Real analysis, real variables Books

Book SynopsisThis is a self-contained book that covers the standard topics in introductory analysis and that in addition constructs the natural, rational, real and complex numbers, and also handles complex-valued functions, sequences, and series. The book teaches how to write proofs. Fundamental proof-writing logic is covered in Chapter 1 and is repeated and enhanced in two appendices. Many examples of proofs appear with words in a different font for what should be going on in the proof writer's head. The book contains many examples and exercises to solidify the understanding. The material is presented rigorously with proofs and with many worked-out examples. Exercises are varied, many involve proofs, and some provide additional learning materials.Table of ContentsHow We will Do Mathematics; Concepts with Which We will Do Mathematics; Construction of the Basic Number Systems; Limits of Functions; Continuity; Differentiation; Integration; Sequences; Infinite Series and Power Series; Exponential and Trigonometric Functions; Appendix A: Advice on Writing Mathematics; Appendix B: What One Should Never Forget;

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Book SynopsisThis textbook introduces readers to real analysis in one and n dimensions. It is divided into two parts: Part I explores real analysis in one variable, starting with key concepts such as the construction of the real number system, metric spaces, and real sequences and series. In turn, Part II addresses the multi-variable aspects of real analysis. Further, the book presents detailed, rigorous proofs of the implicit theorem for the vectorial case by applying the Banach fixed-point theorem and the differential forms concept to surfaces in Rn. It also provides a brief introduction to Riemannian geometry. With its rigorous, elegant proofs, this self-contained work is easy to read, making it suitable for undergraduate and beginning graduate students seeking a deeper understanding of real analysis and applications, and for all those looking for a well-founded, detailed approach to real analysis.Table of ContentsChapter 01- Real Numbers.- Chapter 02- Metric Spaces.- Chapter 03- Real Sequences and Series.- Chapter 04- Real Function Limits.- Chapter 05- Continuous Functions.- Chapter 06- Derivatives.- Chapter 07- The Riemann Integral.- Chapter 08- Differential Analysis in Rn.- Chapter 09- Integration in Rn.- Chapter 10- Topics on Vector Calculus and Vector Analysis.

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Book SynopsisThis fully updated new edition of Wilson Sutherland's classic text, Introduction to Metric and Topological Spaces, establishes the language of metric and topological spaces with continuity as the motivating concept, before developing its discussion to cover compactness, connectedness, and completeness.Trade ReviewThe presentation, description and explanation throughout the seventeen short chapters are excellent, and the text can be described as self-contained, with many suitably chosen examples and exercises ,.. An interesting innovation for the new edition is having a companion web site in which more useful and relevant materials can be found. * Peter Shiu, The Mathematical Gazette *Table of ContentsPREFACE; REFERENCES; INDEX

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Book SynopsisOne Review of Basic Material.- I Numbers and Functions.- II Graphs and Curves.- Two Differentiation and Elementary Functions.- III The Derivative.- IV Sine and Cosine.- V The Mean Value Theorem.- VI Sketching Curves.- VII Inverse Functions.- VIII Exponents and Logarithms.- Three Integration.- IX Integration.- X Properties of the Integral.- XI Techniques of Integration.- XII Applications of Integration.- Four Taylor's Formula and Series.- XIII Taylor's Formula.- XIV Series.- Five Functions of Several Variables.- XV Vectors.- XVI Differentiation of Vectors.- XVII Functions of Several Variables.- XVIII The Chain Rule and the Gradient.- Answer.Table of ContentsI: Review of Basic Material. * Numbers and Functions. * Graphs and Curves. II: Differention and Elementary Functions. * The Derivative. * Sine and Cosine. * The Mean Value Theorem. * Sketching Curves. * Inverse Functions * Exponents and Logarithms. III: Integration. * Integration. * Properties of the Integral. * Techniques of Integration. * Applications of Integration. IV: Taylor's Formula and Series. * Taylor's Formula. * Series. Appendix. V: Functions of Several Variables. * Vectors. * Differention of Vectors. * Functions of Several Variables. * The Chain Rule and the Gradient.

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Book SynopsisTrade Review"...if you use this book frequently it’s definitely worth getting the new edition…" --MAA.org, November 2014 "The integrals are very useful, but this book includes many other features that will be helpful to the reader, especially graduate students. The sections on Hermite and Legendre polynomials are especially helpful for students of Electricity and Magnetism, Quantum Mechanics, and Mathematical physics (they won't have to hunt in several books to find what they need)." --Barry Simon, California Institute of Technology "This book is to the CRC Mathematical Tables as the unabridged Oxford English Dictionary is to Webster's Collegiate. Besides being big, it's easy to find things in, because of the way the integrals are organized into classes...It really helped me through grad school." --Phil Hobbs, Amazon ReviewTable of Contents1. Elementary Functions 2. Indefinite Integrals of Elementary Functions 3. Definite Integrals of Elementary Functions 4. Combinations Involving Trigonometric and Hyperbolic Functions and Power 5. Indefinite Integrals of Special Functions 6. Definite Integrals of Special Functions 7. Associated Legendre Functions 8. Special Functions 9. Hypergeometric Functions 10. Vector Field Theory 11. Algebraic Inequalities 12. Integral Inequalities 13. Matrices and Related Result 14. Determinants 15. Norms 16. Ordinary Differential Equations 17. Fourier, Laplace, and Mellin Transforms 18. The Z-transform

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Book SynopsisFourier analysis is a subject that was born in physics but grew up in mathematics. Now it is part of the standard repertoire for mathematicians, physicists and engineers. This diversity of interest is often overlooked, but in this much-loved book, Tom Körner provides a shop window for some of the ideas, techniques and elegant results of Fourier analysis, and for their applications. These range from number theory, numerical analysis, control theory and statistics, to earth science, astronomy and electrical engineering. The prerequisites are few (a reader with knowledge of second- or third-year undergraduate mathematics should have no difficulty following the text), and the style is lively and entertaining. This edition of Körner''s 1989 text includes a foreword written by Professor Terence Tao introducing it to a new generation of fans.Trade Review'This is an extraordinary and very attractive book … I would like to see the book on the desk of every pure mathematician with an interest in classical analysis, and of every teacher of applied mathematics whose work involves analysis … This is how mathematics ideally should be presented, but too seldom is.' R. P. Boas, SIAM Review'This is a wonderful book … More than anything, this is just fun to read, to browse, to study. … Fourier Analysis is literate, lively and a true classic. I highly recommend it.' William J. Satzer, MAA ReviewsTable of ContentsForeword Terence Tao; Preface; 1. Fourier series; 2. Some differential equations; 3. Orthogonal series; 4. Fourier transforms; 5. Further developments; 6. Other directions; Appendices; Index.

