This elementary presentation exposes readers to both the process of rigor and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim is to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Each chapter begins with the discussion of some motivating examples and concludes with a series of questions.
Understanding Real Analysis, Second Edition offers substantial coverage of foundational material and expands on the ideas of elementary calculus to develop a better understanding of crucial mathematical ideas. The text meets students at their current level and helps them develop a foundation in real analysis. The author brings definitions, proofs, examples and other mathematical tools together to show how they work to create unified theory. These helps students grasp the linguistic conventions of mathematics early in the text. The text allows the instructor to pace the course for students of different mathematical backgrounds. Key Features: Meets and aligns with various student backgrounds Pays explicit attention to basic formalities and technical language Contains varied problems and exercises Drives the narrative through questions
Getting certified to teach high school mathematics typically requires completing a course in real analysis. Yet most teachers point out real analysis content bears little resemblance to secondary mathematics and report it does not influence their teaching in any significant way. This textbook is our attempt to change the narrative. It is our belief that analysis can be a meaningful part of a teacher's mathematical education and preparation for teaching. This book is a companion text. It is intended to be a supplemental resource, used in conjunction with a more traditional real analysis book. The textbook is based on our efforts to identify ways that studying real analysis can provide future teachers with genuine opportunities to think about teaching secondary mathematics. It focuses on how mathematical ideas are connected to the practice of teaching secondary mathematics–and not just the content of secondary mathematics itself. Discussions around pedagogy are premised on the belief that the way mathematicians do mathematics can be useful for how we think about teaching mathematics. The book uses particular situations in teaching to make explicit ways that the content of real analysis might be important for teaching secondary mathematics, and how mathematical practices prevalent in the study of real analysis can be incorporated as practices for teaching. This textbook will be of particular interest to mathematics instructors–and mathematics teacher educators–thinking about how the mathematics of real analysis might be applicable to secondary teaching, as well as to any prospective (or current) teacher who has wondered about what the purpose of taking such courses could be.
This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.
This lively, problem-oriented text, first published in 2004, is designed to coach readers toward mastery of the most fundamental mathematical inequalities. With the Cauchy-Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered - either by one of history's famous mathematicians or by the reader. The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of Hölder, Hilbert, and Hardy. The text is accessible to anyone who knows calculus and who cares about solving problems. It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics.
Designed for courses in advanced calculus and introductory real analysis, Elementary Classical Analysis strikes a careful balance between pure and applied mathematics with an emphasis on specific techniques important to classical analysis without vector calculus or complex analysis. Intended for students of engineering and physical science as well as of pure mathematics.
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.
This book shows that it is possible to provide a fully rigorous treatment of calculus for those planning a career in an area that uses mathematics regularly (e.g., statistics, mathematics, economics, finance, engineering, etc.). It reveals to students on the ways to approach and understand mathematics. It covers efficiently and rigorously the differential and integral calculus, and its foundations in mathematical analysis. It also aims at a comprehensive, efficient, and rigorous treatment by introducing all the concepts succinctly. Experience has shown that this approach, which treats understanding on par with technical ability, has long term benefits for students.
′...effectively fills a long-standing void and will no doubt be hailed as a much-needed new addition to the literature... This text very much exemplifies the strength of Ho-Won Jeong as a theorist and one of the more prolific writers in the larger peace and conflict studies field... the final three chapters on ′De-escalation Dynamics′ (which includes a brief section on third party intervention), on ′Conciliation Strategies,′ and especially the one on ′Ending Conflict,′ which provides a range of outcomes beyond the usual focus on third party intervention (read mediation) epitomizes the value of this new text′ - Journal of Peace Research ′...an awesome tour d′horizon of modern war, violence, and confrontation within and between nations. Illustrating via just about every conflict in every corner of the world, the author invokes an endless array of insights and interpretations, ranging from the micro to the macro, beautifully written in a seamless sequence of closely linked and discursive essays.′ - Professor J. David Singer, University of Michigan ′Ho-Won Jeong has written an illuminatinbg analysis of the dynamics of conflict. He lays out the tools we have to analyze conflict in a literate and comprehensive way. A valuable book for anyone interested in a more comprehensive understanding of conflict, its sources, and its deescalation and termination′ - Janice Gross Stein, Belzberg Professor of Conflict Management, Director, Munk Centre for International Studies, University of Toronto ′Jeong has successfully combined behavioral and structural analysis of the dynamics of social conflict. This volume covers the multiple dimensions - escalation, entrapment, de-escalation, termination, and resolution - both of violent and non-violent confrontation between adversaries, as well as the utility and limitations of external intervention. For students of the social sciences, it should serve as an excellent introduction to the complex realities of social conflict.′ - Milton Esman, John S. Knight Professor of International Studies, Emeritus, Cornell University By examining the dynamic forces which shape and re-shape major conflicts, this timely book provides students with the knowledge base needed to successfully study conflict sources, processes and transformations. Broad in focus, it addresses the multiple social, political and psychological features central to understanding conflict situations and behaviour. A range of both recent and historical examples (including the Arab-Israeli conflict, the ′War on Terrorism′, the Cold War, and the civil wars in Sudan, former Yugoslavia and Sri Lanka) are discussed, illustrating the application of concepts and theories essential to the analysis of inter-group, inter-state and intra-state conflict and conflict resolution in a wider context. Understanding Conflict and Conflict Analysis is key reading for students of international relations, peace and conflict studies, conflict resolution, international security and international law.
