CONTENTS. Preface Preliminaries 1

Transcription

1 Preface xi Preliminaries 1 1 TOOLS FOR ANALYSIS The Completeness Axiom and Some of Its Consequences The Distribution of the Integers and the Rational Numbers Inequalities and Identities 16 2 CONVERGENT SEQUENCES The Convergence of Sequences Sequences and Sets The Monotone Convergence Theorem The Sequential Compactness Theorem Covering Properties of Sets 47 3 CONTINUOUS FUNCTIONS Continuity The Extreme Value Theorem The Intermediate Value Theorem Uniform Continuity The ɛ-δ Criterion for Continuity Images and Inverses; Monotone Functions Limits 81 4 DIFFERENTIATION The Algebra of Derivatives Differentiating Inverses and Compositions The Mean Value Theorem and Its Geometric Consequences The Cauchy Mean Value Theorem and Its Analytic Consequences The Notation of Leibnitz 113 v

2 vi 5 ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS Solutions of Differential Equations The Natural Logarithm and Exponential Functions The Trigonometric Functions The Inverse Trigonometric Functions INTEGRATION: TWO FUNDAMENTAL THEOREMS Darboux Sums; Upper and Lower Integrals The Archimedes Riemann Theorem Additivity, Monotonicity, and Linearity Continuity and Integrability The First Fundamental Theorem: Integrating Derivatives The Second Fundamental Theorem: Differentiating Integrals INTEGRATION: FURTHER TOPICS Solutions of Differential Equations Integration by Parts and by Substitution The Convergence of Darboux and Riemann Sums The Approximation of Integrals APPROXIMATION BY TAYLOR POLYNOMIALS Taylor Polynomials The Lagrange Remainder Theorem The Convergence of Taylor Polynomials A Power Series for the Logarithm The Cauchy Integral Remainder Theorem A Nonanalytic, Infinitely Differentiable Function The Weierstrass Approximation Theorem SEQUENCES AND SERIES OF FUNCTIONS Sequences and Series of Numbers Pointwise Convergence of Sequences of Functions 241

3 vii 9.3 Uniform Convergence of Sequences of Functions The Uniform Limit of Functions Power Series A Continuous Nowhere Differentiable Function THE EUCLIDEAN SPACE R n The Linear Structure of R n and the Scalar Product Convergence of Sequences in R n Open Sets and Closed Sets in R n CONTINUITY, COMPACTNESS, AND CONNECTEDNESS Continuous Functions and Mappings Sequential Compactness, Extreme Values, and Uniform Continuity Pathwise Connectedness and the Intermediate Value Theorem* Connectedness and the Intermediate Value Property* METRIC SPACES Open Sets, Closed Sets, and Sequential Convergence Completeness and the Contraction Mapping Principle The Existence Theorem for Nonlinear Differential Equations Continuous Mappings between Metric Spaces Sequential Compactness and Connectedness DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES Limits Partial Derivatives The Mean Value Theorem and Directional Derivatives LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS First-Order Approximation, Tangent Planes, and Affine Functions Quadratic Functions, Hessian Matrices, and Second Derivatives* Second-Order Approximation and the Second-Derivative Test* 387

4 viii 15 APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS Linear Mappings and Matrices The Derivative Matrix and the Differential The Chain Rule IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM Functions of a Single Variable and Maps in the Plane Stability of Nonlinear Mappings A Minimization Principle and the General Inverse Function Theorem THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS A Scalar Equation in Two Unknowns: Dini s Theorem The General Implicit Function Theorem Equations of Surfaces and Paths in R Constrained Extrema Problems and Lagrange Multipliers INTEGRATING FUNCTIONS OF SEVERAL VARIABLES Integration of Functions on Generalized Rectangles Continuity and Integrability Integration of Functions on Jordan Domains ITERATED INTEGRATION AND CHANGES OF VARIABLES Fubini s Theorem The Change of Variables Theorem: Statements and Examples Proof of the Change of Variables Theorem LINE AND SURFACE INTEGRALS Arclength and Line Integrals Surface Area and Surface Integrals The Integral Formulas of Green and Stokes 543

5 ix A CONSEQUENCES OF THE FIELD AND POSITIVITY AXIOMS 559 A.1 The Field Axioms and Their Consequences 559 A.2 The Positivity Axioms and Their Consequences 563 B LINEAR ALGEBRA 565 Index 581