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Book SynopsisThrough four editions this popular textbook attracted a loyal readership and widespread use. Students find the book to be concise, accessible, and complete. Instructors find the book to be clear, authoritative, and dependable. The primary goal of this new edition remains the same as in previous editions. It is to make real analysis relevant and accessible to a broad audience of students with diverse backgrounds while also maintaining the integrity of the course. This text aims to be the generational touchstone for the subject and the go-to text for developing young scientists. This new edition continues the effort to make the book accessible to a broader audience. Many students who take a real analysis course do not have the ideal background. The new edition offers chapters on background material like set theory, logic, and methods of proof. The more advanced material in the book is made more apparent.This new edition offers a new chapter onTable of ContentsPreface 0 Background Material 0.1 Number Systems 0.1.1 The Natural Numbers 0.1.2 The Integers 0.1.3 The Rational Numbers 02 Logic and Set 0.2.1 And” and “Or” 0.2.2 “not” and “if then”0.2.3 Contrapositive, Converse, and “Iff” 0.2.4 Quantifiers 0.2.5 Set Theory and Venn Diagrams 0.2.6 Relations and Functions 0.2.7 Countable and Uncountable Sets 1 Real and Complex Numbers 1.1 The Real Numbers Appendix: Construction of the Real Numbers 1.2 The Complex Numbers 2 Sequences 712.1 Convergence of Sequences 2.2 Subsequences 2.3 Limsup and Liminf 2.4 Some Special Sequences 3 Series of Numbers 3.1 Convergence of Series 3.2 Elementary Convergence Tests 3.3 Advanced Convergence Tests 3.4 Some Special Series 3.5 Operations on Series 4 Basic Topology 4.1 Open and Closed Sets 4.2 Further Properties of Open and Closed Sets 4.3 Compact Sets 4.4 The Cantor Set 4.5 Connected and Disconnected Sets 4.6 Perfect Sets 5 Limits and Continuity of Functions 5.1 Basic Properties of the Limit of a Function 5.2 Continuous Functions 5.3 Topological Properties and Continuity 5.4 Classifying Discontinuities and Monotonicity 6 Differentiation of Functions 6.1 The Concept of Derivative 6.2 The Mean Value Theorem and Applications 6.3 More on the Theory of Differentiation 7 The Integral7.1 Partitions and the Concept of Integral 7.2 Properties of the Riemann Integral 7.3 Change of Variable and Related Ideas 7.4 Another Look at the Integral 7.5 Advanced Results on Integration Theory 8 Sequences and Series of Functions 8.1 Partial Sums and Pointwise Convergence 8.2 More on Uniform Convergence 8.3 Series of Functions 8.4 The Weierstrass Approximation Theorem 9 Elementary Transcendental Functions 9.1 Power Series 9.2 More on Power Series: Convergence Issues 9.3 The Exponential and Trigonometric Functions 9.4 Logarithms and Powers of Real Numbers 10 Functions of Several Variables 10.1 A New Look at the Basic Concepts of Analysis 10.2 Properties of the Derivative 10.3 The Inverse and Implicit Function Theorems 11 Advanced Topics 11.1 Metric Spaces 11.2 Topology in a Metric Space 11.3 The Baire Category Theorem 11.4 The Ascoli-Arzela Theorem 12 Differential Equations 12.1 Picard’s Existence and Uniqueness Theorem 12.1.1 The Form of a Differential Equation 12.1.2 Picard’s Iteration Technique 12.1.3 Some Illustrative Examples 12.1.4 Estimation of the Picard Iterates 12.2 Power Series Methods 13 Introduction to Harmonic Analysis 13.1 The Idea of Harmonic Analysis 13.2 The Elements of Fourier Series 13.3 An Introduction to the Fourier Transform Appendix: Approximation by Smooth Functions 13.4 Fourier Methods and Differential Equations 13.4.1 Remarks on Different Fourier Notations 13.4.2 The Dirichlet Problem on the Disc 13.4.3 Introduction to the Heat and Wave Equations 13.4.4 Boundary Value Problems 13.4.5 Derivation of the Wave Equation 13.4.6 Solution of the Wave Equation 13.5 The Heat Equation Appendix: Review of Linear Algebra Table of Notation Glossary Bibliography Index

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Book SynopsisThis book sketches a path for newcomers into the theory of harmonic analysis on the real line. It presents a collection of both basic, well-known and some less known results that may serve as a background for future research around this topic. Many of these results are also a necessary basis for multivariate extensions. An extensive bibliography, as well as hints to open problems are included. The book can be used as a skeleton for designing certain special courses, but it is also suitable for self-study.Trade Review“The book under review takes the reader on a journey along a particular path through the vast landscape of modern harmonic analysis in one real variable. From beginning to end, the text is uniquely flavored by the author’s mathematical interests which provides the reader with a good sense of direction. … The book should be accessible to beginning graduate students in analysis and advanced undergraduates with basic knowledge in real analysis … .” (Joris Roos, zbMATH 1514.42001, 2023)“This book is very accurately described by its subtitle ‘a path in the theory’. The book is at times a textbook, an introduction to harmonic analysis, an essay, or a survey, or some combination of these. … Some theorems are stated and proved, some are discussed, and others are quickly mentioned. It's not a standard path, but an engaging one, offering insights and connections that are new or not well known.” (‪Charles N. Moore, Mathematical Reviews, September, 2022)Table of Contents- Introduction. - Classes of Functions. - Fourier Series. - Fourier Transform. - Hilbert Transform. - Hardy Spaces and their Subspaces. - Hardy Inequalities. - Certain Applications.

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Book SynopsisThis book provides a self-contained and rigorous introduction to calculus of functions of one variable, in a presentation which emphasizes the structural development of calculus. Throughout, the authors highlight the fact that calculus provides a firm foundation to concepts and results that are generally encountered in high school and accepted on faith; for example, the classical result that the ratio of circumference to diameter is the same for all circles. A number of topics are treated here in considerable detail that may be inadequately covered in calculus courses and glossed over in real analysis courses.Trade Review“This book would be a valuable asset to a university library and that many instructors would do well to have a copy of this book in their personal libraries. In addition, I believe that some students would benefit if they possessed a copy of this book to use for reference purposes.” (Jonathan Lewin, MAA Reviews, April 15, 2019)Table of ContentsNumbers and Functions.- Sequences.- Continuity and Limits.- Differentiation.- Applications of Differentiation.- Integration.- Elementary Transcendental Functions.- Applications and Approximations of Riemann Integrals.- Infinite Series and Improper Integrals.

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Book SynopsisThis brief explores the Krasnosel'skiĭ-Man (KM) iterative method, which has been extensively employed to find fixed points of nonlinear methods. Table of Contents1. Introduction.- 2. Notation and Mathematical Foundations.-3. The Krasnoselskii-Mann Iteration.- 4. Relations of the Krasnosel'skii-Mann Iteration and the Operator Splitting Methods.- 5. The Inertial Krasnoselskii-Mann Iteration.- 6. The Multi-step Inertial Krasnoselskii-Mann Iteration.- 7. Relaxation Parameters of the Krasnoselskii-Mann Iteration.- 8. Two Applications.

