几何三部曲 第3卷 几何的微分方法 英文PDF|Epub|txt|kindle电子书版本下载

几何三部曲 第3卷 几何的微分方法 英文PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

1 The Genesis of Differential Methods1

1.1 The Static Approach to Curves2

1.2 The Dynamic Approach to Curves4

1.3 Cartesian Versus Parametric10

1.4 Singularities and Multiplicities15

1.5 Chasing the Tangents19

1.6 Tangent：The Differential Approach24

1.7 Rectification of a Curve27

1.8 Length Versus Curve Integral31

1.9 Clocks,Cycloids and Envelopes33

1.10 Radius of Curvature and Evolute38

1.11 Curvature and Normality40

1.12 Curve Squaring42

1.13 Skew Curves46

1.14 Problems51

1.15 Exercises52

2 Plane Curves55

2.1 Parametric Representations55

2.2 Regular Representations61

2.3 The Cartesian Equation of a Curve63

2.4 Tangents67

2.5 Asymptotes69

2.6 Envelopes72

2.7 The Length of an Arc of a Curve82

2.8 Normal Representation86

2.9 Curvature88

2.10 Osculating Circle94

2.11 Evolutes and Involutes96

2.12 Intrinsic Equation of a Plane Curve100

2.13 Closed Curves105

2.14 Piecewise Regular Curves110

2.15 Simple Closed Curves114

2.16 Convex Curves126

2.17 Vertices of a Plane Curve129

2.18 Problems133

2.19 Exercises134

3 A Museum of Curves139

3.1 Some Terminology139

3.2 The Circle142

3.3 The Ellipse143

3.4 The Hyperbola144

3.5 The Parabola145

3.6 The Cycloid146

3.7 The Cardioid146

3.8 The Nephroid148

3.9 The Astroid148

3.10 The Deltoid149

3.11 The Lima？on of Pascal150

3.12 The Lemniscate of Bernoulli151

3.13 The Conchoid of Nicomedes152

3.14 The Cissoid of Diocles153

3.15 The Right Strophoid154

3.16 The Tractrix155

3.17 The Catenary156

3.18 The Spiral of Archimedes157

3.19 The Logarithmic Spiral157

3.20 The Spiral of Cornu158

4 Skew Curves161

4.1 Regular Skew Curves161

4.2 Normal Representations164

4.3 Curvature166

4.4 The Frenet Trihedron168

4.5 Torsion171

4.6 Intrinsic Equations174

4.7 Problems178

4.8 Exercises179

5 The Local Theory of Surfaces181

5.1 Parametric Representation of a Surface182

5.2 Regular Surfaces190

5.3 Cartesian Equation193

5.4 Curves on a Surface194

5.5 The Tangent Plane198

5.6 Tangent Vector Fields202

5.7 Orientation of a Surface206

5.8 Normal Curvature208

5.9 Umbilical Points215

5.10 Pfincipal Directions220

5.11 The Case of Quadrics227

5.12 Approximation by a Quadric230

5.13 The Rodrigues Formula233

5.14 Lines of Curvature235

5.15 Gauss'Approach to Total Cuivature236

5.16 Gaussian Curvature240

5.17 Problems245

5.18 Exercises249

6 Towards Riemannian Geometry253

6.1 What Is Riemannian Geometry？254

6.2 The Metric Tensor258

6.3 Curves on a Riemann Patch261

6.4 Vector Fields Along a Curve263

6.5 The Normal Vector Field to a Curve265

6.6 The Christoffel Symbols267

6.7 Covariant Derivative271

6.8 Parallel Transport276

6.9 Geodesic Cuivature279

6.10 Geodesics282

6.11 The Riemann Tensor286

6.12 What Is a Tensor？289

6.13 Systems of Geodesic Coordinates295

6.14 Cuivature in Geodesic Coordinates303

6.15 The Poincaré Half Plane310

6.16 Embeddable Riemann Patches322

6.17 What Is a Riemann Surface？333

6.18 Problems339

6.19 Exercises341

7 Elements of the Global Theory of Surfaces345

7.1 Surfaces of Revolution345

7.2 Ruled Surfaces354

7.3 Applicability of Surfaces363

7.4 Surfaces with Zero Curvature368

7.5 Developable Surfaces372

7.6 Classification of Developable Surfaces374

7.7 Surfaces with Constant Curvature381

7.8 The Sphere384

7.9 A Counterexample390

7.10 Rotation Numbers392

7.11 Polygonal Domains396

7.12 Polygonal Decompositions401

7.13 The Gauss-Bonnet Theorem405

7.14 Geodesic Triangles410

7.15 The Euler-Poincaré Characteristic411

7.16 Problems415

7.17 Exercises416

Appendix A Topology419

A.1 Open Subsets in Real Spaces419

A.2 Closed Subsets in Real Spaces421

A.3 Compact Subsets in Real Spaces421

A.4 Continuous Mappings of Real Spaces424

A.5 Topological Spaces425

A.6 Closure and Density427

A.7 Compactness428

A.8 Continuous Mappings429

A.9 Homeomorphisms430

A.10 Connectedness432

Appendix B Differential Equations439

B.1 First Order Differential Equations439

B.2 Second Order Differential Equations440

B.3 Variable Initial Conditions441

B.4 Systems of Partial Differential Equations443

References and Further Reading445

Index447