Fundamentals of Analysis with Applications

Description

This book serves as a textbook in real analysis. It focuses on the fundamentals of the structural properties of metric spaces and analytical properties of functions defined between such spaces. Topics include sets, functions and cardinality, real numbers, analysis on R, topology of the real line, metric spaces, continuity and differentiability, sequences and series, Lebesgue integration, and Fourier series. It is primarily focused on the applications of analytical methods to solving partial differential equations rooted in many important problems in mathematics, physics, engineering, and related fields. Both the presentation and treatment of topics are fashioned to meet the expectations of interested readers working in any branch of science and technology. Senior undergraduates in mathematics and engineering are the targeted student readership, and the topical focus with applications to real-world examples will promote higher-level mathematical understanding for undergraduates in sciences and engineering. Preface Introduction Contents List of Figures Chapter 1. Sets, Functions, and Cardinality 1.1 Naive Set Theory 1.2 Relation and Ordering 1.3 Functions 1.4 Cardinality 1.5 Development of Function Concept—A Historical Note Exercises 1 References Chapter 2. The Real Numbers 2.1 Ordered Field Q 2.2 The Complete Ordered Field 2.3 Modulus Metric 2.4 Countable and Uncountable Sets in R Exercises 2 References Chapter 3. Sequences and Series of Numbers 3.1 The Limit of a Sequence 3.2 Algebra of Convergent Sequences 3.3 Convergence Theorems 3.4 Infinite Series Exercises 3 Reference Chapter 4. Analysis on R 4.1 Limit and Continuity 4.2 Differentiability 4.3 Riemann Integration Exercises 4 Reference Chapter 5. Topology of the Real Line 5.1 Open and Closed Set 5.2 Compactness 5.3 Connectedness Exercises 5 Chapter 6. Metric Spaces 6.1 Some Important Metric Spaces 6.2 Topology of Metric Spaces 6.3 Convergence and Completeness 6.4 Compactness 6.5 Connectedness Exercises Reference Chapter 7. Multivariable Analysis 7.1 Limit of Functions 7.2 Continuity of a Function 7.3 Differentiability 7.4 Geometry of Curves and Surfaces 7.5 Two Fundamental Theorems Exercises References Chapter 8. Sequences and Series of Functions 8.1 Pointwise Convergence 8.2 Uniform Convergence 8.3 Power Series Exercises Chapter 9. Measure and Integration 9.1 Measure Space 9.2 Lebesgue Measure 9.3 Lebesgue Integration 9.4 Fundamental Convergence Theorems 9.5 L^p Spaces Exercises References Chapter 10. Fourier Series 10.1 Evolution of Modern Mathematics 10.2 Definitions and Examples 10.3 Convergence Issues 10.4 An Application to Infinite Series Exercises 10 References Appendix. Mathematical Logic Theory of Inference Predicate Calculus Index