分析方法 英文PDF电子书下载

1 Preliminaries 1

1.1 The Logic of Quantifiers 1

1.1.1 Rules of Quantifiers 1

1.1.2 Examples 4

1.1.3 Exercises 7

1.2 Infinite Sets 8

1.2.1 Countable Sets 8

1.2.2 Uncountable Sets 10

1.2.3 Exercises 13

1.3 Proofs 13

1.3.1 How to Discover Proofs 13

1.3.2 How to Understand Proofs 17

1.4 The Rational Number System 18

1.5 The Axiom of Choice 21

2 Construction of the Real Number System 25

2.1 Cauchy Sequences 25

2.1.1 Motivation 25

2.1.2 The Definition 30

2.1.3 Exercises 37

2.2 The Reals as an Ordered Field 38

2.2.1 Defining Arithmetic 38

2.2.2 The Field Axioms 41

2.2.3 Order 45

2.2.4 Exercises 48

2.3 Limits and Completeness 50

2.3.1 Proof of Completeness 50

2.3.2 Square Roots 52

2.3.3 Exercises 54

2.4 Other Versions and Visions 56

2.4.1 Infinite Decimal Expansions 56

2.4.2 Dedekind Cuts 59

2.4.3 Non-Standard Analysis 63

2.4.4 Constructive Analysis 66

2.4.5 Exercises 68

2.5 Summary 69

3 Topology of the Real Line 73

3.1 The Theory of Limits 73

3.1.1 Limits,Sups,and Infs 73

3.1.2 Limit Points 78

3.1.3 Exercises 84

3.2 Open Sets and Closed Sets 86

3.2.1 Open Sets 86

3.2.2 Closed Sets 91

3.2.3 Exercises 98

3.3 Compact Sets 99

3.3.1 Exercises 106

3.4 Summary 107

4 Continuous Functions 111

4.1 Concepts of Continuity 111

4.1.1 Definitions 111

4.1.2 Limits of Functions and Limits of Sequences 119

4.1.3 Inverse Images of Open Sets 121

4.1.4 Related Definitions 123

4.1.5 Exercises 125

4.2 Properties of Continuous Functions 127

4.2.1 Basic Properties 127

4.2.2 Continuous Functions on Compact Domains 131

4.2.3 Monotone Functions 134

4.2.4 Exercises 138

4.3 Summary 140

5 Differential Calculus 143

5.1 Concepts of the Derivative 143

5.1.1 Equivalent Definitions 143

5.1.2 Continuity and Continuous Differentiability 148

5.1.3 Exercises 152

5.2 Properties of the Derivative 153

5.2.1 Local Properties 153

5.2.2 Intermediate Value and Mean Value Theorems 157

5.2.3 Global Properties 162

5.2.4 Exercises 163

5.3 The Calculus of Derivatives 165

5.3.1 Product and Quotient Rules 165

5.3.2 The Chain Rule 168

5.3.3 Inverse Function Theorem 171

5.3.4 Exercises 176

5.4 Higher Derivatives and Taylor's Theorem 177

5.4.1 Interpretations of the Second Derivative 177

5.4.2 Taylor's Theorem 181

5.4.3 L'H？pital's Rule 185

5.4.4 Lagrange Remainder Formula 188

5.4.5 Orders of Zeros 190

5.4.6 Exercises 192

5.5 Summary 195

6 Integral Calculus 201

6.1 Integrals of Continuous Functions 201

6.1.1 Existence of the Integral 201

6.1.2 Fundamental Theorems of Calculus 207

6.1.3 Useful Integration Formulas 212

6.1.4 Numerical Integration 214

6.1.5 Exercises 217

6.2 The Riemann Integral 219

6.2.1 Definition of the Integral 219

6.2.2 Elementary Properties of the Integral 224

6.2.3 Functions with a Countable Number of Discon-tinuities 227

6.2.4 Exercises 231

6.3 Improper Integrals 232

6.3.1 Definitions and Examples 232

6.3.2 Exercises 235

6.4 Summary 236

7 Sequences and Series of Functions 241

7.1 Complex Numbers 241

7.1.1 Basic Properties of C 241

7.1.2 Complex-Valued Functions 247

7.1.3 Exercises 249

7.2 Numerical Series and Sequences 250

7.2.1 Convergence and Absolute Convergence 250

7.2.2 Rearrangements 256

7.2.3 Summation by Parts 260

7.2.4 Exercises 262

7.3 Uniform Convergence 263

7.3.1 Uniform Limits and Continuity 263

7.3.2 Integration and Differentiation of Limits 268

7.3.3 Unrestricted Convergence 272

7.3.4 Exercises 274

7.4 Power Series 276

7.4.1 The Radius of Convergence 276

7.4.2 Analytic Continuation 281

7.4.3 Analytic Functions on Complex Domains 286

7.4.4 Closure Properties of Analytic Functions 288

7.4.5 Exercises 294

7.5 Approximation by Polynomials 296

7.5.1 Lagrange Interpolation 296

7.5.2 Convolutions and Approximate Identities 297

7.5.3 The Weierstrass Approximation Theorem 301

7.5.4 Approximating Derivatives 305

7.5.5 Exercises 307

7.6 Equicontinuity 309

7.6.1 The Definition of Equicontinuity 309

7.6.2 The Arzela-Ascoli Theorem 312

7.6.3 Exercises 314

7.7 Summary 316

8 Transcendental Functions 323

8.1 The Exponential and Logarithm 323

8.1.1 Five Equivalent Definitions 323

8.1.2 Exponential Glue and Blip Functions 329

8.1.3 Functions with Prescribed Taylor Expansions 332

8.1.4 Exercises 335

8.2 Trigonometric Functions 337

