How to Think about Analysis

Author: Lara Alcock

Publisher: Oxford University Press, USA

ISBN: 0198723539

Category: Mathematics

Page: 246

View: 4711

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: 0191035386

Category: Mathematics

Page: 272

View: 5617

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 Like a Mathematician

A Companion to Undergraduate Mathematics

Author: Kevin Houston

Publisher: Cambridge University Press

ISBN: 9781139477055

Category: Mathematics

Page: N.A

View: 8178

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.

How to Study as a Mathematics Major

Author: Lara Alcock

Publisher: OUP Oxford

ISBN: 0191637351

Category: Mathematics

Page: 288

View: 3141

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.

A First Course in Analysis

Author: Donald Yau

Publisher: World Scientific

ISBN: 9814417858

Category: Mathematics

Page: 195

View: 9292

This book is an introductory text on real analysis for undergraduate students. The prerequisite for this book is a solid background in freshman calculus in one variable. The intended audience of this book includes undergraduate mathematics majors and students from other disciplines who use real analysis. Since this book is aimed at students who do not have much prior experience with proofs, the pace is slower in earlier chapters than in later chapters. There are hundreds of exercises, and hints for some of them are included.

Mathematical Analysis

An Introduction

Author: Andrew Browder

Publisher: Springer Science & Business Media

ISBN: 1461207150

Category: Mathematics

Page: 335

View: 1139

Among the traditional purposes of such an introductory course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. To this extent, more than one proof is included for a theorem - where this is considered beneficial - so as to stimulate the students' reasoning for alternate approaches and ideas. The second half of this book, and consequently the second semester, covers differentiation and integration, as well as the connection between these concepts, as displayed in the general theorem of Stokes. Also included are some beautiful applications of this theory, such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions. Throughout, reference is made to earlier sections, so as to reinforce the main ideas by repetition. Unique in its applications to some topics not usually covered at this level.

Introduction to Real Analysis

Author: Michael J. Schramm

Publisher: Courier Corporation

ISBN: 0486131920

Category: Mathematics

Page: 384

View: 5997

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.

Intelligence Analysis: How to Think in Complex Environments

How to Think in Complex Environments

Author: Wayne Michael Hall,Gary Citrenbaum

Publisher: ABC-CLIO

ISBN: 0313382662

Category: Political Science

Page: 440

View: 7513

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

Mathematics Rebooted

A Fresh Approach to Understanding

Author: Lara Alcock

Publisher: Oxford University Press

ISBN: 0198803796

Category: Mathematics

Page: 232

View: 4058

This book is about mathematical thinking, learning and understanding. It is about ways in which good representations capture mathematical ideas, and about building broad and deep knowledge by understanding the links between those ideas. It draws on research in mathematics education and psychology to explain why some misunderstandings and confusions arise for almost everyone, and it describes ways to think about mathematical ideas correctly and with confidence.

Introduction to Analysis

Author: Maxwell Rosenlicht

Publisher: Courier Corporation

ISBN: 0486134687

Category: Mathematics

Page: 272

View: 9409

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.

Real Analysis

Author: John M. Howie

Publisher: Springer Science & Business Media

ISBN: 1447103416

Category: Mathematics

Page: 276

View: 9516

Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, the book covers all the key topics with fully worked examples and exercises with solutions. All the concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject. This book offers a fresh approach to a core subject and manages to provide a gentle and clear introduction without sacrificing rigour or accuracy.

Introductory Real Analysis

Author: A. N. Kolmogorov,S. V. Fomin

Publisher: Courier Corporation

ISBN: 0486134741

Category: Mathematics

Page: 416

View: 6927

Comprehensive, elementary introduction to real and functional analysis covers basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, more. 1970 edition.

Yet Another Introduction to Analysis

Author: Victor Bryant

Publisher: Cambridge University Press

ISBN: 1107717221

Category: Mathematics

Page: 298

View: 8291

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 rather divorced from the calculus which they learnt 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 at 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 the new ideas are related to school topics and are used to extend the reader's understanding of those topics. A first course in analysis at college is always regarded as one of the hardest in the curriculum. However, in this book the reader is led carefully through every step in such a way that he/she will soon be predicting the next step for him/herself. In this way the subject is developed naturally: students will end up not only understanding analysis, but also enjoying it.

A Problem Book in Real Analysis

Author: Asuman G. Aksoy,Mohamed A. Khamsi

Publisher: Springer Science & Business Media

ISBN: 1441912967

Category: Mathematics

Page: 254

View: 5860

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.

