**Pure and Applied Undergraduate Texts**

Volume: 5;
2006;
590 pp;
Hardcover

MSC: Primary 26;

Print ISBN: 978-0-8218-4791-6

Product Code: AMSTEXT/5

List Price: $87.00

AMS Member Price: $69.60

MAA Member Price: $78.30

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**Electronic ISBN: 978-1-4704-1118-3
Product Code: AMSTEXT/5.E**

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# Advanced Calculus: Second Edition

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*Patrick M. Fitzpatrick*

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.

#### Readership

Undergraduate students interested in teaching and learning undergraduate analysis.

#### Reviews & Endorsements

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.

-- Teodora-Liliana Radulescu, Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Advanced Calculus: Second Edition

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents v6 free
- Preface xi12 free
- Preliminaries 120 free
- 1 TOOLS FOR ANALYSIS 524
- 2 CONVERGENT SEQUENCES 2342
- 3 CONTINUOUS FUNCTIONS 5372
- 4 DIFFERENTIATION 87106
- 5* ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS 116135
- 6 INTEGRATION: TWO FUNDAMENTAL THEOREMS 135154
- 6.1 Darboux Sums; Upper and Lower Integrals 135154
- 6.2 The Archimedes- Riemann Theorem 142161
- 6.3 Additivity, Monotonicity, and Linearity 150169
- 6.4 Continuity and Integrability 155174
- 6.5 The First Fundamental Theorem: Integrating Derivatives 160179
- 6.6 The Second Fundamental Theorem: Differentiating Integrals 165184

- 7* INTEGRATION: FURTHER TOPICS 175194
- 8 APPROXIMATION BY TAYLOR POLYNOMIALS 199218
- 8.1 Taylor Polynomials 199218
- 8.2 The Lagrange Remainder Theorem 203222
- 8.3 The Convergence of Taylor Polynomials 209228
- 8.4 A Power Series for the Logarithm 212231
- 8.5 The Cauchy Integral Remainder Theorem 215234
- 8.6 A Nonanalytic, Infinitely Differentiable Function 221240
- 8.7 The Weierstrass Approximation Theorem 223242

- 9 SEQUENCES AND SERIES OF FUNCTIONS 228247
- 10 THE EUCLIDEAN SPACE R[sup(n)] 269288
- 11 CONTINUITY, COMPACTNESS, AND CONNECTEDNESS 290309
- 12* METRIC SPACES 314333
- 12.1 Open Sets, Closed Sets, and Sequential Convergence 314333
- 12.2 Completeness and the Contraction Mapping Principle 322341
- 12.3 The Existence Theorem for Nonlinear Differential Equations 328347
- 12.4 Continuous Mappings between Metric Spaces 337356
- 12.5 Sequential Compactness and Connectedness 342361

- 13 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES 348367
- 14 LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS 372391
- 15 APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS 394413
- 16 IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM 421440
- 17 THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS 440459
- 18 INTEGRATING FUNCTIONS OF SEVERAL VARIABLES 470489
- 19 ITERATED INTEGRATION AND CHANGES OF VARIABLES 498517
- 20 LINE AND SURFACE INTEGRALS 520539
- A: CONSEQUENCES OF THE FIELD AND POSITIVITY AXIOMS 559578
- B: LINEAR ALGEBRA 565584
- Index 581600
- Back Cover Back Cover1610