What is Mathematical Analysis? PDF
Preview What is Mathematical Analysis?
WHAT IS MATHEMATICAL ANALYSIS? DIMENSIONS OF MATHEMATICS Whatls Mathematical Analysis? lohn Baylis Nottingham Polytechnic M MACMILLAN © John Baylis 1991 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting li mi ted copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London WIP 9HE. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. First published 1991 by MACMILLAN EDUCATION LTD Houndmills, Basingstoke, Hampshire RG21 2XS and London Companies and representatives throughout the world ISBN 978-0-333-54064-0 ISBN 978-1-349-12063-5 (eBook) DOI 10.1007/978-1-349-12063-5 A catalogue record for this book is available from the British Library. To Cathy CONTENTS Preface ix I NUMBERS, LINES AND HOLES I The aim of this chapter is to arrive at adefinition of the set of real numbers, the set on which calculus and all subsequent developments are based. We have a brief look at some historical background, and give a fairly detailed account of the shortcomings of the rational number system. Finally, the incompleteness of the ration als is formulated explicitly and the reals - modelIed by the set of infinite decimals - are shown to be complete 2 CURVES - CONTINUOUS, DISCONTINUOUS AND UNIMAGINABLE 17 This chapter examines various ways in wh ich a function can be discontinuous, eventually arriving at the 'official' e-D definition of continuity. The usual pathological cases are considered and there is a discussion about the usefulness of continuous functions 3 ADDING UP FOREVER - PARADOXES AT INFINITY 55 Sequences and numerical series. The historically important examples. The Riemann rearrangement theorem for conditionally convergent series. A brief look at series convergence tests 4 SMOOTH OR SPIKY? - DIFFERENTIATION 77 Differentiability approached through continuity of slope. Desirable properties of functions differentiable over closed vii CONTENTS intervals. Graphical interpretation using the derivative. More pathological examples 5 PUTTING IT ALL TOGETHER - INTEGRATION 96 Integration as area, integration as backwards differentiation: the connection. The integral as an 'easier' concept than area. History of integration as a rigorous theory. Setting up the Riemann integral. Conditions for integrability and some paradoxical examples 6 A BRIEF LOOK AT FURTHER DEVELOPMENTS OF ANALYSIS, AND SUGGESTIONS FOR FURTHER READING 125 Index 128 viii PREFACE To teachers and learners Analysis has a bad image. It is a hard subject which students have always found difficult. Schools (justifiably) and many colleges (wrongly, I believe) have taken the view: analysis is hard so let's not do it. Many universities now seem to modify this slightly to: analysis is hard but unavoidable so let's delay it. The first view produces a generation of students which has avoided so me difficulties but missed out on the considerable satisfactions too. The second runs the risk that when analysis is finally tackled it is done in a great rush, though it is surely a necessary condition for student enjoyment that a long 'sink in' time is given. By way of justification for this book, here are three defensible claims: (i) analysis is interesting, (ii) exposure to analytic ideas should come 'earlier rather than later than is currently the fashion, and (iii) there is a need for sympathetic texts at a level somewhat lower than the first 'formal' course in analysis. A book which is thorough and sufficiently complete to be a basis for a first year course is Mary Hart's Guide to Analysis in the Macmillan Mathematical Guides series. It is also very sympathetic to student needs. What is Mathematical Analysis? aims to put students into a confident and positive frame of mind for tackling such a course. It provides a taste of wh at analysis is about, and therefore must contain so me fairly hard proofs. But these are embedded in far more chat than is customary in more formal texts. I hope that prospective students emerge from this book with a clear idea of why analysis is necessary and of the satisfaction it can provide. Important message to learners You will find the text regularly interrupted by mysterious things called PFrA s. These are 'Pauses For Thought and Action'! They are included in the firm belief that the best way to learn anything is to grapple with it ix PREFACE yourself rather than passively absorb what others tell you. My recommen dation is that you spend a minimum of a few minutes on each one as you co me to them, before turning to the solution which generally appears on the same or the following page. As for previous knowledge required to appreciate this book, a nodding acquaintance with calculus - just basic differentiation and integration - would be useful background. There were several inconsistencies in early treatments of the calculus and one of the motivations for the development of anlysis was to remove these and render calculus 'respectable'. x 1 NUMBERS, LINES AND HOLES In school we learn about numbers, connections between numbers, and how to manipulate them to solve problems - that's arithmetic and algebra. At the same time we learn about shapes and space, shapes being made from straight and curved lines - that's geometry. Then at so me future stage (earlier rather than later according to current trends) these two major threads are woven together with the realisation that equations can describe lines - and we have coordinate geometry. How's that for a whirlwind summary? This fusion of numbers and lines was achieved in the 17th Century by the French philosopher and mathematician Rene Descartes, and later, at the end of the 17th Century, combined with ideas of continuous change, it flowered into the calculus of Isaac Newton in England and Gottfried Leibniz in Germany. It is fair to say that the calculus is the single most powerful mathematical tool we have for understanding the world we live in. Newton quite quickly tamed the universe with a single differential equation and in more recent times the calculus has given invaluable insights into the biological sciences and social sciences like economics. But something was seriously wrong in the state of mathematics. Bishop Berkeley in 1734 noticed that what Newton was doing was basically unsound. Today people gene rally admit that Berkeley was right and Newton wrong, but at the time Berkeley's voice was largely unheeded because the calculus's surge of success was unstoppable. Later, in the 19th Century people like Cauchy (more of hirn later) realised that even the work of Descartes could be seriously flawed. People took more notice then because Cauchy was able to turn his criticisms into positive action, to repair the flaws and make brilliant use of the newly improved system. So what was the nature of these flaws? Were they philosophical, math ematical, logical, aesthetic or just plain factual? A bit of all of these really, but before we explore some of these aspects let us briefly remind ourselves just how useful the geometry/algebra interplay iso 1
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