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