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Book SynopsisThis book provides an introduction to real analysis, a fundamental topic that is an essential requirement in the study of mathematics. It deals with the concepts of infinity and limits, which are the cornerstones in the development of calculus. Beginning with some basic proof techniques and the notions of sets and functions, the book rigorously constructs the real numbers and their related structures from the natural numbers. During this construction, the readers will encounter the notions of infinity, limits, real sequences, and real series. These concepts are then formalised and focused on as stand-alone objects. Finally, they are expanded to limits, sequences, and series of more general objects such as real-valued functions. Once the fundamental tools of the trade have been established, the readers are led into the classical study of calculus (continuity, differentiation, and Riemann integration) from first principles. The book concludes with an introduction to the study of measures and how one can construct the Lebesgue integral as an extension of the Riemann integral. This textbook is aimed at undergraduate students in mathematics. As its title suggests, it covers a large amount of material, which can be taught in around three semesters. Many remarks and examples help to motivate and provide intuition for the abstract theoretical concepts discussed. In addition, more than 600 exercises are included in the book, some of which will lead the readers to more advanced topics and could be suitable for independent study projects. Since the book is fully self-contained, it is also ideal for self-study.Table of ContentsPreface.- 1. Logic and Sets.- 2. Integers.- 3. Construction of the Real Numbers.- 4. The Real Numbers.- 5. Real Sequences.- 6. Some Applications of Real Sequences.- 7. Real Series.- 8. Additional Topics in Real Series.- 9. Functions and Limits.- 10. Continuity.- 11. Function Sequences and Series.- 12. Power Series.- 13. Differentiation.- 14. Some Applications of Differentiation.- 15. Riemann and Darboux Integration.- 16. The Fundamental Theorem of Calculus.- 17. Taylor and MacLaurin Series.- 18. Introduction to Measure Theory.- 19. Lebesgue Integration.- 20. Double Integrals.- Solutions to the Exercises.- Bibliography.- Index.

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Book SynopsisThis concise textbook introduces calculus students to power series through an informal and captivating narrative that avoids formal proofs but emphasizes understanding the fundamental ideas. Power series—and infinite series in general—are a fundamental tool of pure and applied mathematics. The problems focus on ideas, applications, and creative thinking instead of being repetitive and procedural. Calculus is about functions, so the book turns on two fundamental ideas: using polynomials to approximate a function and representing a function in terms of simpler functions. The derivative is reinterpreted in terms of linear approximations, which then leads to Taylor polynomials and the question of convergence. Enough of the theory of convergence is developed to allow a more complete understanding of power series and their applications. A final chapter looks at the distant horizon and discusses other kinds of series representations. SageMath, a free open-source mathematics software system, is used throughout to do computations, provide examples, and create many graphs. While most problems do not require SageMath, students are encouraged to use it where appropriate. An instructor’s guide with solutions to all the problems is available. The book is intended as a supplementary textbook for calculus courses; lecturers and instructors will find innovative and engaging ways to teach this topic. The informal and conversational tone make the book useful to any student seeking to understand this essential aspect of analysis.Table of Contents- To the reader.- Getting close with lines.- Getting closer with polynomials.- Going all the way: Convergence.- Power series.- Distant mountains.- Appendix A: SageMath: A (very) short introduction.- Appendix B: Why I do it this way.- Bibliography.

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Book SynopsisPart 1 begins with an overview of properties of the real numbers and starts to introduce the notions of set theory. The absolute value and in particular inequalities are considered in great detail before functions and their basic properties are handled. From this the authors move to differential and integral calculus. Many examples are discussed. Proofs not depending on a deeper understanding of the completeness of the real numbers are provided. As a typical calculus module, this part is thought as an interface from school to university analysis.Part 2 returns to the structure of the real numbers, most of all to the problem of their completeness which is discussed in great depth. Once the completeness of the real line is settled the authors revisit the main results of Part 1 and provide complete proofs. Moreover they develop differential and integral calculus on a rigorous basis much further by discussing uniform convergence and the interchanging of limits, infinite series (including Taylor series) and infinite products, improper integrals and the gamma function. In addition they discussed in more detail as usual monotone and convex functions.Finally, the authors supply a number of Appendices, among them Appendices on basic mathematical logic, more on set theory, the Peano axioms and mathematical induction, and on further discussions of the completeness of the real numbers. Remarkably, Volume I contains ca. 360 problems with complete, detailed solutions.Table of ContentsIntroductory Calculus: Numbers - Revision; The Absolute Value, Inequalities and Intervals; Mathematical Induction; Functions and Mappings; Functions and Mappings Continued; Derivatives; Derivatives Continued; The Derivative as a Tool to Investigate Functions; The Exponential and Logarithmic Functions; Trigonometric Functions and Their Inverses; Investigating Functions; Integrating Functions; Rules for Integration; Analysis in One Dimension: Problems with the Real Line; Sequences and their Limits; A First Encounter with Series; The Completeness of the Real Numbers; Convergence Criteria for Series, b-adic Fractions; Point Sets in Continuous Functions; Differentiation; Applications of the Derivative; Convex Functions and some Norms on n; Uniform Convergence and Interchanging Limits; The Riemann Integral; The Fundamental Theorem of Calculus; A First Encounter with Differential Equations; Improper Integrals and the GAMMA-Function; Power Series and Taylor Series; Infinite Products and the Gauss Integral; More on the GAMMA-Function; Selected Topics on Functions of a Real Variable;

