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1 Preliminaries1

1.1 The Logic of Quantifiers1

1.1.1 Rules of Quantifiers1

1.1.2 Examples4

1.1.3 Exercises7

1.2 Infinite Sets8

1.2.1 Countable Sets8

1.2.2 Uncountable Sets10

1.2.3 Exercises13

1.3 Proofs13

1.3.1 How to Discover Proofs13

1.3.2 How to Understand Proofs17

1.4 The Rational Number System18

1.5 The Axiom of Choice21

2 Construction of the Real Number System25

2.1 Cauchy Sequences25

2.1.1 Motivation25

2.1.2 The Definition30

2.1.3 Exercises37

2.2 The Reals as an Ordered Field38

2.2.1 Defining Arithmetic38

2.2.2 The Field Axioms41

2.2.3 Order45

2.2.4 Exercises48

2.3 Limits and Completeness50

2.3.1 Proof of Completeness50

2.3.2 Square Roots52

2.3.3 Exercises54

2.4 Other Versions and Visions56

2.4.1 Infinite Decimal Expansions56

2.4.2 Dedekind Cuts59

2.4.3 Non-Standard Analysis63

2.4.4 Constructive Analysis66

2.4.5 Exercises68

2.5 Summary69

3 Topology of the Real Line73

3.1 The Theory of Limits73

3.1.1 Limits,Sups,and Infs73

3.1.2 Limit Points78

3.1.3 Exercises84

3.2 Open Sets and Closed Sets86

3.2.1 Open Sets86

3.2.2 Closed Sets91

3.2.3 Exercises98

3.3 Compact Sets99

3.3.1 Exercises106

3.4 Summary107

4 Continuous Functions111

4.1 Concepts of Continuity111

4.1.1 Definitions111

4.1.2 Limits of Functions and Limits of Sequences119

4.1.3 Inverse Images of Open Sets121

4.1.4 Related Definitions123

4.1.5 Exercises125

4.2 Properties of Continuous Functions127

4.2.1 Basic Properties127

4.2.2 Continuous Functions on Compact Domains131

4.2.3 Monotone Functions134

4.2.4 Exercises138

4.3 Summary140

5 Differential Calculus143

5.1 Concepts of the Derivative143

5.1.1 Equivalent Definitions143

5.1.2 Continuity and Continuous Differentiability148

5.1.3 Exercises152

5.2 Properties of the Derivative153

5.2.1 Local Properties153

5.2.2 Intermediate Value and Mean Value Theorems157

5.2.3 Global Properties162

5.2.4 Exercises163

5.3 The Calculus of Derivatives165

5.3.1 Product and Quotient Rules165

5.3.2 The Chain Rule168

5.3.3 Inverse Function Theorem171

5.3.4 Exercises176

5.4 Higher Derivatives and Taylor's Theorem177

5.4.1 Interpretations of the Second Derivative177

5.4.2 Taylor's Theorem181

5.4.3 L'H？pital's Rule185

5.4.4 Lagrange Remainder Formula188

5.4.5 Orders of Zeros190

5.4.6 Exercises192

5.5 Summary195

6 Integral Calculus201

6.1 Integrals of Continuous Functions201

6.1.1 Existence of the Integral201

6.1.2 Fundamental Theorems of Calculus207

6.1.3 Useful Integration Formulas212

6.1.4 Numerical Integration214

6.1.5 Exercises217

6.2 The Riemann Integral219

6.2.1 Definition of the Integral219

6.2.2 Elementary Properties of the Integral224

6.2.3 Functions with a Countable Number of Discon-tinuities227

6.2.4 Exercises231

6.3 Improper Integrals232

6.3.1 Definitions and Examples232

6.3.2 Exercises235

6.4 Summary236

7 Sequences and Series of Functions241

7.1 Complex Numbers241

7.1.1 Basic Properties of C241

7.1.2 Complex-Valued Functions247

7.1.3 Exercises249

7.2 Numerical Series and Sequences250

7.2.1 Convergence and Absolute Convergence250

7.2.2 Rearrangements256

7.2.3 Summation by Parts260

7.2.4 Exercises262

7.3 Uniform Convergence263

7.3.1 Uniform Limits and Continuity263

7.3.2 Integration and Differentiation of Limits268

7.3.3 Unrestricted Convergence272

7.3.4 Exercises274

7.4 Power Series276

7.4.1 The Radius of Convergence276

7.4.2 Analytic Continuation281

7.4.3 Analytic Functions on Complex Domains286

7.4.4 Closure Properties of Analytic Functions288

7.4.5 Exercises294

7.5 Approximation by Polynomials296

7.5.1 Lagrange Interpolation296

7.5.2 Convolutions and Approximate Identities297

7.5.3 The Weierstrass Approximation Theorem301

7.5.4 Approximating Derivatives305

7.5.5 Exercises307

7.6 Equicontinuity309

7.6.1 The Definition of Equicontinuity309

7.6.2 The Arzela-Ascoli Theorem312

7.6.3 Exercises314

7.7 Summary316

8 Transcendental Functions323

8.1 The Exponential and Logarithm323

8.1.1 Five Equivalent Definitions323

8.1.2 Exponential Glue and Blip Functions329

8.1.3 Functions with Prescribed Taylor Expansions332

8.1.4 Exercises335

8.2 Trigonometric Functions337

