Library of Mathematics – serie

THIS book is intended to provide the university student in the physical sciences with information about the differential calculus which he is likely to need. The techniques described are presented with due regard for their theoretical basis; but the emphasis is on detailed discussion of the ideas of the differ­ ential calculus and on the avoidance of false statements rather than on complete proofs of all results. It is a frequent experi­ ence of the university lecturer that science students 'know how to differentiate', but are less confident when asked to say 'what ix means'. It is with the conviction that a proper understand­ ing of the calculus is actually useful in scientific work and not merely the preoccupation of pedantic mathematicians that this book has been written. The author wishes to thank his colleague and friend, Dr. W. Ledermann, for his invaluable suggestions during the prepara­ tion of this book. P. J. HILTON The University. Manchester . . . Contents PAGE Preface V CHAPTER I Introduction to Coordinate Geometry I 6 2 Rate of Change and Differentiation I. The meaning of 'rate of change' 6 2. Limits 9 3. Rules for differentiating IS 4. Formulae for differentiating 21 Exerc-bses 2 3 3 Maxima and Minima and Taylor's Theorem 34 I. Mean Value Theorem 34 2. Taylor's Theorem 41 3. Maxima and minima 45 4.

THE purpose of this book is to prescnt a straightforward introduction to complex numbers and their properties. Complex numbers, like other kinds of numbers, are essen­ tially objects with which to perform calculations a:cording to certain rules, and when this principle is borne in mind, the nature of complex numbers is no more mysterious than that of the more familiar types of numbers. This formal approach has recently been recommended in a Reportt prepared for the Mathematical Association. We believe that it has distinct advantages in teaching and that it is more in line with modern algebraical ideas than the alternative geometrical or kinematical definitions of v -1 that used to be proposed. On the other hand, an elementary textbook is clearly not the place to enter into a full discussion of such questions as logical consistency, which would have to be included in a rigorous axiomatic treatment. However, the steps that had to be omitted (with due warning) can easily be filled in by the methods of abstract algebra, which do not conflict with the 'naive' attitude adopted here. I should like to thank my friend and colleague Dr. J. A. Green for a number of valuable suggestions, especially in connection with the chapter on convergence, which is a sequel to his volume Sequences and Series in this Library.

THIS book, like its predecessors in the same series, is in­ tended primarily to serve the needs of the university student in the physical sciences. However, it begins where a really elementary treatment of the differential calculus (e. g. , Dif­ ferential Calculus,t in this series) leaves off. The study of physical phenomena inevitably leads to the consideration of functions of more than one variable and their rates of change; the same is also true of the study of statistics, economics, and sociology. The mathematical ideas involved are des­ cribed in this book, and only the student familiar with the corresponding ideas for functions of a single variable should attempt to understand the extension of the method of the differential calculus to several variables. The reader should also be warned that, with the deeper penetration into the subject which is required in studying functions of more than one variable, the mathematical argu­ ments involved also take on a more sophisticated aspect. It should be emphasized that the basic ideas do not differ at all from those described in DC, but they are manipulated with greater dexterity in situations in which they are, perhaps, intuitively not so obvious. This remark may not console the reader bogged down in a difficult proof; but it may well happen (as so often in studying mathematics) that the reader will be given insight into the structure of a proof by follow­ ing the examples provided and attempting the exercises.

THIS book is an introduction both to Laplace's equation and its solutions and to a general method of treating partial differential equations. Chapter 1 discusses vector fields and shows how Laplace's equation arises for steady fields which are irrotational and solenoidal. In the second chapter the method of separation of variables is introduced and used to reduce each partial differential equation, Laplace's equa­ tion in different co-ordinate systems, to three ordinary differential equations. Chapters 3 and 5 are concerned with the solutions of two of these ordinary differential equations, which lead to treatments of Bessel functions and Legendre polynomials. Chapters 4 and 6 show how such solutions are combined to solve particular problems. This general method of approach has been adopted because it can be applied to other scalar and vector fields arising in the physi­ cal sciences; special techniques applicable only to the solu­ tions of Laplace's equation have been omitted. In particular generating functions have been relegated to exercises. After mastering the content of this book, the reader will have methods at his disposal to enable him to look for solutions of other partial differential equations. The author would like to thank Dr. W. Ledermann for his criticism of the first draft of this book. D. R. BLAND The University, Sussex. v Contents Preface page v 1. Occurrence and Derivation of Laplace's Equation 1. Situations in which Laplace's equation arises 1 2. Laplace's equation in orthogonal curvilinear co-ordinates 8 3.