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Book SynopsisIntended for the students interested in the disciplines of mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels, this third volume in a series of titles focuses on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals.Trade Review"We are all fortunate that a mathematician with the experience and vision of E.M. Stein, together with his energetic young collaborator R. Shakarchi, has given us this series of four books on analysis."--Steven George Krantz, Mathematical Reviews "This series is a result of a radical rethinking of how to introduce graduate students to analysis... This volume lives up to the high standard set up by the previous ones."--Fernando Q. Gouvea, MAA Review "As one would expect from these authors, the exposition is, in general, excellent. The explanations are clear and concise with many well-focused examples as well as an abundance of exercises, covering the full range of difficulty... [I]t certainly must be on the instructor's bookshelf as a first-rate reference book."--William P. Ziemer, SIAM ReviewTable of ContentsForeword vii Introduction xv 1Fourier series: completion xvi Limits of continuous functions xvi 3Length of curves xvii 4Differentiation and integration xviii 5The problem of measure xviii Chapter 1. Measure Theory 1 1Preliminaries 1 The exterior measure 10 3Measurable sets and the Lebesgue measure 16 4Measurable functions 7 4.1 Definition and basic properties 27 4.Approximation by simple functions or step functions 30 4.3 Littlewood's three principles 33 5* The Brunn-Minkowski inequality 34 6Exercises 37 7Problems 46 Chapter 2: Integration Theory 49 1The Lebesgue integral: basic properties and convergence theorems 49 2Thespace L 1 of integrable functions 68 3Fubini's theorem 75 3.1 Statement and proof of the theorem 75 3.Applications of Fubini's theorem 80 4* A Fourier inversion formula 86 5Exercises 89 6Problems 95 Chapter 3: Differentiation and Integration 98 1Differentiation of the integral 99 1.1 The Hardy-Littlewood maximal function 100 1.The Lebesgue differentiation theorem 104 Good kernels and approximations to the identity 108 3Differentiability of functions 114 3.1 Functions of bounded variation 115 3.Absolutely continuous functions 127 3.3 Differentiability of jump functions 131 4Rectifiable curves and the isoperimetric inequality 134 4.1* Minkowski content of a curve 136 4.2* Isoperimetric inequality 143 5Exercises 145 6Problems 152 Chapter 4: Hilbert Spaces: An Introduction 156 1The Hilbert space L 2 156 Hilbert spaces 161 2.1 Orthogonality 164 2.2 Unitary mappings 168 2.3 Pre-Hilbert spaces 169 3Fourier series and Fatou's theorem 170 3.1 Fatou's theorem 173 4Closed subspaces and orthogonal projections 174 5Linear transformations 180 5.1 Linear functionals and the Riesz representation theorem 181 5.Adjoints 183 5.3 Examples 185 6Compact operators 188 7Exercises 193 8Problems 202 Chapter 5: Hilbert Spaces: Several Examples 207 1The Fourier transform on L 2 207 The Hardy space of the upper half-plane 13 3Constant coefficient partial differential equations 221 3.1 Weaksolutions 222 3.The main theorem and key estimate 224 4* The Dirichlet principle 9 4.1 Harmonic functions 234 4.The boundary value problem and Dirichlet's principle 43 5Exercises 253 6Problems 259 Chapter 6: Abstract Measure and Integration Theory 262 1Abstract measure spaces 263 1.1 Exterior measures and Caratheodory's theorem 264 1.Metric exterior measures 266 1.3 The extension theorem 270 Integration on a measure space 273 3Examples 276 3.1 Product measures and a general Fubini theorem 76 3.Integration formula for polar coordinates 279 3.3 Borel measures on R and the Lebesgue-Stieltjes integral 281 4Absolute continuity of measures 285 4.1 Signed measures 285 4.Absolute continuity 288 5* Ergodic theorems 292 5.1 Mean ergodic theorem 294 5.Maximal ergodic theorem 296 5.3 Pointwise ergodic theorem 300 5.4 Ergodic measure-preserving transformations 302 6* Appendix: the spectral theorem 306 6.1 Statement of the theorem 306 6.Positive operators 307 6.3 Proof of the theorem 309 6.4 Spectrum 311 7Exercises 312 8Problems 319 Chapter 7: Hausdorff Measure and Fractals 323 1Hausdorff measure 324 Hausdorff dimension 329 2.1 Examples 330 2.Self-similarity 341 3Space-filling curves 349 3.1 Quartic intervals and dyadic squares 351 3.Dyadic correspondence 353 3.3 Construction of the Peano mapping 355 4* Besicovitch sets and regularity 360 4.1 The Radon transform 363 4.Regularity of sets when d 3 370 4.3 Besicovitch sets have dimension 371 4.4 Construction of a Besicovitch set 374 5Exercises 380 6Problems 385 Notes and References 389 Bibliography 391 Symbol Glossary 395 Index 397

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Book SynopsisOne Basic Material.- I Vectors.- II Differentiation of Vectors.- III Functions of Several Variables.- IV The Chain Rule and the Gradient.- Two Maxima, Minima, and Taylor's Formula.- V Maximum and Minimum.- VI Higher Derivatives.- Three Curve Integrals and Double Integrals.- VII Potential Functions.- VIII Curve Integrals.- IX Double Integrals.- X Green's Theorem.- Four Triple and Surface Integrals.- XI Triple Integrals.- XII Surface Integrals.- Five Mappings, Inverse Mappings, and Change of Variables Formula..- XIII Matrices.- XIV Linear Mappings.- XV Determinants.- XVI Applications to Functions of Several Variables.- XVII The Change of Variables Formula.- Appendix Fourier Series.- 1. General Scalar Products.- 2. Computation of Fourier Series.- Answers to Exercises.Table of ContentsI: Basic Material. 1: Vectors. 2: Differentiation of Vectors. 3: Functions of Several Variables. 4: The Chain Rule and the Gradient. II: Maxima, Minima, and Taylor's Formula. 5: Maximum and Minimum. 6: Higher Derivatives. III: Curve Integrals and Double Integrals. 7: Potential Functions. 8: Curve Integrals. 9: Double Integrals. 10: Green's Theorem. IV: Triple and Surface Integrals. 12: Triple Integrals. V: Mappings, Inverse Mappings, and Change of Variables Formula. 13: Matrices. 14: Linear Mappings. 15: Determinants. 16: Applications to Functions of Several Variables. 17: The Change of Variables Formula. Appendix: Fourier Series.

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Book SynopsisThe Dirichlet space is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. It boasts a rich and beautiful theory, yet at the same time remains a source of challenging open problems and a subject of active mathematical research. This book is the first systematic account of the Dirichlet space, assembling results previously only found in scattered research articles, and improving upon many of the proofs. Topics treated include: the Douglas and Carleson formulas for the Dirichlet integral, reproducing kernels, boundary behaviour and capacity, zero sets and uniqueness sets, multipliers, interpolation, Carleson measures, composition operators, local Dirichlet spaces, shift-invariant subspaces, and cyclicity. Special features include a self-contained treatment of capacity, including the strong-type inequality. The book will be valuable to researchers in function theory, and with over 100 exercises it is also suitable for self-study by graduate students.Table of ContentsPreface; 1. Basic notions; 2. Capacity; 3. Boundary behavior; 4. Zero sets; 5. Multipliers; 6. Conformal invariance; 7. Harmonically weighted Dirichlet spaces; 8. Invariant subspaces; 9. Cyclicity; Appendix A. Hardy spaces; Appendix B. The Hardy–Littlewood maximal function; Appendix C. Positive definite matrices; Appendix D. Regularization and the rising-sun lemma; References; Index of notation; Index.

