Advanced Calculus

0821847910

CONTENTS

Preface XI

Preliminaries 1

1 TOOLS FOR ANALYSIS 5
1.1 The Completeness Axiom and Some of Its Consequences 5
1.2 The Distribution of the Integers and the Rational Numbers 12
1.3 Inequalities and Identities 16

2 CONVERGENT SEQUENCES 23
2.1 The Convergence of Sequences 23
2.2 Sequences and Sets 35
2.3 The Monotone Convergence Theorem 38
2.4 The Sequential Compactness Theorem 43
2.5* Covering Properties of Sets 47

3 CONTINUOUS FUNCTIONS 53
3.1 Continuity 53
3.2 The Extreme Value Theorem 58
3.3 The Intermediate Value Theorem 62
3.4 Uniform Continuity 66
3.5 The $\\epsilon$-$\\delta$ Criterion for Continuity 70
3.6 Images and Inverses; Monotone Functions 74
3.7 Limits 81

4 DIFFERENTIATION 87
4.1 The Algebra of Derivatives 87
4.2 Differentiating Inverses and Compositions 96
4.3 The Mean Value Theorem and Its Geometric Consequences 101
4.4 The Cauchy Mean Value Theorem and Its Analytic Consequences 111
4.5 The Notation of Leibnitz 113

5* ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS 116
5.1 Solutions of Differential Equations 116
5.2 The Natural Logarithm and Exponential Functions 118
5.3 The Trigonometric Functions 125
5.4 The Inverse Trigonometric Functions 132

6 INTEGRATION: TWO FUNDAMENTAL THEOREMS 135
6.1 Darboux Sums; Upper and Lower Integrals 135
6.2 The Archimedes-Riemann Theorem 142
6.3 Additivity, Monotonicity, and Linearity 150
6.4 Continuity and Integrability 155
6.5 The First Fundamental Theorem: Integrating Derivatives 160
6.6 The Second Fundamental Theorem: Differentiating Integrals 165

7* INTEGRATION: FURTHER TOPICS 175
7.1 Solutions of Differential Equations 175
7.2 Integration by Parts and by Substitution 178
7.3 The Convergence of Darboux and Riemann Sums 183
7.4 The Approximation of Integrals 190

8 APPROXIMATION BY TAYLOR POLYNOMIALS 199
8.1 Taylor Polynomials 199
8.2 The Lagrange Remainder Theorem 203
8.3 The Convergence of Taylor Polynomials 209
8.4 A Power Series for the Logarithm 212
8.5 The Cauchy Integral Remainder Theorem 215
8.6 A Nonanalytic, Infinitely Differentiable Function 221
8.7 The Weierstrass Approximation Theorem 223

9 SEQUENCES AND SERIES OF FUNCTIONS 228
9.1 Sequences and Series of Numbers 228
9.2 Pointwise Convergence of Sequences of Functions 241
9.3 Uniform Convergence of Sequences of Functions 245
9.4 The Uniform limit of Functions 249
9.5 Power Series 255
9.6 A Continuous Nowhere Differentiable Function 264

10 THE EUCLIDEAN SPACE JRn 269
10.1 The Linear Structure of JRn and the Scalar Product 269
10.2 Convergence of Sequences in ${\\mathbb R}^n$ 277
10.3 Open Sets and Closed Sets in ${\\mathbb R}^n$ 282

11 CONTINUITY, COMPACTNESS, AND CONNECTEDNESS 290
11.1 Continuous Functions and Mappings 290
11.2 Sequential Compactness, Extreme Values, and Uniform Continuity 298
11.3* Pathwise Connectedness and the Intermediate Value Theorem 304
11.4* Connectedness and the Intermediate Value Property 310

12* METRIC SPACES 314
12.1 Open Sets, Closed Sets, and Sequential Convergence 314
12.2 Completeness and the Contraction Mapping Principle 322
12.3 The Existence Theorem for Nonlinear Differential Equations 328
12.4 Continuous Mappings between Metric Spaces 337
12.5 Sequential Compactness and Connectedness 342

13 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES 348
13.1 Limits 348
13.2 Partial Derivatives 353
13.3 The Mean Value Theorem and Directional Derivatives 364

14 LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS 372
14.1 First-Order Approximation, Tangent Planes, and Affine Functions 372
14.2* Quadratic Functions, Hessian Matrices, and Second Derivatives 380
14.3* Second-Order Approximation and the Second-Derivative Test 387

15 APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS 394
15.1 Linear Mappings and Matrices 394
15.2 The Derivative Matrix and the Differential 407
15.3 The Chain Rule 414

16 IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM 421
16.1 Functions of a Single Variable and Maps in the Plane 421
16.2 Stability of Nonlinear Mappings 429
16.3 A Minimization Principle and the General Inverse Function Theorem 433

17 THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS 440
17.1 A Scalar Equation in Two Unknowns: Dini\'s Theorem 440
17.2 The General Implicit Function Theorem 449
17.3 Equations of Surfaces and Paths in ${\\mathbb R}^3$ 454
17.4 Constrained Extrema Problems and Lagrange Multipliers 460

18 INTEGRATING FUNCTIONS OF SEVERAL VARIABLES 470
18.1 Integration of Functions on Generalized Rectangles 470
18.2 Continuity and Integrability 482
18.3 Integration of Functions on Jordan Domains 489

19 ITERATED INTEGRATION AND CHANGES OF VARIABLES 498
19.1 Fubini\'s Theorem 498
19.2 The Change of Variables Theorem: Statements and Examples 505
19.3 Proof of the Change of Variables Theorem 510

20 LINE AND SURFACE INTEGRALS 520
20.1 Arclength and Line Integrals 520
20.2 Surface Area and Surface Integrals 533
20.3 The Integral Formulas of Green and Stokes 543

A CONSEQUENCES OF THE FIELD AND POSITIVITY AXIOMS 559
A.l The Field Axioms and Their Consequences 559
A.2 The Positivity Axioms and Their Consequences 563

B LINEAR ALGEBRA 565

Index 581