Author: Asuman G. Aksoy
Publisher: Springer Science & Business Media
ISBN: 1441912967
Category : Mathematics
Languages : en
Pages : 257
Get Book Here
Book Description
Education is an admirable thing, but it is well to remember from time to time that nothing worth knowing can be taught. Oscar Wilde, “The Critic as Artist,” 1890. Analysis is a profound subject; it is neither easy to understand nor summarize. However, Real Analysis can be discovered by solving problems. This book aims to give independent students the opportunity to discover Real Analysis by themselves through problem solving. ThedepthandcomplexityofthetheoryofAnalysiscanbeappreciatedbytakingaglimpseatits developmental history. Although Analysis was conceived in the 17th century during the Scienti?c Revolution, it has taken nearly two hundred years to establish its theoretical basis. Kepler, Galileo, Descartes, Fermat, Newton and Leibniz were among those who contributed to its genesis. Deep conceptual changes in Analysis were brought about in the 19th century by Cauchy and Weierstrass. Furthermore, modern concepts such as open and closed sets were introduced in the 1900s. Today nearly every undergraduate mathematics program requires at least one semester of Real Analysis. Often, students consider this course to be the most challenging or even intimidating of all their mathematics major requirements. The primary goal of this book is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, we hope that learning analysis becomes less taxing and thereby more satisfying.
Author: Masayoshi Hata
Publisher: World Scientific Publishing Company
ISBN: 9813142847
Category : Mathematics
Languages : en
Pages : 376
Get Book Here
Book Description
This second edition introduces an additional set of new mathematical problems with their detailed solutions in real analysis. It also provides numerous improved solutions to the existing problems from the previous edition, and includes very useful tips and skills for the readers to master successfully. There are three more chapters that expand further on the topics of Bernoulli numbers, differential equations and metric spaces.Each chapter has a summary of basic points, in which some fundamental definitions and results are prepared. This also contains many brief historical comments for some significant mathematical results in real analysis together with many references.Problems and Solutions in Real Analysis can be treated as a collection of advanced exercises by undergraduate students during or after their courses of calculus and linear algebra. It is also instructive for graduate students who are interested in analytic number theory. Readers will also be able to completely grasp a simple and elementary proof of the Prime Number Theorem through several exercises. This volume is also suitable for non-experts who wish to understand mathematical analysis.
Author: Masayoshi Hata
Publisher: World Scientific
ISBN: 981277601X
Category : Mathematics
Languages : en
Pages : 303
Get Book Here
Book Description
This unique book provides a collection of more than 200 mathematical problems and their detailed solutions, which contain very useful tips and skills in real analysis. Each chapter has an introduction, in which some fundamental definitions and propositions are prepared. This also contains many brief historical comments on some significant mathematical results in real analysis together with useful references.Problems and Solutions in Real Analysis may be used as advanced exercises by undergraduate students during or after courses in calculus and linear algebra. It is also useful for graduate students who are interested in analytic number theory. Readers will also be able to completely grasp a simple and elementary proof of the prime number theorem through several exercises. The book is also suitable for non-experts who wish to understand mathematical analysis.
Author: Teodora-Liliana Radulescu
Publisher: Springer Science & Business Media
ISBN: 0387773797
Category : Mathematics
Languages : en
Pages : 462
Get Book Here
Book Description
Problems in Real Analysis: Advanced Calculus on the Real Axis features a comprehensive collection of challenging problems in mathematical analysis that aim to promote creative, non-standard techniques for solving problems. This self-contained text offers a host of new mathematical tools and strategies which develop a connection between analysis and other mathematical disciplines, such as physics and engineering. A broad view of mathematics is presented throughout; the text is excellent for the classroom or self-study. It is intended for undergraduate and graduate students in mathematics, as well as for researchers engaged in the interplay between applied analysis, mathematical physics, and numerical analysis.
Author: Rami Shakarchi
Publisher: Springer Science & Business Media
ISBN: 1461217385
Category : Mathematics
Languages : en
Pages : 369
Get Book Here
Book Description
The present volume contains all the exercises and their solutions for Lang's second edition of Undergraduate Analysis. The wide variety of exercises, which range from computational to more conceptual and which are of vary ing difficulty, cover the following subjects and more: real numbers, limits, continuous functions, differentiation and elementary integration, normed vector spaces, compactness, series, integration in one variable, improper integrals, convolutions, Fourier series and the Fourier integral, functions in n-space, derivatives in vector spaces, the inverse and implicit mapping theorem, ordinary differential equations, multiple integrals, and differential forms. My objective is to offer those learning and teaching analysis at the undergraduate level a large number of completed exercises and I hope that this book, which contains over 600 exercises covering the topics mentioned above, will achieve my goal. The exercises are an integral part of Lang's book and I encourage the reader to work through all of them. In some cases, the problems in the beginning chapters are used in later ones, for example, in Chapter IV when one constructs-bump functions, which are used to smooth out singulari ties, and prove that the space of functions is dense in the space of regu lated maps. The numbering of the problems is as follows. Exercise IX. 5. 7 indicates Exercise 7, §5, of Chapter IX. Acknowledgments I am grateful to Serge Lang for his help and enthusiasm in this project, as well as for teaching me mathematics (and much more) with so much generosity and patience.