8.2.1 Definition of Sine and Cosine 337

8.2.2 Relationship Between Sines,Cosines,and Com-plex Exponentials 344

8.2.3 Exercises 349

8.3 Summary 350

9 Euclidean Space and Metric Spaces 355

9.1 Structures on Euclidean Space 355

9.1.1 Vector Space and Metric Space 355

9.1.2 Norm and Inner Product 358

9.1.3 The Complex Case 364

9.1.4 Exercises 366

9.2 Topology of Metric Spaces 368

9.2.1 Open Sets 368

9.2.2 Limits and Closed Sets 373

9.2.3 Completeness 374

9.2.4 Compactness 377

9.2.5 Exercises 384

9.3 Continuous Functions on Metric Spaces 386

9.3.1 Three Equivalent Definitions 386

9.3.2 Continuous Functions on Compact Domains 391

9.3.3 Connectedness 393

9.3.4 The Contractive Mapping Principle 397

9.3.5 The Stone-Weierstrass Theorem 399

9.3.6 Nowhere Differentiable Functions,and Worse 403

9.3.7 Exercises 409

9.4 Summary 412

10 Differential Calculus in Euclidean Space 419

10.1 The Differehtial 419

10.1.1 Definition of Differentiability 419

10.1.2 Partial Derivatives 423

10.1.3 The Chain Rule 428

10.1.4 Differentiation of Integrals 432

10.1.5 Exercises 435

10.2 Higher Derivatives 437

10.2.1 Equality of Mixed Partials 437

10.2.2 Local Extrema 441

10.2.3 Taylor Expansions 448

10.2.4 Exercises 452

10. 3 Summary 454

11 Ordinary Differential Equations 459

11.1 Existence and Uniqueness 459

11.1.1 Motivation 459

11.1.2 Picard Iteration 467

11.1.3 Linear Equations 473

11.1.4 Local Existence and Uniqueness 476

11.1.5 Higher Order Equations 481

11.1.6 Exercises 483

11.2 Other Methods of Solution 485

11.2.1 Difference Equation Approximation 485

11.2.2 Peano Existence Theorem 490

11.2.3 Power-Series Solutions 494

11.2.4 Exercises 500

11.3 Vector Fields and Flows 501

11.3.1 Integral Curves 501

11.3.2 Hamiltonian Mechanics 505

11.3.3 First-Order Linear P.D.E.'s 506

11.3.4 Exercises 507

11.4 Summary 509

12 Fourier Series 515

12.1 Origins of Fourier Series 515

12.1.1 Fourier Series Solutions of P.D.E.'s 515

12.1.2 Spectral Theory 520

12.1.3 Harmonic Analysis 525

12.1.4 Exercises 528

12.2 Convergence of Fourier Series 531

12.2.1 Uniform Convergence for C1 Functions 531

12.2.2 Summability of Fourier Series 537

12.2.3 Convergence in the Mean 543

12.2.4 Divergence and Gibb's Phenomenon 550

12.2.5 Solution of the Heat Equation 555

12.2.6 Exercises 559

12.3 Summary 562

13 Implicit Functions,Curves,and Surfaces 567

13.1 The Implicit Function Theorem 567

13.1.1 Statement of the Theorem 567

13.1.2 The Proof 573

13.1.3 Exercises 580

13.2 Curves and Surfaces 581

13.2.1 Motivation and Examples 581

13.2.2 Immersions and Embeddings 585

13.2.3 Parametric Description of Surfaces 591

13.2.4 Implicit Description of Surfaces 597

13.2.5 Exercises 600

13.3 Maxima and Minima on Surfaces 602

13.3.1 Lagrange Multipliers 602

13.3.2 A Second Derivative Test 605

13.3.3 Exercises 609

13.4 Arc Length 610

13.4.1 Rectifiable Curves 610

13.4.2 The Integral Formula for Arc Length 614

13.4.3 Arc Length Parameterization 616

13.4.4 Exercises 617

13.5 Summary 618

14 The Lebesgue Integral 623

14.1 The Concept of Measure 623

14.1.1 Motivation 623

14.1.2 Properties of Length 627

14.1.3 Measurable Sets 631

14.1.4 Basic Properties of Measures 634

14.1.5 A Formula for Lebesgue Measure 636

14.1.6 Other Examples of Measures 639

14.1.7 Exercises 641

14.2 Proof of Existence of Measures 643

14.2.1 Outer Measures 643

14.2.2 Metric Outer Measure 647

14.2.3 Hausdorff Measures 650

14.2.4 Exercises 654

14.3 The Integral 655

14.3.1 Non-negative Measurable Functions 655

14.3.2 The Monotone Convergence Theorem 660

14.3.3 Integrable Functions 664

14.3.4 Almost Everywhere 667

14.3.5 Exercises 668

14.4 The Lebesgue Spaces L1 and L2 670

14.4.1 L1 as a Banach Space 670

14.4.2 L2 as a Hilbert Space 673

14.4.3 Fourier Series for L2 Functions 676

14.4.4 Exercises 681

14.5 Summary 682

15 Multiple Integrals 691

15.1 Interchange of Integrals 691

15.1.1 Integrals of Continuous Functions 691

15.1.2 Fubini's Theorem 694

15.1.3 The Monotone Class Lemma 700

15.1.4 Exercises 703

15.2 Change of Variable in Multiple Integrals 705

15.2.1 Determinants and Volume 705

15.2.2 The Jacobian Factor 709

15.2.3 Polar Coordinates 714

15.2.4 Change of Variable for Lebesgue Integrals 717

15.2.5 Exercises 720

15.3 Summary 722

Index 727