几何三部曲 第2卷 几何的代数方法 英文版PDF电子书下载

1 The Birth of Analytic Geometry 1

1.1 Fermat's Analytic Geometry 2

1.2 Descartes'Analytic Geometry 5

1.3 Moreon Cartesian Systems of Coordinates 6

1.4 Non-Cartesian Systems of Coordinates 9

1.5 Computing Distances and Angles 11

1.6 Planes and Lines in Solid Geometry 15

1.7 The Cross Product 17

1.8 Forgetting the Origin 19

1.9 The Tangent to a Curve 24

1.10 The Conics 27

1.11 The Ellipse 29

1.12 The Hyperbola 31

1.13 The Parabola 34

1.14 The Quadrics 37

1.15 The Ruled Quadrics 43

1.16 Problems 47

1.17 Exercises 49

2 Affine Geometry 51

2.1 Affine Spaces over a Field 52

2.2 Examples of Affine Spaces 55

2.3 Affine Subspaces 56

2.4 Parallel Subspaces 58

2.5 Generated Subspaces 59

2.6 Supplementary Subspaces 60

2.7 Lines and Planes 61

2.8 Barycenters 63

2.9 Barycentric Coordinates 65

2.10 Triangles 66

2.11 Parallelograms 70

2.12 Affine Transformations 73

2.13 Affine Isomorphisms 75

2.14 Transiations 78

2.15 Projections 79

2.16 Symmetries 80

2.17 Homotheties and Affinities 83

2.18 The Intercept Thales Theorem 84

2.19 Affine Coordinates 86

2.20 Change of Coordinates 87

2.21 The Equations of a Subspace 88

2.22 The Matrix of an Affine Transformation 89

2.23 The Quadrics 91

2.24 The Reduced Equation of a Quadric 93

2.25 The Symmetries of a Quadric 96

2.26 The Equation of a Non-degenerate Quadric 100

2.27 Problems 108

2.28 Exercises 110

3 More on Real Affine Spaces 119

3.1 About Left,Right and Between 119

3.2 Orientation of a Real Affine Space 121

3.3 Direct and Inverse Affine Isomorphisms 125

3.4 Parallelepipeds and Half Spaces 125

3.5 Pasch's Theorem 128

3.6 Affine Classification of Real Quadrics 129

3.7 Problems 134

3.8 Exercises 135

4 Euclidean Geometry 137

4.1 Metric Geometry 137

4.2 Defining Lengths and Angles 138

4.3 Metric Properties of Euclidean Spaces 140

4.4 Rectangles,Diamonds and Squares 144

4.5 Examples of Euclidean Spaces 146

4.6 Orthonormal Bases 149

4.7 Polar Coordinates 152

4.8 Orthogonal Projections 154

4.9 Some Approximation Problems 156

4.10 Isometries 161

4.11 Classification of Isometries 163

4.12 Rotations 165

4.13 Similarities 170

4.14 Euclidean Quadrics 173

4.15 Problems 174

4.16 Exercises 176

5 Hermitian Spaces 181

5.1 Hermitian Products 181

5.2 Orthonormal Bases 184

5.3 The Metric Structure of Hermitian Spaces 187

5.4 Complex Quadrics 189

5.5 Problems 192

5.6 Exercises 193

6 Projective Geometry 195

6.1 Projective Spaces over a Field 195

6.2 Projective Subspaces 198

6.3 The Duality Principle 200

6.4 Homogeneous Coordinates 202

6.5 Projective Basis 205

6.6 The Anharmonic Ratio 207

6.7 Projective Transformations 209

6.8 Desargues'Theorem 215

6.9 Pappus'Theorem 219

6.10 Fano's Theorem 223

6.11 Harmonic Quadruples 225

6.12 The Axioms of Projective Geometry 226

6.13 Projective Quadrics 227

6.14 Duality with Respect to a Quadric 231

6.15 Poles and Polar Hyperplanes 232

6.16 Tangent Space to a Quadric 235

6.17 Projective Conics 236

6.18 The Anharmonic Ratio Along a Conic 242

6.19 The Pascal and Brianchon Theorems 246

6.20 Affine Versus Projective 250

6.21 Real Quadrics 256

6.22 The Topology of Projective Real Spaces 261

6.23 Problems 263

6.24 Exercises 264

7 Algebraic Curves 267

7.1 Looking for the Right Context 268

7.2 The Equation of an Algebraic Curve 270

7.3 The Degree of a Curve 273

7.4 Tangents and Multiple Points 276

7.5 Examples of Singularities 283

7.6 Inflexion Points 287

7.7 The Bezout Theorem 292

7.8 Curves Through Points 303

7.9 The Number of Multiplicities 307

7.10 Conics 310

7.11 Cubics and the Cramer Paradox 311

7.12 Inflexion Points of a Cubic 316

7.13 The Group of a Cubic 322

7.14 Rational Curves 326

7.15 A Criterion of Rationality 331

7.16 Problems 337

7.17 Exercises 339

Appendix A Polynomials over a Field 341

A.1 Polynomials Versus Polynomial Functions 341

A.2 Euclidean Division 342

A.3 The Bezout Theorem 344

A.4 Irreducible Polynomials 346

A.5 The Greatest Common Divisor 347

A.6 Roots of a Polynomial 349

A.7 Adding Roots to a Polynomial 351

A.8 The Derivative of a Polynomial 354

Appendix B Polynomials in Several Variables 359

B.1 Roots 359

B.2 Polynomial Domains 362

B.3 Quotient Field 364

B.4 Irreducible Polynomials 366

B.5 Partial Derivatives 370

Appendix C Homogeneous Polynomials 373

C.1 Basic Properties 373

C.2 Homogeneous Versus Non-homogeneous 376

Appendix D Resultants 379

D.1 The Resultant of two Polynomials 379

D.2 Roots Versus Divisibility 384

D.3 The Resultant of Homogeneous Polynomials 387

Appendix E Symmetric Polynomials 391

E.1 Elementary Symmetric Polynomials 391

E.2 The Structural Theorem 392

Appendix F Complex Numbers 397

F.1 The Field of Complex Numbers 397

F.2 Modulus Argument and Exponential 398

F.3 The Fundamental Theorem of Algebra 401

F.4 More on Complex and Real Polynomials 404

Appendix G Quadratic Forms 407

G.1 Quadratic Forms over a Field 407

G.2 Conjugation and Isotropy 409

G.3 Real Quadratic Forms 411

G.4 Quadratic Forms on Euclidean Spaces 414

G.5 On Complex Quadratic Forms 415

Appendix H Dual Spaces 417

H.1 The Dual of a Vector Space 417

H.2 Mixed Orthogonality 420

References and Further Reading 423

Index 425