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Book SynopsisWith a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus.  Besides the classical capstones--the change of variables formula, implicit and inverse function theorems, the integral theorems of Gauss and Stokes--the text treats other important topics in differential analysis, such as Morse''s lemma and the Poincaré lemma.  The ideas behind most topics can be understood with just two or three variables.  The book incorporates modern computational tools to give visualization real power.  Using 2D and 3D graphics, the book offers new insights into fundamental elements of the calculus of differentiable maps.  The geometric theme continues with an analysis of the physical meaning of the divergence and the curl at a level of detail not found in other advanced calculus books.  This is a textbook for undergraduates and graduate students in mathematics, the physical sciences, and economics.  Prerequisites are an introduction to linear algebra and multivariable calculus.  There is enough material for a year-long course on advanced calculus and for a variety of semester courses--including topics in geometry.  The measured pace of the book, with its extensive examples and illustrations, make it especially suitable for independent study.Trade ReviewFrom the reviews:“Many concepts in calculus and linear algebra have obvious geometric interpretations. … This book differs from other advanced calculus works … it can serve as a useful reference for professors. … it is the adopted course resource, its inclusion in a college library’s collection should be determined by the size and interests of the mathematics faculty. Summing Up … . Upper-division undergraduate through professional collections.” (C. Bauer, Choice, Vol. 48 (8), April, 2011)“The author of this book sees an opportunity to bring back a more geometric, visual and physically-motivated approach to the subject. … The author makes exceptionally good use of two and three-dimensional graphics. Drawings and figures are abundant and strongly support his exposition. Exercises are plentiful and they cover a range from routine computational work to proofs and extensions of results from the text. … Strong students … are likely to be attracted by the approach and the serious meaty content.” (William J. Satzer, The Mathematical Association of America, January, 2011)“A new geometric and visual approach to advanced calculus is presented. … The book can be useful a textbook for beginners as well as a source of supplementary material for university teachers in calculus and analysis. … the book meets a wide auditorium among undergraduate and graduate students in mathematics, physics, economics and in other fields which essentially use mathematical models. It is also very interesting for teachers and instructors in Calculus and Mathematical Analysis.” (Sergei V. Rogosin, Zentralblatt MATH, Vol. 1205, 2011)Table of Contents1 Starting Points.-1.1 Substitution.- Exercises.- 1.2 Work and path integrals.- Exercises.- 1.3 Polar coordinates.- Exercises.- 2 Geometry of Linear Maps.- 2.1 Maps from R2 to R2.- Exercises.- 2.2 Maps from Rn to Rn.- Exercises.- 2.3 Maps from Rn to Rp, n 6= p.- Exercises.- 3 Approximations.- 3.1 Mean-value theorems.- Exercises.- 3.2 Taylor polynomials in one variable.- Exercises.- 3.3 Taylor polynomials in several variables.- Exercises.- 4 The Derivative.- 4.1 Differentiability.- Exercises.- 4.2 Maps of the plane.- Exercises.- 4.3 Parametrized surfaces.- Exercises.- 4.4 The chain rule.- Exercises.- 5 Inverses.- 5.1 Solving equations.- Exercises.- 5.2 Coordinate Changes.- Exercises.- 5.3 The Inverse Function Theorem.- Exercises.- 6 Implicit Functions.- 6.1 A single equation.- Exercises.- 6.2 A pair of equations.- Exercises.- 6.3 The general case.- Exercises.- 7 Critical Points.- 7.1 Functions of one variable.- Exercises.- 7.2 Functions of two variables.- Exercises.- 7.3 Morse’s lemma.- Exercises.- 8 Double Integrals.- 8.1 Example: gravitational attraction.- Exercises.- 8.2 Area and Jordan content.- Exercises.- 8.3 Riemann and Darboux integrals.- Exercises.- 9 Evaluating Double Integrals.- 9.1 Iterated integrals.- Exercises.- 9.2 Improper integrals.- Exercises.- 9.3 The change of variables formula.- 9.4 Orientation.- Exercises.- 9.5 Green’s Theorem.- Exercises.- 10 Surface Integrals.- 10.1 Measuring flux.- Exercises.- 10.2 Surface area and scalar integrals.- Exercises.- 10.3 Differential forms.- Exercises.- 11 Stokes’ Theorem.- 11.1 Divergence.- Exercises.- 11.2 Circulation and Vorticity.- Exercises.- 11.3 Stokes’ Theorem.- 11.4 Closed and Exact Forms.- Exercises

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Book SynopsisReal analysis provides the fundamental underpinnings for calculus, arguably the most useful and influential mathematical idea ever invented. It is a core subject in any mathematics degree, and also one which many students find challenging. A Sequential Introduction to Real Analysis gives a fresh take on real analysis by formulating all the underlying concepts in terms of convergence of sequences. The result is a coherent, mathematically rigorous, but conceptually simple development of the standard theory of differential and integral calculus ideally suited to undergraduate students learning real analysis for the first time.This book can be used as the basis of an undergraduate real analysis course, or used as further reading material to give an alternative perspective within a conventional real analysis course.Table of ContentsBasic Properties of the Set or Real Numbers; Real Sequences; Limit Theorems; Subsequences; Series; Continuous Functions; Some Symbolic Logic; Limits of Functions; Differentiable Functions; Power Series; Integration; Logarithms and Irrational Powers; What are the Reals?;

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Book SynopsisAutomorphic forms on the upper half plane have been studied for a long time. Most attention has gone to the holomorphic automorphic forms, with numerous applications to number theory. Maass, [34], started a systematic study of real analytic automorphic forms. He extended Hecke’s relation between automorphic forms and Dirichlet series to real analytic automorphic forms. The names Selberg and Roelcke are connected to the spectral theory of real analytic automorphic forms, see, e. g. , [50], [51]. This culminates in the trace formula of Selberg, see, e. g. , Hejhal, [21]. Automorphicformsarefunctionsontheupperhalfplanewithaspecialtra- formation behavior under a discontinuous group of non-euclidean motions in the upper half plane. One may ask how automorphic forms change if one perturbs this group of motions. This question is discussed by, e. g. , Hejhal, [22], and Phillips and Sarnak, [46]. Hejhal also discusses the e?ect of variation of the multiplier s- tem (a function on the discontinuous group that occurs in the description of the transformation behavior of automorphic forms). In [5]–[7] I considered variation of automorphic forms for the full modular group under perturbation of the m- tiplier system. A method based on ideas of Colin de Verdi` ere, [11], [12], gave the meromorphic continuation of Eisenstein and Poincar´ e series as functions of the eigenvalue and the multiplier system jointly. The present study arose from a plan to extend these results to much more general groups (discrete co?nite subgroups of SL (R)).Trade ReviewFrom reviews: "It is made abundantly clear that this viewpoint, of families of automorphic functions depending on varying eigenvalue and multiplier systems, is both deep and fruitful." - MathSciNetTable of ContentsModular introduction.- Modular introduction.- General theory.- Automorphic forms on the universal covering group.- Discrete subgroups.- Automorphic forms.- Poincaré series.- Selfadjoint extension of the Casimir operator.- Families of automorphic forms.- Transformation and truncation.- Pseudo Casimir operator.- Meromorphic continuation of Poincaré series.- Poincaré families along vertical lines.- Singularities of Poincaré families.- Examples.- Automorphic forms for the modular group.- Automorphic forms for the theta group.- Automorphic forms for the commutator subgroup.

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Book SynopsisWhile most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself.By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics.Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.Trade Review“This is a book of both analysis and set theory, and the analysis begins at an elementary level with the necessary treatment of completeness of the reals. … the analysis makes it valuable to the serious student, say a senior or first-year graduate student. … Stillwell’s book can work well as a text for the course in foundations, with its good treatment of the cardinals and ordinals. … This enjoyable book makes the connection clear.” (James M. Cargal, The UMAP Journal, Vol. 38 (1), 2017)“This book is an interesting introduction to set theory and real analysis embedded in properties of the real numbers. … The 300-plus problems are frequently challenging and will interest both upper-level undergraduate students and readers with a strong mathematical background. … A list of approximately 100 references at the end of the book will help students to further explore the topic. … Summing Up: Recommended. Lower-division undergraduates.” (D. P. Turner, Choice, Vol. 51 (11), August, 2014)“This is an informal look at the nature of the real numbers … . There are extensive historical notes about the evolution of real analysis and our understanding of real numbers. … Stillwell has deliberately set out to provide a different sort of construction where you understand what the foundation is supporting and why it is important. I think this is very successful, and his book … is much more informative and enjoyable.” (Allen Stenger, MAA Reviews, February, 2014)“This book will be fully appreciated by either professional mathematicians or those students, who already have passed a course in analysis or set theory. … The book contains a quantity of motivation examples, worked examples and exercises, what makes it suitable also for self-study.” (Vladimír Janiš, zbMATH, 2014)“The book offers a rigorous foundation of the real number system. It is intended for senior undergraduates who have already studied calculus, but a wide range of readers will find something interesting, new, or instructive in it. … This is an extremely reader-friendly book. It is full of interesting examples, very clear explanations, historical background, applications. Each new idea comes after proper motivation.” (László Imre Szabó, Acta Scientiarum Mathematicarum (Szeged), Vol. 80 (1-2), 2014)Table of ContentsThe Fundamental Questions.- From Discrete to Continuous.- Infinite Sets.- Functions and Limits.- Open Sets and Continuity.- Ordinals.- The Axiom of Choice.- Borel Sets.- Measure Theory.- Reflections.- Bibliography.- Index.

