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.
Author(s): Robert S. Borden
Series: Dover Books on Mathematics
Publisher: Dover Publications
Year: 1998
Language: English
Pages: 430
Tags: Advanced Calculus, Real Analysis, Analysis, Manifolds
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