Understanding Analysis

Author: Stephen Abbott

Publisher: Springer

ISBN: 1493927124

Category: Mathematics

Page: 312

View: 1517

This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one. Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

Numbers and Functions

Steps into Analysis

Author: R. P. Burn

Publisher: Cambridge University Press

ISBN: 1316033783

Category: Mathematics

Page: N.A

View: 5496

The transition from studying calculus in schools to studying mathematical analysis at university is notoriously difficult. In this third edition of Numbers and Functions, Professor Burn invites the student reader to tackle each of the key concepts in turn, progressing from experience through a structured sequence of more than 800 problems to concepts, definitions and proofs of classical real analysis. The sequence of problems, of which most are supplied with brief answers, draws students into constructing definitions and theorems for themselves. This natural development is informed and complemented by historical insight. Carefully corrected and updated throughout, this new edition also includes extra questions on integration and an introduction to convergence. The novel approach to rigorous analysis offered here is designed to enable students to grow in confidence and skill and thus overcome the traditional difficulties.

Bridging the Gap to University Mathematics

Author: Edward Hurst,Martin Gould

Publisher: Springer Science & Business Media

ISBN: 9781848002906

Category: Mathematics

Page: 344

View: 7477

Helps to ease the transition between school/college and university mathematics by (re)introducing readers to a range of topics that they will meet in the first year of a degree course in the mathematical sciences, refreshing their knowledge of basic techniques and focussing on areas that are often perceived as the most challenging. Each chapter starts with a "Test Yourself" section so that readers can monitor their progress and readily identify areas where their understanding is incomplete. A range of exercises, complete with full solutions, makes the book ideal for self-study.

How to Think

A Survival Guide for a World at Odds

Author: Alan Jacobs

Publisher: Currency

ISBN: 0451499603

Category: Business & Economics

Page: 160

View: 2281

As a celebrated cultural critic and a writer for national publications like The Atlantic and Harper's, Alan Jacobs has spent his adult life belonging to communities that often clash in America's culture wars. And in his years of confronting the big issues that divide us--political, social, religious--Jacobs has learned that many of our fiercest disputes occur not because we're doomed to be divided, but because the people involved simply aren't thinking. Most of us don't want to think. Thinking is trouble. Thinking can force us out of familiar, comforting habits, and it can complicate our relationships with like-minded friends. Finally, thinking is slow, and that's a problem when our habits of consuming information (mostly online) leave us lost in the spin cycle of social media, partisan bickering, and confirmation bias. In this smart, endlessly entertaining book, Jacobs diagnoses the many forces that act on us to prevent thinking--forces that have only worsened in the age of Twitter, "alternative facts, " and information overload--and he also dispels the many myths we hold about what it means to think well. (For example: It'simpossible to "think for yourself.").

How to Think About Algorithms

Author: Jeff Edmonds

Publisher: Cambridge University Press

ISBN: 1139471759

Category: Computers

Page: N.A

View: 8926

This textbook, for second- or third-year students of computer science, presents insights, notations, and analogies to help them describe and think about algorithms like an expert, without grinding through lots of formal proof. Solutions to many problems are provided to let students check their progress, while class-tested PowerPoint slides are on the web for anyone running the course. By looking at both the big picture and easy step-by-step methods for developing algorithms, the author guides students around the common pitfalls. He stresses paradigms such as loop invariants and recursion to unify a huge range of algorithms into a few meta-algorithms. The book fosters a deeper understanding of how and why each algorithm works. These insights are presented in a careful and clear way, helping students to think abstractly and preparing them for creating their own innovative ways to solve problems.

Making Thinking Visible

How to Promote Engagement, Understanding, and Independence for All Learners

Author: Ron Ritchhart,Mark Church,Karin Morrison

Publisher: John Wiley & Sons

ISBN: 9781118015032

Category: Education

Page: 320

View: 4411

A proven program for enhancing students' thinking and comprehension abilities Visible Thinking is a research-based approach to teaching thinking, begun at Harvard's Project Zero, that develops students' thinking dispositions, while at the same time deepening their understanding of the topics they study. Rather than a set of fixed lessons, Visible Thinking is a varied collection of practices, including thinking routines?small sets of questions or a short sequence of steps?as well as the documentation of student thinking. Using this process thinking becomes visible as the students' different viewpoints are expressed, documented, discussed and reflected upon. Helps direct student thinking and structure classroom discussion Can be applied with students at all grade levels and in all content areas Includes easy-to-implement classroom strategies The book also comes with a DVD of video clips featuring Visible Thinking in practice in different classrooms.