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Book SynopsisThis book explains the notion of Brakke’s mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 ≤ k < n). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke’s mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke’s existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard’s regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.

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Book SynopsisThis book aims to establish a foundation for fractional derivatives and fractional differential equations. The theory of fractional derivatives enables considering any positive order of differentiation. The history of research in this field is very long, with its origins dating back to Leibniz. Since then, many great mathematicians, such as Abel, have made contributions that cover not only theoretical aspects but also physical applications of fractional calculus. The fractional partial differential equations govern phenomena depending both on spatial and time variables and require more subtle treatments. Moreover, fractional partial differential equations are highly demanded model equations for solving real-world problems such as the anomalous diffusion in heterogeneous media. The studies of fractional partial differential equations have continued to expand explosively. However we observe that available mathematical theory for fractional partial differential equations is not still complete. In particular, operator-theoretical approaches are indispensable for some generalized categories of solutions such as weak solutions, but feasible operator-theoretic foundations for wide applications are not available in monographs.To make this monograph more readable, we are restricting it to a few fundamental types of time-fractional partial differential equations, forgoing many other important and exciting topics such as stability for nonlinear problems. However, we believe that this book works well as an introduction to mathematical research in such vast fields.Trade Review“The book is written nicely and useful as an introductory book on time fractional derivatives in abstract spaces.” (Syed Abbas, zbMATH 1485.34002, 2022)Table of Contents

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Book SynopsisUseful as a text for students and a reference for the more advanced mathematician, this book presents a unified treatment of that part of measure theory most useful for its application in modern analysis. Coverage includes sets and classes, measures and outer measures, Haar measure and measure and topology in groups.Trade ReviewP.R. Halmos Measure Theory "As with the first edition, this considerably improved volume will serve the interested student to find his way to active and creative work in the field of Hilbert space theory."—MATHEMATICAL REVIEWSTable of ContentsPreface; 0. Prerequisites; 1. Sets and Classes; 2. Measures and Outer Measures; 3. Extension of Measures; 4. Measurable Functions; 5. Integration; 6. General Set Functions; 7. Product Spaces; 8. Transformations and Functions; 9. Probability; 10. Locally Compact Spaces; 11. Haar Measure; 12. Measure and Topology in Groups; References; Bibliography; List of Frequently Used Symbols; Index.

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Book SynopsisOne of the ways in which topology has influenced other branches of mathematics in the past few decades is by putting the study of continuity and convergence into a general setting. This new edition of Wilson Sutherland''s classic text introduces metric and topological spaces by describing some of that influence. The aim is to move gradually from familiar real analysis to abstract topological spaces, using metric spaces as a bridge between the two. The language of metric and topological spaces is established with continuity as the motivating concept. Several concepts are introduced, first in metric spaces and then repeated for topological spaces, to help convey familiarity. The discussion develops to cover connectedness, compactness and completeness, a trio widely used in the rest of mathematics. Topology also has a more geometric aspect which is familiar in popular expositions of the subject as `rubber-sheet geometry'', with pictures of Möbius bands, doughnuts, Klein bottles and the liTrade ReviewThe first Edition of this work was highly praised ... It is reassuring to note that the Second Edition is equally impressive. The changes that have been made have only served to enhance the book. * Rob Ashmore, Mathematics today *The presentation, description and explanation throughout the seventeen short chapters are excellent, and the text can be described as self-contained, with many suitably chosen examples and exercises ,.. An interesting innovation for the new edition is having a companion web site in which more useful and relevant materials can be found. * Peter Shiu, The Mathematical Gazette *Table of ContentsPREFACE; REFERENCES; INDEX

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Book SynopsisFractal Examples.- Metric Topology.- Topological Dimension.- Self-Similarity.- Measure Theory.- Fractal Dimension.- Additional Topics.Trade ReviewFrom the reviews of the second edition: "As a non-specialist, I found this book very helpful. It gave me a better understanding of the nature of fractals, and of the technical issues involved in the theory. I think it will be valuable as a textbook for undergraduate students in mathematics, and also for researchers wanting to learn fractal geometry from scratch. The material is well-organized and the proofs are clear; the abundance of examples is an asset for a book on measure theory and topology." (Fabio Mainardi, MathDL, February, 2008) "This is the second edition of a well-known textbook in the field … . The book may serve as a textbook for a one-semester (advanced) undergraduate course in mathematics. … the book is also suitable for readers interested in theoretical fractal geometry coming from other disciplines (e.g. physics, computer sciences) with a basic knowledge of mathematics. The presentation of the material is appealing … and the style is clear and motivating. … the book will remain as a standard reference in the field." (José-Manuel Rey, Zentralblatt MATH, Vol. 1152, 2009)Table of ContentsFractal Examples.- Metric Topology.- Topological Dimension.- Self-Similarity.- Measure Theory.- Fractal Dimension.- Additional Topics.

