几何三部曲 第2卷 几何的代数方法 英文版PDF|Epub|txt|kindle电子书版本网盘下载

几何三部曲 第2卷 几何的代数方法 英文版PDF格式电子书版下载

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1 The Birth of Analytic Geometry1

1.1 Fermat's Analytic Geometry2

1.2 Descartes'Analytic Geometry5

1.3 Moreon Cartesian Systems of Coordinates6

1.4 Non-Cartesian Systems of Coordinates9

1.5 Computing Distances and Angles11

1.6 Planes and Lines in Solid Geometry15

1.7 The Cross Product17

1.8 Forgetting the Origin19

1.9 The Tangent to a Curve24

1.10 The Conics27

1.11 The Ellipse29

1.12 The Hyperbola31

1.13 The Parabola34

1.14 The Quadrics37

1.15 The Ruled Quadrics43

1.16 Problems47

1.17 Exercises49

2 Affine Geometry51

2.1 Affine Spaces over a Field52

2.2 Examples of Affine Spaces55

2.3 Affine Subspaces56

2.4 Parallel Subspaces58

2.5 Generated Subspaces59

2.6 Supplementary Subspaces60

2.7 Lines and Planes61

2.8 Barycenters63

2.9 Barycentric Coordinates65

2.10 Triangles66

2.11 Parallelograms70

2.12 Affine Transformations73

2.13 Affine Isomorphisms75

2.14 Transiations78

2.15 Projections79

2.16 Symmetries80

2.17 Homotheties and Affinities83

2.18 The Intercept Thales Theorem84

2.19 Affine Coordinates86

2.20 Change of Coordinates87

2.21 The Equations of a Subspace88

2.22 The Matrix of an Affine Transformation89

2.23 The Quadrics91

2.24 The Reduced Equation of a Quadric93

2.25 The Symmetries of a Quadric96

2.26 The Equation of a Non-degenerate Quadric100

2.27 Problems108

2.28 Exercises110

3 More on Real Affine Spaces119

3.1 About Left,Right and Between119

3.2 Orientation of a Real Affine Space121

3.3 Direct and Inverse Affine Isomorphisms125

3.4 Parallelepipeds and Half Spaces125

3.5 Pasch's Theorem128

3.6 Affine Classification of Real Quadrics129

3.7 Problems134

3.8 Exercises135

4 Euclidean Geometry137

4.1 Metric Geometry137

4.2 Defining Lengths and Angles138

4.3 Metric Properties of Euclidean Spaces140

4.4 Rectangles,Diamonds and Squares144

4.5 Examples of Euclidean Spaces146

4.6 Orthonormal Bases149

4.7 Polar Coordinates152

4.8 Orthogonal Projections154

4.9 Some Approximation Problems156

4.10 Isometries161

4.11 Classification of Isometries163

4.12 Rotations165

4.13 Similarities170

4.14 Euclidean Quadrics173

4.15 Problems174

4.16 Exercises176

5 Hermitian Spaces181

5.1 Hermitian Products181

5.2 Orthonormal Bases184

5.3 The Metric Structure of Hermitian Spaces187

5.4 Complex Quadrics189

5.5 Problems192

5.6 Exercises193

6 Projective Geometry195

6.1 Projective Spaces over a Field195

6.2 Projective Subspaces198

6.3 The Duality Principle200

6.4 Homogeneous Coordinates202

6.5 Projective Basis205

6.6 The Anharmonic Ratio207

6.7 Projective Transformations209

6.8 Desargues'Theorem215

6.9 Pappus'Theorem219

6.10 Fano's Theorem223

6.11 Harmonic Quadruples225

6.12 The Axioms of Projective Geometry226

6.13 Projective Quadrics227

6.14 Duality with Respect to a Quadric231

6.15 Poles and Polar Hyperplanes232

6.16 Tangent Space to a Quadric235

6.17 Projective Conics236

6.18 The Anharmonic Ratio Along a Conic242

6.19 The Pascal and Brianchon Theorems246

6.20 Affine Versus Projective250

6.21 Real Quadrics256

6.22 The Topology of Projective Real Spaces261

6.23 Problems263

6.24 Exercises264

7 Algebraic Curves267

7.1 Looking for the Right Context268

7.2 The Equation of an Algebraic Curve270

7.3 The Degree of a Curve273

7.4 Tangents and Multiple Points276

7.5 Examples of Singularities283

7.6 Inflexion Points287

7.7 The Bezout Theorem292

7.8 Curves Through Points303

7.9 The Number of Multiplicities307

7.10 Conics310

7.11 Cubics and the Cramer Paradox311

7.12 Inflexion Points of a Cubic316

7.13 The Group of a Cubic322

7.14 Rational Curves326

7.15 A Criterion of Rationality331

7.16 Problems337

7.17 Exercises339

Appendix A Polynomials over a Field341

A.1 Polynomials Versus Polynomial Functions341

A.2 Euclidean Division342

A.3 The Bezout Theorem344

A.4 Irreducible Polynomials346

A.5 The Greatest Common Divisor347

A.6 Roots of a Polynomial349

A.7 Adding Roots to a Polynomial351

A.8 The Derivative of a Polynomial354

Appendix B Polynomials in Several Variables359

B.1 Roots359

B.2 Polynomial Domains362

B.3 Quotient Field364

B.4 Irreducible Polynomials366

B.5 Partial Derivatives370

Appendix C Homogeneous Polynomials373

C.1 Basic Properties373

C.2 Homogeneous Versus Non-homogeneous376

Appendix D Resultants379

D.1 The Resultant of two Polynomials379

D.2 Roots Versus Divisibility384

D.3 The Resultant of Homogeneous Polynomials387

Appendix E Symmetric Polynomials391

E.1 Elementary Symmetric Polynomials391

E.2 The Structural Theorem392

Appendix F Complex Numbers397

F.1 The Field of Complex Numbers397

F.2 Modulus Argument and Exponential398

F.3 The Fundamental Theorem of Algebra401

F.4 More on Complex and Real Polynomials404

Appendix G Quadratic Forms407

G.1 Quadratic Forms over a Field407

G.2 Conjugation and Isotropy409

G.3 Real Quadratic Forms411

G.4 Quadratic Forms on Euclidean Spaces414

G.5 On Complex Quadratic Forms415

Appendix H Dual Spaces417

H.1 The Dual of a Vector Space417

H.2 Mixed Orthogonality420

References and Further Reading423

Index425