TY  - BOOK
AU  - Fitzpatrick,Patrick
ED  - American Mathematical Society.
TI  - Advanced calculus
T2  - Pure and applied undergraduate texts
SN  - 9780821847916
AV  - QA303.2 .F58 2009
PY  - 2009///?]
CY  - Providence, R.I.
PB  - American Mathematical Society
KW  - Calculus
KW  - Textbooks
N1  - 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
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