Oxford Users' Guide to Mathematics PDF
Preview Oxford Users' Guide to Mathematics
Oxford Users' Guide to Mathematics Oxford Users' Guide to Mathematics E. Zeidler with W. Hackbusch and H.R. Schwarz Oxford Users' Guide to Mathematics Edited by Eberhard Zeidler Max Planck Institute for Mathematics in the Sciences Leipzig Translated and typeset by Bruce Hunt OXFORD UNIVERSITY PRESS OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dares Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York Translated by Bruce Hunt English translation © Oxford University Press 2004 The moral rights of the author have been asserted Database right Oxford University Press (maker) Title of the original: Teubner-Taschenbuch der Mathematik, Vol 1 © 1996 B. G. Teubner, Stuttgart and Leipzig Translation arranged with the approval of the Publisher B. G. Teubner from the original German edition All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this title is available from the British Library Library of Congress Cataloging in Publication Data (Data available) ISBN 0 19 850763 1 1 0 9 8 7 6 5 4 3 2 1 Typeset inL ATEXb y Bruce Hunt Printed in Hong Kong on acid-free paper by Nordica Printing Co., Ltd. Foreword We live in an age in which mathematics plays a more and more important role, to the extent that it is hard to think of an aspect of human life to which it either has not provided, or does not have the potential to provide, crucial insights. Mathematics is the language in which quantitative models of the world around us are described. As subjects become more understood, they become more mathematical. A good exam- ple is medecine, where the Radon transform is what makes X-ray tomography work, where statistics form the basis of evaluating the success or failure of treatments, and where mathematical models of organs such as the heart, of tumour growth, and of nerve impulses are of key importance. To apply mathematics in a different area requires of the mathematician the ability and willingness to learn about that area and understand its own language, and it requires of the specialist in that area a similar ability and willingness to learn the language of the appropriate parts of mathematics. Mathematics has its own internal language barriers too, so that to move from one part of the subject to another can require a major effort. It is to these needs of furnishing basic understanding and overcoming language barriers that the Oxford Users' Guide to Mathematics responds. In editing it, Eberhard Zeidler has given us a remarkable panoramic overview of mathematics, ranging from elementary facts and concepts, to advanced and sophisticated techniques, lucidly and economically explained. The outcome of many years of work, it will provide readers of diverse back- grounds with the fundamental concepts and language on which deeper understanding and significant applications can be based. Oxford, December 2003 John Ball This page intentionally left blank Preface In the past few years, mathematics has made enormous strides forward. An eminent factor for this progress has been the construction and application of ever stronger and faster computers. Moreover, the extremely complicated problems of modern technology which pose themselves to engineers and natural scientists require highly sophisticated mathematics, in which routine knowledge no longer suffices and the boundaries between pure and applied mathematics are starting to melt. The Oxford Users' Guide to Mathematics responds to the high standards required by the growing influence of computer science within the mathematical sciences and the increasingly close relationships between mathematics and the natural and engineering sciences. It conveys a lively, modern picture of mathematics aimed at a wide readership, including: - students of high schools and undergraduates, - graduate students of mathematics, - students of engineering, natural sciences, economy and other directions of study which require mathematical background, - practitioners who work in these fields, - teachers, both in schools and at universities. The needs of as broad an audience as this will be taken into account in our presentation by the consideration of a wide range of aspects, starting from elementary facts all the way up to modern, highly sophisticated results and methods. In addition, the presentation is very broad in its consideration of very diverse areas of mathematical research. In this respect, the book is both vertically and horizontally quite deep. At the same time, we go to great pains to motivate the material completely and explain the basic ideas in depth, both aspects of which are emphasized more in the text than are technical details. Also, applications of the ideas and methods play an important role in the development. There are many interludes on historical background of the results or more generally on the period in which the results were first obtained. In addition to these remarks throughout the text, there is, at the end of the volume, a detailed sketch of the history of mathematics. This exemplifies the point of view that mathematics is more than a dry collection of formulas, definitions, theorems and manipulations with symbols. Rather, mathematics is an integral part of our culture and a wonderful medium for thought and discovery, which makes it possible to make progress on frontiers like the modern theory of elementary particles and cosmology, areas which cannot be understood without a mathematical model, as they so lie so far from our usual realm of perception and understanding. In the introductory Chapter 0 we collect basic mathematical notions and facts which are often required by students, scientists and other practitioners, in the form of a reference viii Preface book. A student of medicine, for example, can find here an elementary introduction to the methods of mathematical statistics, which will hopeful be of use in the writing of a statistical part of a thesis. The following three chapters are devoted to the three basic pillars of mathematics: - analysis, - algebra, and - geometry. These chapters are followed by a chapter devoted to - foundations of mathematics (logic and set theory), which takes into account in particular the needs and difficulties of beginning students. The last three chapters are then devoted to the most important fields of applications of mathematics, namely - theory of variations and optimization, - stochastics (probability theory and mathematical statistics), and - scientific computation. The possibilities which modern supercomputers offer have radically changed scientific computation. Whether mathematician, engineer or natural scientist, the practitioner today is in a position to carry out extensive experiments on the computer which make it possible to collect experience from examples in hitherto underdeveloped areas of math- ematics. In this way, completely new questions arise and give new impulses for the development of mathematical theories. The last chapter is the first appearance, in pocket book form, of the modern theory of scientific computation, which, as mentioned above, has revolutionized the engineering sciences. The past decade has seen the appearance of software systems which make it possible to carry out many routine jobs in mathematics on a standard PC. This is mentioned at the corresponding places in the text, where these methods are motivated and described. The bibliography at the end of the book was very carefully put together and gives you an idea of where to turn should you have questions above and beyond what is directly treated in the text. The level of references varies from introductory texts to classics and goes on to advanced monographs, reflecting the frontiers of modern research. This book has a long history. The Pocketbook of Mathematics by I. N. Bronstein and K. A. Semendjajew was originally translated from Russian into German by Viktor Ziegler. It appeared in 1958, published by the B. G. Teubner Verlag in Leipzig, and has become in the mean time a standard in the German language with 18 editions until 1978. Toward the end of the last century it was decided to bring this classic up to date, not only with respect to the presented material, but also with respect to the breadth and kind of presentation; this was carried out under the supervision of Eberhard Zeidler, who wrote all chapters except that on scientific computing, which was authored by Wolfgang Hackbusch and Hans Rudolf Schwarz. This appeared, again by Teubner-Verlag, in 1996. The work was so fundamentally different than its predecessors that Oxford University Press felt it would be worth translating it, and they assigned Bruce Hunt that job of doing so. The translator has done his best to keep the spirit of the book as in the original, at the same time including a series of corrections which had been reported by astute Preface ix readers or which were spotted in the process of translating the volume. Furthermore the translator has gone to great pains to improve the graphical quality of the text, as this improves the ease with which the material can be absorbed by the reader. Acknowledgments: In addition to the pure translation of the volume, it was agreed with the publisher to typeset the entire book from scratch; in particular this meant retyping all the formulas and tables, as well as redoing all the illustrations. For help with these aspects, as well as extensive proof reading of the translation, both the translator and the editor are indebted to Micaela Krieger-Hauwede (illustrations), Lars Uhlmann (equations and tables) and Kerstin Folting (equations, tables and a meticulous proof- reading of the entire text). Without their help the translation would have taken much longer than it did. The editor likes to thank the translator for his excellent job. Frankfurt/Main Bruce Hunt Leipzig Eberhard Zeidler Fall 2003
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