Table of Contents
Preface vii
1 The Basics of Mathematical Reasoning 1
1.1 Statements and predicates 1
1.2 Quantifiers 4
1.3 Sets 7
1.4 Functions 10
1.5 Exercises 14
1.6 More challenging problems 16
2 The Real Number System 17
2.1 The algebraic axioms of the real numbers 18
2.2 The order axiom of the real numbers 20
2.3 The completeness axiom 25
2.4 Visualizing the real numbers 27
2.5 Exercises 30
3 Special Classes of Real Numbers 33
3.1 The natural numbers and the induction principle 33
3.2 Applications of the induction principle 37
3.3 Archimedes' Principle 41
3.4 Rational and irrational numbers 44
3.5 Exercises 50
3.6 More challenging problems 53
4 Limits of Sequences 57
4.1 Sequences 57
4.2 Convergent sequences 59
4.3 The arithmetic of limits 64
4.4 Convergence of monotone sequences 69
4.5 Fundamental sequences and Cauchy's characterization of convergence 74
4.6 Series 76
4.7 Power series 87
4.8 Some fundamental sequences and series 89
4.9 Exercises 90
4.10 More challenging problems 97
5 Limits of Functions 101
5.1 Definition and basic properties 101
5.2 Exponentials and logarithms 105
5.3 Limits involving infinities 113
5.4 One-sided limits 116
5.5 Some fundamental limits 118
5.6 Trigonometric functions: a less than completely rigorous definition 120
5.7 Useful trig identities 126
5.8 Landau notation 126
5.9 Exercises 128
5.10 More challenging problems 131
6 Continuity 133
6.1 Definition and examples 133
6.2 Fundamental properties of continuous functions 137
6.3 Uniform continuity 145
6.4 Exercises 149
6.5 More challenging problems 152
7 Differential Calculus 155
7.1 Linear approximation and derivative 155
7.2 Fundamental examples 160
7.3 The basic rules of differential calculus 164
7.4 Fundamental properties of differentiable functions 172
7.5 Table of derivatives 182
7.6 Exercises 183
7.7 More challenging problems 188
8 Applications of Differential Calculus 191
8.1 Taylor approximations 191
8.2 L'Hôpital's Rule 197
8.3 Convexity 200
8.3.1 Basic facts about convex functions 201
8.3.2 Some classical applications of convexity 207
8.4 How to sketch the graph of a function 217
8.5 Antiderivatives 221
8.6 Exercises 232
8.7 More challenging problems 237
9 Integral Calculus 239
9.1 The integral as area: a first look 239
9.2 The Riemann integral 241
9.3 Darboux sums and Riemann integrability 245
9.4 Examples of Riemann integrable functions 253
9.5 Basic properties of the Riemann integral 262
9.6 How to compute a Riemann integral 265
9.6.1 Integration by parts 267
9.6.2 Change of variables 273
9.7 Improper integrals 281
9.7.1 Euler's Gamma and Beta functions 291
9.8 Length, area and volume 292
9.8.1 Length 292
9.8.2 Area 295
9.8.3 Solids of revolution 298
9.9 Exercises 301
9.10 More challenging problems 308
10 Complex Numbers and Some of Their Applications 311
10.1 The field of complex numbers 311
10.1.1 The geometric interpretation of complex numbers 313
10.2 Analytic properties of complex numbers 316
10.3 Complex power series 323
10.4 Exercises 327
11 The Geometry and the Topology of Euclidean Spaces 329
11.1 Basic affine geometry 329
11.2 Basic Euclidean geometry 347
11.3 Basic Euclidean topology 354
11.4 Convergence 359
11.5 Normed vector spaces 366
11.6 Exercises 368
11.7 More challenging problems 375
12 Continuity 377
12.1 Limits and continuity 379
12.2 Connectedness and compactness 386
12.2.1 Connectedness 386
12.2.2 Compactness 387
12.3 Topological properties of continuous maps 394
12.4 Continuous partitions of unity 397
12.5 Exercises 401
12.6 More challenging problems 408
13 Multi-variable Differential Calculus 411
13.1 The differential of a map at a point 412
13.2 Partial derivatives and Fréchet differentials 416
13.3 The chain rule 426
13.4 Higher order partial derivatives 437
13.5 Exercises 442
13.6 More challenging problems 446
14 Applications of Multi-variable Differential Calculus 449
14.1 Taylor formula 449
14.2 Extrema of functions of several variables 451
14.3 Diffeomorphisms and the inverse function theorem 456
14.4 The implicit function theorem 464
14.5 Submanifolds of Rn 471
14.5.1 Definition and basic examples 472
14.5.2 Tangent spaces 481
14.5.3 Lagrange multipliers 488
14.6 Exercises 491
14.7 More challenging problems 497
15 Multidimensional Riemann Integration 499
15.1 Riemann integrable functions of several variables 499
15.1.1 The Riemann integral over a box 499
15.1.2 A conditional Fubini theorem 511
15.1.3 Functions Riemann integrable over Rn 515
15.1.4 Volume and Jordan measurability 518
15.1.5 The Riemann integral over arbitrary regions 521
15.2 Fubini theorem and iterated integrals 522
15.2.1 An unconditional Fubini theorem 523
15.2.2 Some applications 526
15.3 Change in variables formula 529
15.3.1 Formulation and some classical examples 529
15.3.2 Proof of the change of variables formula 545
15.4 Improper integrals 551
15.4.1 Locally integrable functions 551
15.4.2 Absolutely integrable functions 554
15.4.3 Examples 557
15.5 Exercises 562
15.6 More challenging problems 571
16 Integration over Submanifolds 575
16.1 Integration along curves 575
16.1.1 Integration of functions along curves 575
16.1.2 Integration of differential 1-forms over paths 583
16.1.3 Integration of 1-forms over oriented curves 589
16.1.4 The 2-dimensional Stokes' formula: a baby case 593
16.2 Integration over surfaces 599
16.2.1 The area of a parallelogram 599
16.2.2 Compact surfaces (with boundary) 601
16.2.3 Integrals over surfaces 604
16.2.4 Orientable surfaces in R3 614
16.2.5 The flux of a vector field through an oriented surface in R3 616
16.2.6 Stokes' Formula 620
16.3 Differential forms and their calculus 623
16.3.1 Differential forms on Euclidean spaces 623
16.3.2 Orientable submanifolds 633
16.3.3 Integration along oriented submanifolds 635
16.3.4 The general Stokes' formula 639
16.3.5 What are these differential forms anyway 645
16.4 Exercises 648
16.5 More challenging problems 654
Bibliography 655
Index 657
