Problems and Solutions for Undergraduate Real Analysis II

This book Problems and Solutions for Undergraduate Real Analysis II is the continuum of the first book Problems and Solutions for Undergraduate Real Analysis I . Its aim is the same as its first book: We want to assist undergraduate students or first-year students who study mathematics in learning their first rigorous real analysis course. The wide variety of problems, which are of varying difficulty, include the following topics: Sequences and Series of Functions Improper Integrals Lebesgue Measure Lebesgue Measurable Functions Lebesgue Integration Differential Calculus of Functions of Several Variables Integral Calculus of Functions of Several Variables Furthermore, the main features of this book are listed as follows: The book contains 226 problems, which cover the topics mentioned above, with detailed and complete solutions. Particularly, we include over 100 problems for the Lebesgue integration theory which, I believe, is totally new to all undergraduate students. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step. Different colors are used frequently in order to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) This book **Problems and Solutions for Undergraduate Real Analysis II** is the continuum of the first book **Problems and Solutions for Undergraduate Real Analysis I**. Its aim is the same as its first book: We want to assist undergraduate students or first-year students who study mathematics in learning their first rigorous real analysis course. The wide variety of problems, which are of varying difficulty, include the following topics: * Sequences and Series of Functions * Improper Integrals * Lebesgue Measure * Lebesgue Measurable Functions * Lebesgue Integration * Differential Calculus of Functions of Several Variables * Integral Calculus of Functions of Several Variables Furthermore, the main features of this book are listed as follows: 1. The book contains 226 problems, which cover the topics mentioned above, with detailed and complete solutions. Particularly, we include over 100 problems for the Lebesgue integration theory which, I believe, is totally new to all undergraduate students. 2. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic. 3. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step. 4. Different colors are used frequently in order to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) This book "Problems and Solutions for Undergraduate Real Analysis II " is the continuum of the first book "Problems and Solutions for Undergraduate Real Analysis I ". Its aim is the same as its first book: We want to assist undergraduate students or first-year students who study mathematics in learning their first rigorous real analysis course. The wide variety of problems, which are of varying difficulty, include the following topics: Sequences and Series of Functions, Improper Integrals, Lebesgue Measure, Lebesgue Measurable Functions, Lebesgue Integration, Differential Calculus of Functions of Several Variables and Integral Calculus of Functions of Several Variables.Furthermore, the main features of this book are listed as follows: 1. The book contains 226 problems, which cover the topics mentioned above, with detailed and complete solutions. Particularly, we include over 100 problems for the Lebesgue integration theory which, I believe, is totally new to all undergraduate students. 2. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic. 3. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step. 4. Different colors are used frequently in order to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) Preface List of Figures Sequences and Series of Functions Fundamental Concepts Uniform Convergence for Sequences of Functions Uniform Convergence for Series of Functions Equicontinuous Families of Functions Approximation by Polynomials Improper Integrals Fundamental Concepts Evaluations of Improper Integrals Convergence of Improper Integrals Miscellaneous Problems on Improper Integrals Lebesgue Measure Fundamental Concepts Lebesgue Outer Measure Lebesgue Measurable Sets Necessary and Sufficient Conditions for Measurable Sets Lebesgue Measurable Functions Fundamental Concepts Lebesgue Measurable Functions Applications of Littlewood's Three Principles Lebesgue Integration Fundamental Concepts Properties of Integrable Functions Applications of Fatou's Lemma Applications of Convergence Theorems Differential Calculus of Functions of Several Variables Fundamental Concepts Differentiation of Functions of Several Variables The Mean Value Theorem for Differentiable Functions The Inverse Function Theorem and the Implicit Function Theorem Higher Order Derivatives Integral Calculus of Functions of Several Variables Fundamental Concepts Jordan Measurable Sets Integration on Rn Applications of the Mean Value Theorem Applications of the Change of Variables Theorem Index Bibliography