How to Think about Analysis

Author: Lara Alcock
Publisher: Oxford University Press, USA
ISBN: 9780198723530
Release Date: 2014
Genre: Mathematics

Analysis is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared.

How to Think About Analysis

Author: Lara Alcock
Publisher: OUP Oxford
ISBN: 9780191035371
Release Date: 2014-09-25
Genre: Mathematics

Analysis (sometimes called Real Analysis or Advanced Calculus) is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared. It is not like other Analysis books. It is not a textbook containing standard content. Rather, it is designed to be read before arriving at university and/or before starting an Analysis course, or as a companion text once a course is begun. It provides a friendly and readable introduction to the subject by building on the student's existing understanding of six key topics: sequences, series, continuity, differentiability, integrability and the real numbers. It explains how mathematicians develop and use sophisticated formal versions of these ideas, and provides a detailed introduction to the central definitions, theorems and proofs, pointing out typical areas of difficulty and confusion and explaining how to overcome these. The book also provides study advice focused on the skills that students need if they are to build on this introduction and learn successfully in their own Analysis courses: it explains how to understand definitions, theorems and proofs by relating them to examples and diagrams, how to think productively about proofs, and how theories are taught in lectures and books on advanced mathematics. It also offers practical guidance on strategies for effective study planning. The advice throughout is research based and is presented in an engaging style that will be accessible to students who are new to advanced abstract mathematics.

How to Think About Analysis

Author: Lara Alcock
Publisher: OUP Oxford
ISBN: 9780191035388
Release Date: 2014-09-25
Genre: Mathematics

Analysis (sometimes called Real Analysis or Advanced Calculus) is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared. It is not like other Analysis books. It is not a textbook containing standard content. Rather, it is designed to be read before arriving at university and/or before starting an Analysis course, or as a companion text once a course is begun. It provides a friendly and readable introduction to the subject by building on the student's existing understanding of six key topics: sequences, series, continuity, differentiability, integrability and the real numbers. It explains how mathematicians develop and use sophisticated formal versions of these ideas, and provides a detailed introduction to the central definitions, theorems and proofs, pointing out typical areas of difficulty and confusion and explaining how to overcome these. The book also provides study advice focused on the skills that students need if they are to build on this introduction and learn successfully in their own Analysis courses: it explains how to understand definitions, theorems and proofs by relating them to examples and diagrams, how to think productively about proofs, and how theories are taught in lectures and books on advanced mathematics. It also offers practical guidance on strategies for effective study planning. The advice throughout is research based and is presented in an engaging style that will be accessible to students who are new to advanced abstract mathematics.

How to Study as a Mathematics Major

Author: Lara Alcock
Publisher: OUP Oxford
ISBN: 9780191637353
Release Date: 2013-01-10
Genre: Mathematics

Every year, thousands of students in the USA declare mathematics as their major. Many are extremely intelligent and hardworking. However, even the best will encounter challenges, because upper-level mathematics involves not only independent study and learning from lectures, but also a fundamental shift from calculation to proof. This shift is demanding but it need not be mysterious — research has revealed many insights into the mathematical thinking required, and this book translates these into practical advice for a student audience. It covers every aspect of studying as a mathematics major, from tackling abstract intellectual challenges to interacting with professors and making good use of study time. Part 1 discusses the nature of upper-level mathematics, and explains how students can adapt and extend their existing skills in order to develop good understanding. Part 2 covers study skills as these relate to mathematics, and suggests practical approaches to learning effectively while enjoying undergraduate life. As the first mathematics-specific study guide, this friendly, practical text is essential reading for any mathematics major.

How to Think Like a Mathematician

Author: Kevin Houston
Publisher: Cambridge University Press
ISBN: 1139477056
Release Date: 2009-02-12
Genre: Mathematics

Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.

Intelligence Analysis How to Think in Complex Environments

Author: Wayne Michael Hall
Publisher: ABC-CLIO
ISBN: 9780313382666
Release Date: 2009-12-22
Genre: Political Science

This book offers a vast conceptual and theoretical exploration of the ways intelligence analysis must change in order to succeed against today's most dangerous combatants and most complex irregular theatres of conflict. • Includes quotations from a wide range of acclaimed thinkers • Offers an extensive bibliography of works cited and resources for further reading • Presents a comprehensive index

Introduction to Real Analysis

Author: Michael J. Schramm
Publisher: Courier Corporation
ISBN: 9780486131924
Release Date: 2012-05-11
Genre: Mathematics

This text forms a bridge between courses in calculus and real analysis. Suitable for advanced undergraduates and graduate students, it focuses on the construction of mathematical proofs. 1996 edition.

