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Book DetailsPure and Applied Undergraduate TextsVolume: 5; 2006; 590 ppMSC: Primary 26;
Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is selfcontained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclidean space. These are the basis of a rigorous treatment of differential calculus (including the Implicit Function Theorem and Lagrange Multipliers) for mappings between Euclidean spaces and integration for functions of several real variables.
Special attention has been paid to the motivation for proofs. Selected topics, such as the Picard Existence Theorem for differential equations, have been included in such a way that selections may be made while preserving a fluid presentation of the essential material.
Supplemented with numerous exercises, Advanced Calculus is a perfect book for undergraduate students of analysis.
An instructor's manual for this title is available electronically. Please send email to textbooks@ams.org for more information.ReadershipUndergraduate students interested in teaching and learning undergraduate analysis.

Table of Contents

Cover

Preface

Tools for Analysis

The Completeness Axiom and Some of Its Consequences

The Distribution of the Integers and the Rational Numbers

Inequalities and Identities

Convergent Sequences

The Convergence of Sequences

Sequences and Sets

The Monotone Convergence Theorem

The Sequential Compactness Theorem

Covering Properties of Sets

Continuous Functions

Continuity

The Extreme Value Theorem

The Intermediate Value Theorem

Uniform Continuity

The Criterion for Continuity

Images and Inverses; Monotone Functions

Limits

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

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 ArchimedesRiemann 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

Uniform Convergence of Sequences of Functions

The Uniform Limit of Functions

Power Series

A Continuous Nowhere Differentiable Function

The Euclidean Space

The Linear Structure of and the Scalar Product

Convergence of Sequences in Rn

Open Sets and Closed Sets in Rn

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 RealValued Functions

FirstOrder Approximation, Tangent Planes, and Affine Functions

Quadratic Functions, Hessian Matrices, and Second Derivatives

SecondOrder Approximation and the SecondDerivative Test

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

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

Appendix A Consequences Of The Field And Positivity Axioms

Appendix B Linear Algebra

Index

Back Cover


Additional Material

Reviews

This is a wellwritten and wellstructured book with clearly explained proofs and a good supply of exercises, some of them are quite challenging. It is this reviewer's opinion that the volume should be an excellent and useful tool for undergraduate students.
TeodoraLiliana Radulescu, Zentralblatt MATH


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 Book Details
 Table of Contents
 Additional Material
 Reviews
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Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is selfcontained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclidean space. These are the basis of a rigorous treatment of differential calculus (including the Implicit Function Theorem and Lagrange Multipliers) for mappings between Euclidean spaces and integration for functions of several real variables.
Special attention has been paid to the motivation for proofs. Selected topics, such as the Picard Existence Theorem for differential equations, have been included in such a way that selections may be made while preserving a fluid presentation of the essential material.
Supplemented with numerous exercises, Advanced Calculus is a perfect book for undergraduate students of analysis.
An instructor's manual for this title is available electronically. Please send email to textbooks@ams.org for more information.
Undergraduate students interested in teaching and learning undergraduate analysis.

Cover

Preface

Tools for Analysis

The Completeness Axiom and Some of Its Consequences

The Distribution of the Integers and the Rational Numbers

Inequalities and Identities

Convergent Sequences

The Convergence of Sequences

Sequences and Sets

The Monotone Convergence Theorem

The Sequential Compactness Theorem

Covering Properties of Sets

Continuous Functions

Continuity

The Extreme Value Theorem

The Intermediate Value Theorem

Uniform Continuity

The Criterion for Continuity

Images and Inverses; Monotone Functions

Limits

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

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 ArchimedesRiemann 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

Uniform Convergence of Sequences of Functions

The Uniform Limit of Functions

Power Series

A Continuous Nowhere Differentiable Function

The Euclidean Space

The Linear Structure of and the Scalar Product

Convergence of Sequences in Rn

Open Sets and Closed Sets in Rn

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 RealValued Functions

FirstOrder Approximation, Tangent Planes, and Affine Functions

Quadratic Functions, Hessian Matrices, and Second Derivatives

SecondOrder Approximation and the SecondDerivative Test

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

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

Appendix A Consequences Of The Field And Positivity Axioms

Appendix B Linear Algebra

Index

Back Cover

This is a wellwritten and wellstructured book with clearly explained proofs and a good supply of exercises, some of them are quite challenging. It is this reviewer's opinion that the volume should be an excellent and useful tool for undergraduate students.
TeodoraLiliana Radulescu, Zentralblatt MATH