Author: B. Gelbaum
Publisher: Springer Science & Business Media
ISBN: 1461576792
Category : Mathematics
Languages : en
Pages : 232
Get Book Here
Book Description
These problems and solutions are offered to students of mathematics who have learned real analysis, measure theory, elementary topology and some theory of topological vector spaces. The current widely used texts in these subjects provide the background for the understanding of the problems and the finding of their solutions. In the bibliography the reader will find listed a number of books from which the necessary working vocabulary and techniques can be acquired. Thus it is assumed that terms such as topological space, u-ring, metric, measurable, homeomorphism, etc., and groups of symbols such as AnB, x EX, f: IR 3 X 1-+ X 2 - 1, etc., are familiar to the reader. They are used without introductory definition or explanation. Nevertheless, the index provides definitions of some terms and symbols that might prove puzzling. Most terms and symbols peculiar to the book are explained in the various introductory paragraphs titled Conventions. Occasionally definitions and symbols are introduced and explained within statements of problems or solutions. Although some solutions are complete, others are designed to be sketchy and thereby to give their readers an opportunity to exercise their skill and imagination. Numbers written in boldface inside square brackets refer to the bib liography. I should like to thank Professor P. R. Halmos for the opportunity to discuss with him a variety of technical, stylistic, and mathematical questions that arose in the writing of this book. Buffalo, NY B.R.G.
Author: Charambolos D. Aliprantis
Publisher:
ISBN:
Category :
Languages : en
Pages : 463
Get Book Here
Book Description
Author: M. G. Goluzina
Publisher: American Mathematical Soc.
ISBN: 9780821897386
Category : Mathematics
Languages : en
Pages : 386
Get Book Here
Book Description
This book is intended for students wishing to deepen their knowledge of mathematical analysis and for those teaching courses in this area. It differs from other problem books in the greater difficulty of the problems, some of which are well-known theorems in analysis. Nonetheless, no special preparation is required to solve the majority of the problems. Brief but detailed solutions to most of the problems are given in the second part of the book. This book is unique in that the authors have aimed to systematize a range of problems that are found in sources that are almost inaccessible (especially to students) and in mathematical folklore.
Author: Charles Chapman Pugh
Publisher: Springer Science & Business Media
ISBN: 0387216847
Category : Mathematics
Languages : en
Pages : 445
Get Book Here
Book Description
Was plane geometry your favourite math course in high school? Did you like proving theorems? Are you sick of memorising integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician. In this new introduction to undergraduate real analysis the author takes a different approach from past studies of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians like Dieudonne, Littlewood and Osserman. The author has taught the subject many times over the last 35 years at Berkeley and this book is based on the honours version of this course. The book contains an excellent selection of more than 500 exercises.
Author: Donald R. Sherbert
Publisher:
ISBN:
Category :
Languages : en
Pages : 417
Get Book Here
Book Description
Introduction to Real Analysis, Fourth Edition by Robert G. BartleDonald R. Sherbert The first three editions were very well received and this edition maintains the samespirit and user-friendly approach as earlier editions. Every section has been examined.Some sections have been revised, new examples and exercises have been added, and a newsection on the Darboux approach to the integral has been added to Chapter 7. There is morematerial than can be covered in a semester and instructors will need to make selections andperhaps use certain topics as honors or extra credit projects.To provide some help for students in analyzing proofs of theorems, there is anappendix on ''Logic and Proofs'' that discusses topics such as implications, negations,contrapositives, and different types of proofs. However, it is a more useful experience tolearn how to construct proofs by first watching and then doing than by reading abouttechniques of proof.Results and proofs are given at a medium level of generality. For instance, continuousfunctions on closed, bounded intervals are studied in detail, but the proofs can be readilyadapted to a more general situation. This approach is used to advantage in Chapter 11where topological concepts are discussed. There are a large number of examples toillustrate the concepts, and extensive lists of exercises to challenge students and to aid themin understanding the significance of the theorems.Chapter 1 has a brief summary of the notions and notations for sets and functions thatwill be used. A discussion of Mathematical Induction is given, since inductive proofs arisefrequently. There is also a section on finite, countable and infinite sets. This chapter canused to provide some practice in proofs, or covered quickly, or used as background materialand returning later as necessary.Chapter 2 presents the properties of the real number system. The first two sections dealwith Algebraic and Order properties, and the crucial Completeness Property is given inSection 2.3 as the Supremum Property. Its ramifications are discussed throughout theremainder of the chapter.In Chapter 3, a thorough treatment of sequences is given, along with the associatedlimit concepts. The material is of the greatest importance. Students find it rather naturalthough it takes time for them to become accustomed to the use of epsilon. A briefintroduction to Infinite Series is given in Section 3.7, with more advanced materialpresented in Chapter 9 Chapter 4 on limits of functions and Chapter 5 on continuous functions constitute theheart of the book. The discussion of limits and continuity relies heavily on the use ofsequences, and the closely parallel approach of these chapters reinforces the understandingof these essential topics. The fundamental properties of continuous functions on intervalsare discussed in Sections 5.3 and 5.4. The notion of a gauge is introduced in Section 5.5 andused to give alternate proofs of these theorems. Monotone functions are discussed inSection 5.6.The basic theory of the derivative is given in the first part of Chapter 6. This material isstandard, except a result of Caratheodory is used to give simpler proofs of the Chain Ruleand the Inversion Theorem. The remainder of the chapter consists of applications of theMean Value Theorem and may be explored as time permits.In Chapter 7, the Riemann integral is defined in Section 7.1 as a limit of Riemannsums. This has the advantage that it is consistent with the students' first exposure to theintegral in calculus, and since it is not dependent on order properties, it permits immediategeneralization to complex- and vector-values functions that students may encounter in latercourses. It is also consistent with the generalized Riemann integral that is discussed inChapter 10. Sections 7.2 and 7.3 develop properties of the integral and establish theFundamental Theorem and many more
