Advanced calculus / Patrick M. Fitzpatrick

Originally published: 2nd edition Belmont, CA : Thomson Brooks/Cole, c2006

Includes index

Table Of Contents: Preface Preliminaries 1 TOOLS FOR ANALYSIS 1.1 The Completeness Axiom and Some of Its Consequences 1.2 The Distribution of the Integers and the Rational Numbers 1.3 Inequalities and Identities 2 CONVERGENT SEQUENCES 2.1 The Convergence of Sequences 2.2 Sequences and Sets 2.3 The Monotone Convergence Theorem 2.4 The Sequential Compactness Theorem 2.5 Covering Properties of Sets* 3 CONTINUOUS FUNCTIONS 3.1 Continuity 3.2 The Extreme Value Theorem 3.3 The Intermediate Value Theorem 3.4 Uniform Continuity 3.5 The element of-δ Criterion for Continuity 3.6 Images and Inverses; Monotone Functions 3.7 Limits 4 DIFFERENTIATION 4.1 The Algebra of Derivatives 4.2 Differentiating Inverses and Compositions 4.3 The Mean Value Theorem and Its Geometric Consequences 4.4 The Cauchy Mean Value Theorem and Its Analytic Consequences 4.5 The Notation of Leibnitz 5 ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS 5.1 Solutions of Differential Equations 5.2 The Natural Logarithm and Exponential Functions 5.3 The Trigonometric Functions 5.4 The Inverse Trigonometric Functions 6 INTEGRATION: TWO FUNDAMENTAL THEOREMS 6.1 Darboux Sums; Upper and Lower Integrals 6.2 The Archimedes—Riemann Theorem 6.3 Additivity, Monotonicity, and Linearity 6.4 Continuity and Integrability 6.5 The First Fundamental Theorem: Integrating Derivatives 6.6 The Second Fundamental Theorem: Differentiating Integrals 7 INTEGRATION: FURTHER TOPICS 7.1 Solutions of Differential Equations 7.2 Integration by Parts and by Substitution 7.3 The Convergence of Darboux and Riemann Sums 7.4 The Approximation of Integrals 8 APPROXIMATION BY TAYLOR POLYNOMIALS 8.1 Taylor Polynomials 8.2 The Lagrange Remainder Theorem 8.3 The Convergence of Taylor Polynomials 8.4 A Power Series for the Logarithm 8.5 The Cauchy Integral Remainder Theorem 8.6 A Nonanalytic, Infinitely Differentiable Function 8.7 The Weierstrass Approximation Theorem 9 SEQUENCES AND SERIES OF FUNCTIONS 9.1 Sequences and Series of Numbers 9.2 Pointwise Convergence of Sequences of Functions 9.3 Uniform Convergence of Sequences of Functions 9.4 The Uniform Limit of Functions 9.5 Power Series 9.6 A Continuous Nowhere Differentiable Function 10 THE EUCLIDEAN SPACE Rn 10.1 The Linear Structure of Rn and the Scalar Product 10.2 Convergence of Sequences in Rn 10.3 Open Sets and Closed Sets in Rn 11 CONTINUITY, COMPACTNESS, AND CONNECTEDNESS 11.1 Continuous Functions and Mappings 11.2 Sequential Compactness, Extreme Values, and Uniform Continuity 11.3 Pathwise Connectedness and the Intermediate Value Theorem* 11.4 Connectedness and the Intermediate Value Property* 12 METRIC SPACES 12.1 Open Sets, Closed Sets, and Sequential Convergence 12.2 Completeness and the Contraction Mapping Principle 12.3 The Existence Theorem for Nonlinear Differential Equations 12.4 Continuous Mappings between Metric Spaces 12.5 Sequential Compactness and Connectedness 13 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES 13.1 Limits 13.2 Partial Derivatives 13.3 The Mean Value Theorem and Directional Derivatives 14 LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS 14.1 First-Order Approximation, Tangent Planes, and Affine Functions 14.2 Quadratic Functions, Hessian Matrices, and Second Derivatives* 14.3 Second-Order Approximation and the Second-Derivative Test* 15 APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS 15.1 Linear Mappings and Matrices 15.2 The Derivative Matrix and the Differential 15.3 The Chain Rule 16 IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM 16.1 Functions of a Single Variable and Maps in the Plane 16.2 Stability of Nonlinear Mappings 16.3 A Minimization Principle and the General Inverse Function Theorem 17 THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS 17.1 A Scalar Equation in Two Unknowns: Dini's Theorem 17.2 The General Implicit Function Theorem 17.3 Equations of Surfaces and Paths in R³ 17.4 Constrained Extrema Problems and Lagrange Multipliers 18 INTEGRATING FUNCTIONS OF SEVERAL VARIABLES 18.1 Integration of Functions on Generalized Rectangles 18.2 Continuity and Integrability 18.3 Integration of Functions on Jordan Domains 19 ITERATED INTEGRATION AND CHANGES OF VARIABLES 19.1 Fubini's Theorem 19.2 The Change of Variables Theorem: Statements and Examples 19.3 Proof of the Change of Variables Theorem 20 LINE AND SURFACE INTEGRALS 20.1 Arclength and Line Integrals 20.2 Surface Area and Surface Integrals 20.3 The Integral Formulas of Green and Stokes A CONSEQUENCES OF THE FIELD AND POSITIVITY AXIOMS A.1 The Field Axioms and Their Consequences A.2 The Positivity Axioms and Their Consequences B LINEAR ALGEBRA Index