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Patrick M. Fitzpatrick University of Maryland, College Park, MD
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Hardcover ISBN: 978-0-8218-4791-6
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AMS Member Price: $111.60 Click above image for expanded view Advanced Calculus: Second Edition Patrick M. Fitzpatrick University of Maryland, College Park, MD Available Formats:  Hardcover ISBN: 978-0-8218-4791-6 Product Code: AMSTEXT/5  List Price:$93.00 MAA Member Price: $83.70 AMS Member Price:$74.40
 Electronic ISBN: 978-1-4704-1118-3 Product Code: AMSTEXT/5.E
 List Price: $87.00 MAA Member Price:$78.30 AMS Member Price: $69.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$139.50 MAA Member Price: $125.55 AMS Member Price:$111.60
• Book Details

Volume: 52006; 590 pp
MSC: 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 self-contained 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.

• 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 Archimedes-Riemann Theorem
• 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 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
• 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

• Reviews

• This is a well-written and well-structured 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.

• Requests

Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 52006; 590 pp
MSC: 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 self-contained 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.

• 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 Archimedes-Riemann Theorem
• 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 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
• 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 well-written and well-structured 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.