Cover images 0 Murray Robertson/visual elements 1998-99, taken from the 109 Visual Elements Periodic Table, available at www.chemsoc.org/viselements
ISBN 0-85404-677-1
A catalogue record for this book is available from the British Library
0The Royal Society of Chemistry 2003 All rights reserved Apart-from any fair dealing for the purposes of research or private study, or criticism or reviews as permitted under the terms of the U K Copyright, Designs and Patents Act, 1988, this publication may not be reproduced, stored or trunsmitted, in any form or by any means, without the prior permission in writing of The Royal Society of Chemistry, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the U K , or in accordance with the terms OJ the licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to The Royal Society of Chemistry at the address printed on this page. Published by The Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 OWF, UK Registered Charity No. 207890 For further information see our web site at www.rsc.org Typeset in Great Britain by Alden Bookset, Northampton Printed and bound in Italy by Rotolito Lombarda
Preface These two introductory texts provide a sound foundation in the key mathematical topics required for degree level chemistry courses. While they are primarily aimed at students with limited backgrounds in mathematics, the texts should prove accessible and useful to all chemistry undergraduates. We have chosen from the outset to place the mathematics in a chemical context - a challenging approach because the context can often make the problem appear more difficult than it actually is. However, it is equally important to convince students of the relevance of mathematics in all branches of chemistry. Our approach links the key mathematical principles with the chemical context by introducing the basic concepts first, and then demonstrates how they translate into a chemical setting. Historically, physical chemistry has been the target for mathematical support; however, in all branches of chemistry - be they the more traditional areas of inorganic, organic and physical, or the newer areas of biochemistry, analytical and environmental chemistry - mathematical tools are required to build models of varying degrees of complexity, in order to develop a language for providing insight and understanding together with, ideally, some predictive capability. Since the target student readership possesses a wide range of mathematical experience, we have created a course of study in which selected key topics are treated without going too far into the finer mathematical details. The first two chapters of Volume 1 focus on numbers, algebra and functions in some detail, as these topics form an important foundation for further mathematical developments in calculus, and for working with quantitative models in chemistry. There then follow chapters on limits, differential calculus, differentials and integral calculus. Volume 2 covers power series, complex numbers, and the properties and applications of determinants, matrices and vectors. We avoid discussing the statistical treatment of error analysis, in part because of the limitations imposed by the format of this series of tutorial texts, but also because the procedures used in the processing of experimental results are commonly provided by departments of chemistry as part of their programme of practical chemistry courses. However, the propagation of errors, resulting from the use of formulae, forms part of the chapter on differentials in Volume 1. Martin Cockett Graham Doggett York
iii
EDITOR-IN-CHIEF
EXECUTIVE EDITORS
EDUCATIONAL CONSULTANT
Professor E W Abel
Professor A G Davies Professor D Phillips Professor J D Woollins
Mr M Berry
This series of books consists of short, single-topic or modular texts, concentrating on the fundamental areas of chemistry taught in undergraduate science courses. Each book provides a concise account of the basic principles underlying a given subject, embodying an independentlearning philosophy and including worked examples. The one topic, one book approach ensures that the series is adaptable to chemistry courses across a variety of institutions. TITLES IN THE SERIES
FORTHCOMING TITLES
Stereochemistry D G Morris Reactions and Characterization of Solids S E Dann Main Group Chemistry W Henderson d- and f-Block Chemistry C J Jones Structure and Bonding J Barrett Functional Group Chemistry J R Hanson Organotransition Metal Chemistry A F Hill Heterocyclic Chemistry M Sainsbury Atomic Structure and Periodicity J Barrett Thermodynamics and Statistical Mechanics J M Seddon and J D Gale Basic Atomic and Molecular Spectroscopy J M Hollas Organic Synthetic Methods J R Hanson Aromatic Chemistry J D Hepworth, D R Waring and A4 J Waring Quantum Mechanics for Chemists D 0 Hayward Peptides and Proteins S Doonan Reaction Kinetics M Robson Wright Natural Products: The Secondary Metabolites J R Hanson Maths for Chemists, Volume 1, Numbers, Functions and Calculus M Cockett and G Doggett Maths for Chemists, Volume 2, Power Series, Complex Numbers and Linear Algebra M Cockett and G Doggett
Mechanisms in Organic Reactions Molecular Interactions Lanthanide and Actinide Elements Bioinorganic Chemistry Chemistry of Solid Surfaces Biology for Chemists Multi-element NMR EPR Spectroscopy Biophysical Chemistry
Further information about this series is available at www.rsc.orgltct Order and enquiries should be sent to: Sales and Customer Care, Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 OWF, UK
Tel: +44 1223 432360; Fax: +44 1223 426017; Email:
[email protected] Contents
1.1 Real Numbers 1.2 Algebra
2 19
2.1 2.2 2.3 2.4
Defining Function Representation of Fgnctions Some Special Mathematical Functions Equations
30 38 46 65
3.1 Mathematical and Chemical Examples 3.2 Defining the Limiting Process
77 81
4.1 4.2 4.3 4.4 4.5
The Average Rate of Change The Instantaneous Rate of Change Higher Order Derivatives Maxima, Minima and Points of Inflection The Differentiation of Functions of Two or More Variables
90 91 97 102 106
V
vi
Contents
5.1 The Effects of Incremental Change 5.2 The Differential of a Function of Two or More Variables 5.3 The Propagation of Errors
110 114 115
Reversing the Effects of Differentiation The Definite Integral The Indefinite Integral General Strategies for Solving More Complicated Integrals 6.5 The Connection between the Definite and Idefinite Integral
120 121 125
6.1 6.2 6.3 6.4
7.1 Using the Derivative of a Function to Create a Differential Equation 7.2 Some Examples of Differential Equations Arising in Classical and Chemical Contexts 7.3 First-order Differential Equations 7.4 Second-order Differential Equations
127 132
136 137 139 150
Symbols
> 3
>> < s
