**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: $93.00

AMS Member Price: $74.40

MAA Member Price: $83.70

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

List Price: $87.00

AMS Member Price: $69.60

MAA Member Price: $78.30

# 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 11 free
- Preface 1212 free
- Tools for Analysis 2424
- Convergent Sequences 4242
- Continuous Functions 7272
- Differentiation 106106
- Elementary Functions as Solutions of Differential Equations 135135
- Integration: Two Fundamental Theorems 154154
- Integration: Further Topics 194194
- Approximation By Taylor Polynomials 218218
- Sequences And Series Of Functions 247247
- The Euclidean Space 288288
- Continuity, Compactness, And Connectedness 309309
- Metric Spaces 333333
- Differentiating Functions Of Several Variables 367367
- Local Approximation Of Real-Valued Functions 391391
- Approximating Nonlinear Mappings By Linear Mappings 313313
- Images And Inverses: The Inverse Function Theorem 440440
- The Implicit Function Theorem And Its Applications 459459
- Integrating Functions Of Several Variables 489489
- Iterated Integration And Changes Of Variables 517517
- Line And Surface Integrals 539539
- Appendix A Consequences Of The Field And Positivity Axioms 578578
- Appendix B Linear Algebra 584584
- Index 600600
- Back Cover 610610