Advanced Calculus. International Series in Pure and Applied Mathematics. Third Edition

Description

Includes study material for understanding sets and functions, continuity, intergration, convergence, differentiation and other functions of advanced calculus. 1 SETS AND FUNCTIONS 1.1 Introduction 1.2 R and R* 1.3 Distance 1.4 Functions 1.5 Topological Terminology 1.6 Sequences 1.7 Consequences of the Monotonic-Sequence Property 1.8 Compact Sets 2 CONTINUITY 2.1 Preview 2.2 Basic Definitions 2.3 Uniform Continuity 2.4 Implications of Continuity 2.5 Limits of Functions 2.6 Discontinuities 2.7 Inverses for Functions of One Variable 3 DIFFERENTIATION 3.1 Preview 3.2 Mean Value Theorems and L'Hospital's Rule 3.3 Derivatives for Functions on R^n 3.4 Differentiation of Composite Functions 3.5 Taylor's Theorem 3.6 Extremal Problems 4 INTEGRATION 4.1 Preview 4.2 The Definite Integral 4.3 Evaluation of Definite Integrals 4.4 Substitution in Multiple Integrals 4.5 Improper Integrals 5 SERIES 5.1 Preview 5.2 Infinite Series 5.3 Conditionally Convergent Series 5.4 Double Series 5.5 Some Sums 6 UNIFORM CONVERGENCE 6.1 Preview 6.2 Series and Sequences of Functions 6.3 Power Series 6.4 Improper Integrals with Parameters 6.5 The Gamma Function 6.6 Fourier Series 7 DIFFERENTIATION OF TRANSFORMATIONS 7.1 Preview 7.2 Transformations 7.3 Linear Functions and Transformations 7.4 Differentials of Transformations 7.5 Inverses of Transformations 7.6 The Implicit Function Theorems 7.7 Functional Dependence 8 APPLICATIONS TO GEOMETRY AND ANALYSIS 8.1 Preview 8.2 Set Functions 8.3 Transformations of Multiple Integrals 8.4 Curves and Arc Length 8.5 Surfaces arid Surface Area 8.6 Integrals over Curves and Surfaces 9 DIFFERENTIAL GEOMETRY AND VECTOR CALCULUS 9.1 Preview 9.2 Differential Forms 9.3 Vector Analysis 9.4 The Theorems of Green, Gauss, and Stokes 9.5 Exact Forms and Closed Forms 9.6 Applications 10 NUMERICAL METHODS 10.1 Preview 10.2 Locating Zeros 10.3 Fixed-Point Methods 10.4 Extremal Problems 10.5 Miscellaneous Approximation Methods Back Matter APPENDIXES 1 Logic and Set Theory 2 Foundations of the Real Number System 3 Linear Algebra 4 Applications of Mathematics 5 Introduction to Complex Analysis 6 Further Topics in Real Analysis SUGGESTED READING HINTS AND ANSWERS LIST OF SYMBOLS INDEX Back Cover