvature and Evolute 38

1.11 Curvature and Normality 40

1.12 Curve Squaring 42

1.13 Skew Curves 46

1.14 Problems 51

1.15 Exercises 52

2 Plane Curves 55

2.1 Parametric Representations 55

2.2 Regular Representations 61

2.3 The Cartesian Equation of a Curve 63

2.4 Tangents 67

2.5 Asymptotes 69

2.6 Envelopes 72

2.7 The Length of an Arc of a Curve 82

2.8 Normal Representation 86

2.9 Curvature 88

2.10 Osculating Circle 94

2.11 Evolutes and Involutes 96

2.12 Intrinsic Equation of a Plane Curve 100

2.13 Closed Curves 105

2.14 Piecewise Regular Curves 110

2.15 Simple Closed Curves 114

2.16 Convex Curves 126

2.17 Vertices of a Plane Curve 129

2.18 Problems 133

2.19 Exercises 134

3 A Museum of Curves 139

3.1 Some Terminology 139

3.2 The Circle 142

3.3 The Ellipse 143

3.4 The Hyperbola 144

3.5 The Parabola 145

3.6 The Cycloid 146

3.7 The Cardioid 146

3.8 The Nephroid 148

3.9 The Astroid 148

3.10 The Deltoid 149

3.11 The Lima？on of Pascal 150

3.12 The Lemniscate of Bernoulli 151

3.13 The Conchoid of Nicomedes 152

3.14 The Cissoid of Diocles 153

3.15 The Right Strophoid 154

3.16 The Tractrix 155

3.17 The Catenary 156

3.18 The Spiral of Archimedes 157

3.19 The Logarithmic Spiral 157

3.20 The Spiral of Cornu 158

4 Skew Curves 161

4.1 Regular Skew Curves 161

4.2 Normal Representations 164

4.3 Curvature 166

4.4 The Frenet Trihedron 168

4.5 Torsion 171

4.6 Intrinsic Equations 174

4.7 Problems 178

4.8 Exercises 179

5 The Local Theory of Surfaces 181

5.1 Parametric Representation of a Surface 182

5.2 Regular Surfaces 190

5.3 Cartesian Equation 193

5.4 Curves on a Surface 194

5.5 The Tangent Plane 198

5.6 Tangent Vector Fields 202

5.7 Orientation of a Surface 206

5.8 Normal Curvature 208

5.9 Umbilical Points 215

5.10 Pfincipal Directions 220

5.11 The Case of Quadrics 227

5.12 Approximation by a Quadric 230

5.13 The Rodrigues Formula 233

5.14 Lines of Curvature 235

5.15 Gauss'Approach to Total Cuivature 236

5.16 Gaussian Curvature 240

5.17 Problems 245

5.18 Exercises 249

6 Towards Riemannian Geometry 253

6.1 What Is Riemannian Geometry？ 254

6.2 The Metric Tensor 258

6.3 Curves on a Riemann Patch 261

6.4 Vector Fields Along a Curve 263

6.5 The Normal Vector Field to a Curve 265

6.6 The Christoffel Symbols 267

6.7 Covariant Derivative 271

6.8 Parallel Transport 276

6.9 Geodesic Cuivature 279

6.10 Geodesics 282

6.11 The Riemann Tensor 286

6.12 What Is a Tensor？ 289

6.13 Systems of Geodesic Coordinates 295

6.14 Cuivature in Geodesic Coordinates 303

6.15 The Poincaré Half Plane 310

6.16 Embeddable Riemann Patches 322

6.17 What Is a Riemann Surface？ 333

6.18 Problems 339

6.19 Exercises 341

7 Elements of the Global Theory of Surfaces 345

7.1 Surfaces of Revolution 345

7.2 Ruled Surfaces 354

7.3 Applicability of Surfaces 363

7.4 Surfaces with Zero Curvature 368

7.5 Developable Surfaces 372

7.6 Classification of Developable Surfaces 374

7.7 Surfaces with Constant Curvature 381

7.8 The Sphere 384

7.9 A Counterexample 390

7.10 Rotation Numbers 392

7.11 Polygonal Domains 396

7.12 Polygonal Decompositions 401

7.13 The Gauss-Bonnet Theorem 405

7.14 Geodesic Triangles 410

7.15 The Euler-Poincaré Characteristic 411

7.16 Problems 415

7.17 Exercises 416

Appendix A Topology 419

A.1 Open Subsets in Real Spaces 419

A.2 Closed Subsets in Real Spaces 421

A.3 Compact Subsets in Real Spaces 421

A.4 Continuous Mappings of Real Spaces 424

A.5 Topological Spaces 425

A.6 Closure and Density 427

A.7 Compactness 428

A.8 Continuous Mappings 429

A.9 Homeomorphisms 430

A.10 Connectedness 432

Appendix B Differential Equations 439

B.1 First Order Differential Equations 439

B.2 Second Order Differential Equations 440

B.3 Variable Initial Conditions 441

B.4 Systems of Partial Differential Equations 443

References and Further Reading 445

Index 447