A Course in Advanced Calculus

✍ Scribed by Robert S. Borden

• Publisher: Dover Publications
• Year: 1998
• Tongue: English
• Leaves: 430
• Series: Dover Books on Mathematics
• Category: Library

✦ Synopsis

This remarkable undergraduate-level text offers a study in calculus that simultaneously unifies the concepts of integration in Euclidean space while at the same time giving students an overview of other areas intimately related to mathematical analysis. The author achieves this ambitious undertaking by shifting easily from one related subject to another. Thus, discussions of topology, linear algebra, and inequalities yield to examinations of innerproduct spaces, Fourier series, and the secret of Pythagoras. Beginning with a look at sets and structures, the text advances to such topics as limit and continuity in En, measure and integration, differentiable mappings, sequences and series, applications of improper integrals, and more.
Carefully chosen problems appear at the end of each chapter, and this new edition features an additional appendix of tips and solutions for selected problems.

Reprint of Elsevier Science Publishing Co., Inc., New York, 1983 edition.

✦ Table of Contents

CONTENTS

Preface

CHAPTER 1 SETS AND STRUCTURES

1.1 Sets

1.2 Algebraic Structures

1.3 Morphisms

1.4 Order Structures

Problems

CHAPTER 2 LIMIT AND CONTINUITY IN En

2.1 Limit of a Function

2.2 Sequences in En

2.3 Limit Superior and Limit Inferior of a Function

Problems

CHAPTER 3 INEQUALITIES

3.1 Some Basic Inequalities

Problems

CHAPTER 4 LINEAR SPACES

4.1 Linear and Affine Mappings

4.2 Continuity of Linear Maps

4.3 Determinants

4.4 The Grassmann Algebra

Problems

CHAPTER 5 FORMS IN En

5.1 Orientation of Parallelotopes

5.2 1-Forms in En

5.3 Some Applications of 1-Forms

5.4 0-Forms in En

5.5 2-Forms in En

5.6 An Application in E³

5.7 A Substantial Example

5.8 k-Forms in En

5.9 Another Example

Problems

CHAPTER 6 TOPOLOGY

6.1 The Open-Set Topology

6.2 Continuity and Limit

6.3 Metrics and Norms

6.4 Product Topologies

6.5 Compactness

6.6 Dense Sets, Connected Sets, Separability, and Category

6.7 Some Properties of Continuous Maps

6.8 Normal Spaces and the Tietze Extension Theorem

6.9 The Cantor Ternary Set

Problems

CHAPTER 7 INNER-PRODUCT SPACES

7.1 Real Inner Products

7.2 Orthogonality and Orthonormal Sets

7.3 An Example: The Space L²(0, 2π)

7.4 Fourier Series and Convergence

7.5 The Gram–Schmidt Process

7.6 Approximation by Projection

7.7 Complex Inner-Product Spaces

7.8 The Gram Determinant and Measures of k-Parallelotopes

7.9 Vector Products in E³

Problems

CHAPTER 8 MEASURE AND INTEGRATION

8.1 Measure

8.2 Measure Spaces and a Darboux Integral

8.3 The Measure Space (En,M, μ) and Lebesgue Measure

8.4 The Lebesgue Integral in En

8.5 Signed Measures

8.6 Affine Maps on (En,M, μ)

8.7 Integration by Pullbacks; the Affine Case

8.8 A Non measurable Set in E¹

8.9 The Riemann–Stieltjes Integral in E¹

8.10 Fubini’s Theorem

8.11 Approximate Continuity

Problems

CHAPTER 9 DIFFERENTIABLE MAPPINGS

9.1 The Derivative of a Map

9.2 Taylor’s Formula

9.3 The Inverse Function Theorem

9.4 The Implicit Function Theorem

9.5 Lagrange Multipliers

9.6 Some Particular Parametric Maps

9.7 A Fixed-Point Theorem

Problems

CHAPTER 10 SEQUENCES AND SERIES

10.1 Convergence of Sequences of Functions

10.2 Series of Functions and Convergence

10.3 Power Series

10.4 Arithmetic with Series

10.5 Infinite Products

Problems

CHAPTER 11 APPLICATIONS OF IMPROPER INTEGRALS

11.1 Improper Integrals

11.2 Some Further Convergence Theorems

11.3 Some Special Functions

11.4 Dirac Sequences and Convolutions

11.5 The Fourier Transform

11.6 The Laplace Transform

11.7 Generalized Functions

Problems

CHAPTER 12 THE GENERALIZED STOKES THEOREM

12.1 Manifolds and Partitions of Unity

12.2 The Stokes Theorem

Problems

Tips and Solutions for Selected Problems

Bibliography

✦ Subjects

Advanced Calculus, Real Analysis, Analysis, Manifolds