Razdan A. Fundamentals of Analysis with Applications 2022 torrent

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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
List of Figures
Sets, Functions, and Cardinality
Naive Set Theory
Relation and Ordering
Functions
Cardinality
Development of Function Concept—A Historical Note
Exercises
References
The Real Numbers
Ordered Field Q
The Complete Ordered Field
Modulus Metric
Countable and Uncountable Sets in R
Exercises
References
Sequences and Series of Numbers
The Limit of a Sequence
Algebra of Convergent Sequences
Convergence Theorems
Infinite Series
Exercises
Reference
Analysis on R
Limit and Continuity
Differentiability
Riemann Integration
Exercises
Reference
Topology of the Real Line
Open and Closed Set
Compactness
Connectedness
Exercises
Metric Spaces
Some Important Metric Spaces
Topology of Metric Spaces
Convergence and Completeness
Compactness
Connectedness
Exercises
Reference
Multivariable Analysis
Limit of Functions
Continuity of a Function
Differentiability
Geometry of Curves and Surfaces
Two Fundamental Theorems
Exercises
References
Sequences and Series of Functions
Pointwise Convergence
Uniform Convergence
Power Series
Exercises
Measure and Integration
Measure Space
Lebesgue Measure
Lebesgue Integration
Fundamental Convergence Theorems
L^p Spaces
Exercises
References
Fourier Series
Evolution of Modern Mathematics
Definitions and Examples
Convergence Issues
An Application to Infinite Series
Exercises
References
Appendix Mathematical Logic
Theory of Inference
Predicate Calculus
Index