8.2.1 Definition of Sine and Cosine337

8.2.2 Relationship Between Sines,Cosines,and Com-plex Exponentials344

8.2.3 Exercises349

8.3 Summary350

9 Euclidean Space and Metric Spaces355

9.1 Structures on Euclidean Space355

9.1.1 Vector Space and Metric Space355

9.1.2 Norm and Inner Product358

9.1.3 The Complex Case364

9.1.4 Exercises366

9.2 Topology of Metric Spaces368

9.2.1 Open Sets368

9.2.2 Limits and Closed Sets373

9.2.3 Completeness374

9.2.4 Compactness377

9.2.5 Exercises384

9.3 Continuous Functions on Metric Spaces386

9.3.1 Three Equivalent Definitions386

9.3.2 Continuous Functions on Compact Domains391

9.3.3 Connectedness393

9.3.4 The Contractive Mapping Principle397

9.3.5 The Stone-Weierstrass Theorem399

9.3.6 Nowhere Differentiable Functions,and Worse403

9.3.7 Exercises409

9.4 Summary412

10 Differential Calculus in Euclidean Space419

10.1 The Differehtial419

10.1.1 Definition of Differentiability419

10.1.2 Partial Derivatives423

10.1.3 The Chain Rule428

10.1.4 Differentiation of Integrals432

10.1.5 Exercises435

10.2 Higher Derivatives437

10.2.1 Equality of Mixed Partials437

10.2.2 Local Extrema441

10.2.3 Taylor Expansions448

10.2.4 Exercises452

10. 3 Summary454

11 Ordinary Differential Equations459

11.1 Existence and Uniqueness459

11.1.1 Motivation459

11.1.2 Picard Iteration467

11.1.3 Linear Equations473

11.1.4 Local Existence and Uniqueness476

11.1.5 Higher Order Equations481

11.1.6 Exercises483

11.2 Other Methods of Solution485

11.2.1 Difference Equation Approximation485

11.2.2 Peano Existence Theorem490

11.2.3 Power-Series Solutions494

11.2.4 Exercises500

11.3 Vector Fields and Flows501

11.3.1 Integral Curves501

11.3.2 Hamiltonian Mechanics505

11.3.3 First-Order Linear P.D.E.'s506

11.3.4 Exercises507

11.4 Summary509

12 Fourier Series515

12.1 Origins of Fourier Series515

12.1.1 Fourier Series Solutions of P.D.E.'s515

12.1.2 Spectral Theory520

12.1.3 Harmonic Analysis525

12.1.4 Exercises528

12.2 Convergence of Fourier Series531

12.2.1 Uniform Convergence for C1 Functions531

12.2.2 Summability of Fourier Series537

12.2.3 Convergence in the Mean543

12.2.4 Divergence and Gibb's Phenomenon550

12.2.5 Solution of the Heat Equation555

12.2.6 Exercises559

12.3 Summary562

13 Implicit Functions,Curves,and Surfaces567

13.1 The Implicit Function Theorem567

13.1.1 Statement of the Theorem567

13.1.2 The Proof573

13.1.3 Exercises580

13.2 Curves and Surfaces581

13.2.1 Motivation and Examples581

13.2.2 Immersions and Embeddings585

13.2.3 Parametric Description of Surfaces591

13.2.4 Implicit Description of Surfaces597

13.2.5 Exercises600

13.3 Maxima and Minima on Surfaces602

13.3.1 Lagrange Multipliers602

13.3.2 A Second Derivative Test605

13.3.3 Exercises609

13.4 Arc Length610

13.4.1 Rectifiable Curves610

13.4.2 The Integral Formula for Arc Length614

13.4.3 Arc Length Parameterization616

13.4.4 Exercises617

13.5 Summary618

14 The Lebesgue Integral623

14.1 The Concept of Measure623

14.1.1 Motivation623

14.1.2 Properties of Length627

14.1.3 Measurable Sets631

14.1.4 Basic Properties of Measures634

14.1.5 A Formula for Lebesgue Measure636

14.1.6 Other Examples of Measures639

14.1.7 Exercises641

14.2 Proof of Existence of Measures643

14.2.1 Outer Measures643

14.2.2 Metric Outer Measure647

14.2.3 Hausdorff Measures650

14.2.4 Exercises654

14.3 The Integral655

14.3.1 Non-negative Measurable Functions655

14.3.2 The Monotone Convergence Theorem660

14.3.3 Integrable Functions664

14.3.4 Almost Everywhere667

14.3.5 Exercises668

14.4 The Lebesgue Spaces L1 and L2670

14.4.1 L1 as a Banach Space670

14.4.2 L2 as a Hilbert Space673

14.4.3 Fourier Series for L2 Functions676

14.4.4 Exercises681

14.5 Summary682

15 Multiple Integrals691

15.1 Interchange of Integrals691

15.1.1 Integrals of Continuous Functions691

15.1.2 Fubini's Theorem694

15.1.3 The Monotone Class Lemma700

15.1.4 Exercises703

15.2 Change of Variable in Multiple Integrals705

15.2.1 Determinants and Volume705

15.2.2 The Jacobian Factor709

15.2.3 Polar Coordinates714

15.2.4 Change of Variable for Lebesgue Integrals717

15.2.5 Exercises720

15.3 Summary722

Index727