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Book SynopsisOrientation Quizzes.- R Review of Fundamentals.- R.1 Basic Algebra: Real Numbers and Inequalities.- R.2 Intervals and Absolute Values.- R.3 Laws of Exponents.- R.4 Straight Lines.- R.5 Circles and Parabolas.- R.6 Functions and Graphs.- 1 Derivatives and Limits.- 1.1 Introduction to the Derivative.- 1.2 Limits.- 1.3 The Derivative as a Limit and the Leibniz Notation.- 1.4 Differentiating Polynomials.- 1.5 Products and Quotients.- 1.6 The Linear Approximation and Tangent Lines.- 2 Rates of Change and the Chain Rule.- 2.1 Rates of Change and the Second Derivative.- 2.2 The Chain Rule.- 2.3 Fractional Powers and Implicit Differentiation.- 2.4 Related Rates and Parametric Curves.- 2.5 Antiderivatives.- 3 Graphing and MaximumMinimum Problems.- 3.1 Continuity and the Intermediate Value Theorem.- 3.2 Increasing and Decreasing Functions.- 3.3 The Second Derivative and Concavity.- 3.4 Drawing Graphs.- 3.5 MaximumMinimum Problems.- 3.6 The Mean Value Theorem.- 4 The Integral.- 4.1 Summation.- 4.2Table of ContentsOrientation Quizzes.- R Review of Fundamentals.- R.1 Basic Algebra: Real Numbers and Inequalities.- R.2 Intervals and Absolute Values.- R.3 Laws of Exponents.- R.4 Straight Lines.- R.5 Circles and Parabolas.- R.6 Functions and Graphs.- 1 Derivatives and Limits.- 1.1 Introduction to the Derivative.- 1.2 Limits.- 1.3 The Derivative as a Limit and the Leibniz Notation.- 1.4 Differentiating Polynomials.- 1.5 Products and Quotients.- 1.6 The Linear Approximation and Tangent Lines.- 2 Rates of Change and the Chain Rule.- 2.1 Rates of Change and the Second Derivative.- 2.2 The Chain Rule.- 2.3 Fractional Powers and Implicit Differentiation.- 2.4 Related Rates and Parametric Curves.- 2.5 Antiderivatives.- 3 Graphing and Maximum—Minimum Problems.- 3.1 Continuity and the Intermediate Value Theorem.- 3.2 Increasing and Decreasing Functions.- 3.3 The Second Derivative and Concavity.- 3.4 Drawing Graphs.- 3.5 Maximum—Minimum Problems.- 3.6 The Mean Value Theorem.- 4 The Integral.- 4.1 Summation.- 4.2 Sums and Areas.- 4.3 The Definition of the Integral.- 4.4 The Fundamental Theorem of Calculus.- 4.5 Definite and Indefinite Integrals.- 4.6 Applications of the Integral.- 5 Trigonometric Functions.- 5.1 Polar Coordinates and Trigonometry.- 5.2 Differentiation of the Trigonometric Functions.- 5.3 Inverse Functions.- 5.4 The Inverse Trigonometric Functions.- 5.5 Graphing and Word Problems.- 5.6 Graphing in Polar Coordinates.- 6 Exponentials and Logarithms.- 6.1 Exponential Functions.- 6.2 Logarithms.- 6.3 Differentiation of the Exponential and Logarithmic Functions.- 6.4 Graphing and Word Problems.- Answers A.1.- Index I.1.

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Book Synopsis7 Basic Methods of Integration.- 7.1 Calculating Integrals.- 7.2 Integration by Substitution.- 7.3 Changing Variables in the Definite Integral.- 7.4 Integration by Parts.- 8 Differential Equations.- 8.1 Oscillations.- 8.2 Growth and Decay.- 8.3 The Hyperbolic Functions.- 8.4 The Inverse Hyperbolic Functions.- 8.5 Separable Differential Equations.- 8.6 Linear First-Order Equations.- 9 Applications of Integration.- 9.1 Volumes by the Slice Method.- 9.2 Volumes by the Shell Method.- 9.3 Average Values and the Mean Value Theorem for Integrals.- 9.4 Center of Mass.- 9.5 Energy, Power, and Work.- 10 Further Techniques and Applications of Integration.- 10.1 Trigonometric Integrals.- 10.2 Partial Fractions.- 10.3 Arc Length and Surface Area.- 10.4 Parametric Curves.- 10.5 Length and Area in Polar Coordinates.- 11 Limits, L'Hôpital's Rule, and Numerical Methods.- 11.1 Limits of Functions.- 11.2 L'Hôpital's Rule.- 11.3 Improper Integrals.- 11.4 Limits of Sequences and Newton's Method.- 11.5 NumeTable of Contents7 Basic Methods of Integration.- 7.1 Calculating Integrals.- 7.2 Integration by Substitution.- 7.3 Changing Variables in the Definite Integral.- 7.4 Integration by Parts.- 8 Differential Equations.- 8.1 Oscillations.- 8.2 Growth and Decay.- 8.3 The Hyperbolic Functions.- 8.4 The Inverse Hyperbolic Functions.- 8.5 Separable Differential Equations.- 8.6 Linear First-Order Equations.- 9 Applications of Integration.- 9.1 Volumes by the Slice Method.- 9.2 Volumes by the Shell Method.- 9.3 Average Values and the Mean Value Theorem for Integrals.- 9.4 Center of Mass.- 9.5 Energy, Power, and Work.- 10 Further Techniques and Applications of Integration.- 10.1 Trigonometric Integrals.- 10.2 Partial Fractions.- 10.3 Arc Length and Surface Area.- 10.4 Parametric Curves.- 10.5 Length and Area in Polar Coordinates.- 11 Limits, L’Hôpital’s Rule, and Numerical Methods.- 11.1 Limits of Functions.- 11.2 L’Hôpital’s Rule.- 11.3 Improper Integrals.- 11.4 Limits of Sequences and Newton’s Method.- 11.5 Numerical Integration.- 12 Infinite Series.- 12.1 The Sum of an Infinite Series.- 12.2 The Comparison Test and Alternating Series.- 12.3 The Integral and Ratio Tests.- 12.4 Power Series.- 12.5 Taylor’s Formula.- 12.6 Complex Numbers.- 12.7 Second-Order Linear Differential Equations.- 12.8 Series Solutions of Differential Equations.- Answers.

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Book Synopsis13 Vectors.- 13.1 Vectors in the Plane.- 13.2 Vectors in Space.- 13.3 Lines and Distance.- 13.4 The Dot Product.- 13.5 The Cross Product.- 13.6 Matrices and Determinants.- 14 Curves and Surfaces.- 14.1 The Conic Sections.- 14.2 Translation and Rotation of Axes.- 14.3 Functions, Graphs, and Level Surfaces.- 14.4 Quadric Surfaces.- 14.5 Cylindrical and Spherical Coordinates.- 14.6 Curves in Space.- 14.7 The Geometry and Physics of Space Curves.- 15 Partial Differentiation.- 15.1 Introduction to Partial Derivatives.- 15.2 Linear Approximations and Tangent Planes.- 15.3 The Chain Rule.- 15.4 Matrix Multiplication and the Chain Rule.- 16 Gradients, Maxima, and Minima.- 16.1 Gradients and Directional Derivatives.- 16.2 Gradients, Level Surfaces, and Implicit Differentiation.- 16.3 Maxima and Minima.- 16.4 Constrained Extrema and Lagrange Multipliers.- 17 Multiple Integration.- 17.1 The Double Integral and Iterated Integral.- 17.2 The Double Integral Over General Regions.- 17.3 Applications oTable of Contents13 Vectors.- 13.1 Vectors in the Plane.- 13.2 Vectors in Space.- 13.3 Lines and Distance.- 13.4 The Dot Product.- 13.5 The Cross Product.- 13.6 Matrices and Determinants.- 14 Curves and Surfaces.- 14.1 The Conic Sections.- 14.2 Translation and Rotation of Axes.- 14.3 Functions, Graphs, and Level Surfaces.- 14.4 Quadric Surfaces.- 14.5 Cylindrical and Spherical Coordinates.- 14.6 Curves in Space.- 14.7 The Geometry and Physics of Space Curves.- 15 Partial Differentiation.- 15.1 Introduction to Partial Derivatives.- 15.2 Linear Approximations and Tangent Planes.- 15.3 The Chain Rule.- 15.4 Matrix Multiplication and the Chain Rule.- 16 Gradients, Maxima, and Minima.- 16.1 Gradients and Directional Derivatives.- 16.2 Gradients, Level Surfaces, and Implicit Differentiation.- 16.3 Maxima and Minima.- 16.4 Constrained Extrema and Lagrange Multipliers.- 17 Multiple Integration.- 17.1 The Double Integral and Iterated Integral.- 17.2 The Double Integral Over General Regions.- 17.3 Applications of the Double Integral.- 17.4 Triple Integrals.- 17.5 Integrals in Polar, Cylindrical, and Spherical Coordinates.- 17.6 Applications of Triple Integrals.- 18 Vector Analysis.- 18.1 Line Integrals.- 18.2 Path Independence.- 18.3 Exact Differentials.- 18.4 Green’s Theorem.- 18.5 Circulation and Stokes’ Theorem.- 18.6 Flux and the Divergence Theorem.- Answers.

