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Preliminaries1

1 TOOLS FOR ANALYSIS5

1.1 The Completeness Axiom and Some of Its Consequences5

1.2 The Distribution of the Integers and the Rational Numbers12

1.3 Inequalities and Identities16

2 CONVERGENT SEQUENCES23

2.1 The Convergence of Sequences23

2.2 Sequences and Sets35

2.3 The Monotone Convergence Theorem38

2.4 The Sequential Compactness Theorem43

2.5 Covering Properties of Sets47

3 CONTINUOUS FUNCTIONS53

3.1 Continuity53

3.2 The Extreme Value Theorem58

3.3 The Intermediate Value Theorem62

3.4 Uniform Continuity66

3.5 The∈-δ Criterion for Continuity70

3.6 Images and Inverses;Monotone Functions74

3.7 Limits81

4 DIFFERENTIATION87

4.1 The Algebra of Derivatives87

4.2 Differentiating Inverses and Compositions96

4.3 The Mean Value Theorem and Its Geometric Consequences101

4.4 The Cauchy Mean Value Theorem and Its Analytic Consequences111

4.5 The Notation of Leibnitz113

5 ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS116

5.1 Solutions of Differential Equations116

5.2 The Natural Logarithm and Exponential Functions118

5.3 The Trigonometric Functions125

5.4 The Inverse Trigonometric Functions132

6 INTEGRATION:TWO FUNDAMENTAL THEOREMS135

6.1 Darboux Sums;Upper and Lower Integrals135

6.2 The Archimedes-Riemann Theorem142

6.3 Additivity,Monotonicity,and Linearity150

6.4 Continuity and Integrability155

6.5 The First Fundamental Theorem:Integrating Derivatives160

6.6 The Second Fundamental Theorem:Differentiating Integrals165

7 INTEGRATION:FURTHER TOPICS175

7.1 Solutions of Differential Equations175

7.2 Integration by Parts and by Substitution178

7.3 The Convergence of Darboux and Riemann Sums183

7.4 The Approximation of Integrals190

8 APPROXIMATION BY TAYLOR POLYNOMIALS199

8.1 Taylor Polynomials199

8.2 The Lagrange Remainder Theorem203

8.3 The Convergence of Taylor Polynomials209

8.4 A Power Series for the Logarithm212

8.5 The Cauchy Integral Remainder Theorem215

8.6 A Nonanalytic,Infinitely Differentiable Function221

8.7 The Weierstrass Approximation Theorem223

9 SEQUENCES AND SERIES OF FUNCTIONS228

9.1 Sequences and Series of Numbers228

9.2 Pointwise Convergence of Sequences of Functions241

9.3 Uniform Convergence of Sequences of Functions245

9.4 The Uniform Limit of Functions249

9.5 Power Series255

9.6 A Continuous Nowhere Differentiable Function264

10 THE EUCLIDEAN SPACE Rn269

10.1 The Linear Structure of Rn and the Scalar Product269

10.2 Convergence of Sequences in Rn277

10.3 Open Sets and Closed Sets in Rn282

11 CONTINUITY,COMPACTNESS,AND CONNECTEDNESS290

11.1 Continuous Functions and Mappings290

11.2 Sequential Compactness,Extreme Values,and Uniform Continuity298

11.3 Pathwise Connectedness and the Intermediate Value Theorem304

11.4 Connectedness and the Intermediate Value Property310

12 METRIC SPACES314

12.1 Open Sets,Closed Sets,and Sequential Convergence314

12.2 Completeness and the Contraction Mapping Principle322

12.3 The Existence Theorem for Nonlinear Differential Equations328

12.4 Continuous Mappings between Metric Spaces337

12.5 Sequential Compactness and Connectedness342

13 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES348

13.1 Limits348

13.2 Partial Derivatives353

13.3 The Mean Value Theorem and Directional Derivatives364

14 LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS372

14.1 First-Order Approximation,Tangent Planes,and Affine Functions372

14.2 Quadratic Functions,Hessian Matrices,and Second Derivatives380

14.3 Second-Order Approximation and the Second-Derivative Test387

15 APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS394

15.1 Linear Mappings and Matrices394

15.2 The Derivative Matrix and the Differential407

15.3 The Chain Rule414

16 IMAGES AND INVERSES:THE INVERSE FUNCTION THEOREM421

16.1 Functions of a Single Variable and Maps in the Plane421

16.2 Stability of Nonlinear Mappings429

16.3 A Minimization Principle and the General Inverse Function Theorem433

17 THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS440

17.1 A Scalar Equation in Two Unknowns:Dini's Theorem440

17.2 The General Implicit Function Theorem449

17.3 Equations of Surfaces and Paths in R3454

17.4 Constrained Extrema Problems and Lagrange Multipliers460

18 INTEGRATING FUNCTIONS OF SEVERAL VARIABLES470

18.1 Integration of Functions on Generalized Rectangles470

18.2 Continuity and Integrability482

18.3 Integration of Functions on Jordan Domains489

19 ITERATED INTEGRATION AND CHANGES OF VARIABLES498

19.1 Fubini's Theorem498

19.2 The Change of Variables Theorem:Statements and Examples505

19.3 Proof of the Change of Variables Theorem510

20 LINE AND SURFACE INTEGRALS520

20.1 Arclength and Line Integrals520

20.2 Surface Area and Surface Integrals533

20.3 The Integral Formulas of Green and Stokes543

A CONSEQUENCES OF THE FIELD AND POSITIVITY AXIOMS559

A.1 The Field Axioms and Their Consequences559

A.2 The Positivity Axioms and Their Consequences563

B LINEAR ALGEBRA565

Index581