A Course in Advanced Calculus

Description

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. 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