The Path

Author: Michael Puett
Publisher: Simon and Schuster
ISBN: 9781476777832
Release Date: 2016-04-05
Genre: Body, Mind & Spirit

For the first time an award-winning Harvard professor shares the lessons from his wildly popular course on classical Chinese philosophy, showing you how these ancient ideas can guide you on the path to a good life today. The lessons taught by ancient Chinese philosophers surprisingly still apply, and they challenge our fundamental assumptions about how to lead a fulfilled, happy, and successful life. Self-discovery, it turns out, comes through looking outward, not inward. Power comes from holding back. Good relationships come from small gestures. Spontaneity comes from practice. And excellence comes from what you choose to do, not your “natural” abilities. Counterintuitive. Countercultural. Even revolutionary. These powerful ideas have made Professor Michael Puett's course the third most popular at Harvard University in recent years, with enrollment surging every year since it was first offered in 2006. It's clear students are drawn by a bold promise Professor Puett makes on the first day of class: “These ideas will change your life.” Now he offers his course to the world.

Numbers and Functions

Author: R. P. Burn
Publisher: Cambridge University Press
ISBN: 9781107444539
Release Date: 2015-02-19
Genre: Mathematics

Revised and updated edition. Hundreds of exercises to help students gradually transition from school to university-level calculus.

Introduction to Analysis

Author: Maxwell Rosenlicht
Publisher: Courier Corporation
ISBN: 9780486134680
Release Date: 2012-05-04
Genre: Mathematics

Written for junior and senior undergraduates, this remarkably clear and accessible treatment covers set theory, the real number system, metric spaces, continuous functions, Riemann integration, multiple integrals, and more. 1968 edition.

A Problem Book in Real Analysis

Author: Asuman G. Aksoy
Publisher: Springer Science & Business Media
ISBN: 9781441912961
Release Date: 2010-03-10
Genre: Mathematics

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.

Counterexamples in Analysis

Author: Bernard R. Gelbaum
Publisher: Courier Corporation
ISBN: 9780486134918
Release Date: 2012-07-12
Genre: Mathematics

These counterexamples deal mostly with the part of analysis known as "real variables." Covers the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, functions of 2 variables, plane sets, more. 1962 edition.

Yet Another Introduction to Analysis

Author: Victor Bryant
Publisher: Cambridge University Press
ISBN: 052138835X
Release Date: 1990-06-28
Genre: Mathematics

Mathematics education in schools has seen a revolution in recent years. Students everywhere expect the subject to be well-motivated, relevant and practical. When such students reach higher education, the traditional development of analysis, often divorced from the calculus they learned at school, seems highly inappropriate. Shouldn't every step in a first course in analysis arise naturally from the student's experience of functions and calculus in school? And shouldn't such a course take every opportunity to endorse and extend the student's basic knowledge of functions? In Yet Another Introduction to Analysis, the author steers a simple and well-motivated path through the central ideas of real analysis. Each concept is introduced only after its need has become clear and after it has already been used informally. Wherever appropriate, new ideas are related to common topics in math curricula and are used to extend the reader's understanding of those topics. In this book the readers are led carefully through every step in such a way that they will soon be predicting the next step for themselves. In this way students will not only understand analysis, but also enjoy it.

Elementary Analysis

Author: Kenneth A. Ross
Publisher: Springer Science & Business Media
ISBN: 9781461462712
Release Date: 2013-04-16
Genre: Mathematics

For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus. Proofs are given in full, and the large number of well-chosen examples and exercises range from routine to challenging. The second edition preserves the book’s clear and concise style, illuminating discussions, and simple, well-motivated proofs. New topics include material on the irrationality of pi, the Baire category theorem, Newton's method and the secant method, and continuous nowhere-differentiable functions.

Proofs from THE BOOK

Author: Martin Aigner
Publisher: Springer
ISBN: 9783662442050
Release Date: 2014-08-06
Genre: Mathematics

This revised and enlarged fifth edition features four new chapters, which contain highly original and delightful proofs for classics such as the spectral theorem from linear algebra, some more recent jewels like the non-existence of the Borromean rings and other surprises. From the Reviews "... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999 "... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..." LMS Newsletter, January 1999 "Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questio ns so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... " SIGACT News, December 2011.