Guide to Mathematical Methods
A second edition of this text for science and engineering undergraduates which introduces the mathematical techniques and tools needed to solve the mathematical problems they will face on the first year of their course. Updated and revised by Camilla Jord
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Palgrave Mathematical Guides Consultant Editor: David A. Towers Senior Lecturer in Mathematics, University of Lancaster Linear Algebra D. Towers Abstract Algebra C. Whitehead Analysis F. Hart Numerical Analysis J. Turner Mathematical Modelling D. Edwards and M. Hanson Mathematical Methods (2nd edn) J. Gilbert and C. Jordan Mechanics P. Dyke and R. Whitworth
Guide to Mathematical Methods
Second Edition
John Gilbert Department of Mathematics University of Lancaster Lancaster and
Camilla Jordan Staff Tutor in the Mathematics and Computing Faculty of the Open University
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© John Gilbert & Camilla Jordan 1991, 2002 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 limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988. First edition 1991 Reprinted six times Second edition 2002 Published by PALGRAVE MACMILLAN Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y. 10010 Companies and representatives throughout the world PALGRAVE MACMILLAN is the global academic imprint of the Palgrave Macmillan division of St. Martin’s Press, LLC and of Palgrave Macmillan Ltd. Macmillan® is a registered trademark in the United States, United Kingdom and other countries. Palgrave is a registered trademark in the European Union and other countries. ISBN 978-0-333-79444-9 ISBN 978-1-4039-1456-9 (eBook) DOI 10.1007/978-1-4039-1456-9 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. A catalogue record for this book is available from the British Library. 10 11
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Contents Preface to first edition
xi
Preface to second edition
xiii
Symbols, notation and Greek letters
xiv
1 Preliminaries 1.1
Basic algebraic skills . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Powers and indices . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Algebraic fractions and rationalising . . . . . . . . . . . . . . .
6
1.4
Factorising . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.5
Completing the square . . . . . . . . . . . . . . . . . . . . . .
11
1.6
Miscellaneous exercises . . . . . . . . . . . . . . . . . . . . . .
13
1.7
Answers to exercises . . . . . . . . . . . . . . . . . . . . . . . .
14
2 Functions
v
1
17
2.1
Sets and intervals . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
Functions . . . . . . . . .
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