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Book Synopsis1 Real Numbers.- 1.1 Sets, Relations, Functions.- 1.2 Numbers.- 1.3 Infinite Sets.- 1.4 Incommensurability.- 1.5 Ordered Fields.- 1.6 Functions on R.- 1.7 Intervals in R.- 1.8 Algebraic and Transcendental Numbers.- 1.9 Existence of R.- 1.10 Exercises.- 1.11 Notes.- 2 Sequences and Series.- 2.1 Sequences.- 2.2 Continued Fractions.- 2.3 Infinite Series.- 2.4 Rearrangements of Series.- 2.5 Unordered Series.- 2.6 Exercises.- 2.7 Notes.- 3 Continuous Functions on Intervals.- 3.1 Limits and Continuity.- 3.2 Two Fundamental Theorems.- 3.3 Uniform Continuity.- 3.4 Sequences of Functions.- 3.5 The Exponential function.- 3.6 Trigonometric Functions.- 3.7 Exercises.- 3.8 Notes.- 4 Differentiation.- 4.1 Derivatives.- 4.2 Derivatives of Some Elementary Functions.- 4.3 Convex Functions.- 4.4 The Differential Calculus.- 4.5 L'Hospital's Rule.- 4.6 Higher Order Derivatives.- 4.7 Analytic Functions.- 4.8 Exercises.- 4.9 Notes.- 5 The Riemann Integral.- 5.1 Riemann Sums.- 5.2 Existence Results.- 5.3 ProTrade ReviewThis is a very good textbook presenting a modern course in analysis both at the advanced undergraduate and at the beginning graduate level. It contains 14 chapters, a bibliography, and an index. At the end of each chapter interesting exercises and historical notes are enclosed.\par From the cover: ``The book begins with a brief discussion of sets and mappings, describes the real number field, and proceeds to a treatment of real-valued functions of a real variable. Separate chapters are devoted to the ideas of convergent sequences and series, continuous functions, differentiation, and the Riemann integral (of a real-valued function defined on a compact interval). The middle chapters cover general topology and a miscellany of applications: the Weierstrass and Stone-Weierstrass approximation theorems, the existence of geodesics in compact metric spaces, elements of Fourier analysis, and the Weyl equidistribution theorem. Next comes a discussion of differentiation of vector-valued functions of several real variables, followed by a brief treatment of measure and integration (in a general setting, but with emphasis on Lebesgue theory in Euclidean spaces). The final part of the book deals with manifolds, differential forms, and Stokes' theorem [in the spirit of M. Spivak's: ``Calculus on manifolds'' (1965; Zbl 141.05403)] which is applied to prove Brouwer's fixed point theorem and to derive the basic properties of harmonic functions, such as the Dirichlet principle''. ZENTRALBLATT MATH A. Browder Mathematical Analysis An Introduction "Everything needed is clearly defined and formulated, and there is a reasonable number of examples…. Anyone teaching a year course at this level to should seriously consider this carefully written book. In the reviewer's opinion, it would be a real pleasure to use this text with such a class."—MATHEMATICAL REVIEWSTable of Contents1 Real Functions 2 Sequences and Series 3 Continuous Functions on Intervals 4 Differentiation 5 The Riemann Integral 6 Topology 7 Function Spaces 8 Differentiable Maps 9 Measures 10 Integration 11 Manifolds 12 Multilinear Algebra 13 Differential Forms 14 Integration on Manifolds

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Book Synopsis1 The Real Number System.- 1.1 Axioms for a Field.- 1.2 Natural Numbers and Sequences.- 1.3 Inequalities.- 1.4 Mathematical Induction.- 2 Continuity And Limits.- 2.1 Continuity.- 2.2 Limits.- 2.3 One-Sided Limits.- 2.4 Limits at Infinity; Infinite Limits.- 2.5 Limits of Sequences.- 3 Basic Properties of Functions on ?1.- 3.1 The Intermediate-Value Theorem.- 3.2 Least Upper Bound; Greatest Lower Bound.- 3.3 The BolzanoWeierstrass Theorem.- 3.4 The Boundedness and Extreme-Value Theorems.- 3.5 Uniform Continuity.- 3.6 The Cauchy Criterion.- 3.7 The Heine-Borel and Lebesgue Theorems.- 4 Elementary Theory of Differentiation.- 4.1 The Derivative in ?1.- 4.2 Inverse Functions in ?1.- 5 Elementary Theory of Integration.- 5.1 The Darboux Integral for Functions on ?1.- 5.2 The Riemann Integral.- 5.3 The Logarithm and Exponential Functions.- 5.4 Jordan Content and Area.- 6 Elementary Theory of Metric Spaces.- 6.1 The Schwarz and Triangle Inequalities; Metric Spaces.- 6.2 Elements of Point Set TopTable of Contents1: The Real Number System. 2: Continuity and Limits. 3: Basic Properties of Functions on R. 4: Elementary Theory of Differentiation. 5: Elementary Theory of Integration. 6: Elementary Theory of Metric Spaces. 7: Differentiation in R. 8: Integration in R. 9: Infinite Sequences and Infinite Series. 10: Fourier Series. 11: Functions Defined by Integrals; Improper Integrals. 12: The Riemann-Stieltjes Integral and Functions of Bounded Variation. 13: Contraction Mappings, Newton's Method, and Differential Equations. 14: Implicit Function Theorems and Lagrange Multipliers. 15: Functions on Metric Spaces; Approximation. 16: Vector Field Theory; the Theorems of Green and Stokes. Appendices.

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Book SynopsisContains the exercises and their solutions for Lang's second edition of "Undergraduate Analysis." The variety of exercises, which range from computational to more conceptual and which are of varying difficulty, cover several subjects. This volume also serves as an independent source for those interested in learning analysis or linear algebra.Table of Contents0 Sets and Mappings.- I Real Numbers.- II Limits and Continuous Functions.- III Differentiation.- IV Elementary Functions.- V The Elementary Real Integral.- VI Normed Vector Spaces.- VII Limits.- VIII Compactness.- IX Series.- X The Integral in One Variable.- XI Approximation with Convolutions.- XII Fourier Series.- XIII Improper Integrals.- XIV The Fourier Integral.- XV Functions on n-Space.- XVI The Winding Number and Global Potential Functions.- XVII Derivatives in Vector Spaces.- XVIII Inverse Mapping Theorem.- XIX Ordinary Differential Equations.- XX Multiple Integrals.- XXI Differential Forms.

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