The aim of this book is to give an elementary treatment of multiple integrals. The notions of integrals extended over a curve, a plane region, a surface and a solid are introduced in tum, and methods for evaluating these integrals are presented in detail. Especial reference is made to the results required in Physics and other mathematical sciences, in which multiple integrals are an indispensable tool. A full theoretical discussion of this topic would involve deep problems of analysis and topology, which are outside the scope of this volume, and concessions had to be made in respect of completeness without, it is hoped, impairing precision and a reasonable standard of rigour. As in the author's Integral Calculus (in this series), the main existence theorems are first explained informally and then stated exactly, but not proved. Topological difficulties are circumvented by imposing some­ what stringent, though no unrealistic, restrictions on the regions of integration. Numerous examples are worked out in the text, and each chapter is followed by a set of exercises. My thanks are due to my colleague Dr. S. Swierczkowski, who read the manuscript and made valuable suggestions. w. LEDERMANN The University of Sussex, Brighton.

Linear programming is a relatively modern branch of Mathe­ matics, which is a result of the more scientific approach to management and planning of the post-war era. The purpose of this book is to present a mathematical theory of the subject, whilst emphasising the applications and the techniques of solution. An introduction to the theory of games is given in chapter five and the relationship between matrix games and linear programmes is established. The book assumes that the reader is familiar with matrix algebra and the background knowledge required is covered in the book, Linear Equations by P.M. Cohn, of this series. In fact the notation used in this text conforms with that intro­ duced by Cohn. The book is based on a course of about 18 lectures given to Mathematics and Physics undergraduates. Several examples are worked out in the text and each chapter is followed by a set of examples. I am grateful to my husband for many valuable suggestions and advice, and also to Professor W. Ledermann, for encourag­ ing me to write this book.

I have tried to provide an introduction, at an elementary level, to some of the important topics in real analysis, without avoiding reference to the central role which the completeness of the real numbers plays throughout. Many elementary textbooks are written on the assumption that an appeal to the complete­ ness axiom is beyond their scope; my aim here has been to give an account of the development from axiomatic beginnings, without gaps, while keeping the treatment reasonably simple. Little previous knowledge is assumed, though it is likely that any reader will have had some experience of calculus. I hope that the book will give the non-specialist, who may have considerable facility in techniques, an appreciation of the foundations and rigorous framework of the mathematics that he uses in its applications; while, for the intending mathe­ matician, it will be more of a beginner's book in preparation for more advanced study of analysis. I should finally like to record my thanks to Professor Ledermann for the suggestions and comments that he made after reading the first draft of the text.

LINEAR equations play an important part, not only in mathe­ matics itself, but also in many fields in which mathematics is used. Whether we deal with elastic deformations or electrical networks, the flutter of aeroplane wings or the estimation of errors by the method of least squares, at some stage in the cal­ culation we encounter a system of linear equations. In each case the problem of solving the equations is the same, and it is with the mathematical treatment of this question that this book is concerned. By meeting the problem in its pure state the reader will gain an insight which it is hoped will help him when he comes to apply it to his field of work. The actual pro­ cess of setting up the equations and of interpreting the solution is one which more properly belongs to that field, and in any case is a problem of a different nature altogether. So we need not concern ourselves with it here and are able to concentrate on the mathematical aspect of the situation. The most important tools for handling linear equations are vectors and matrices, and their basic properties are developed in separate chapters. The method by which the nature of the solution is described is one which leads immediately to a solu­ tion in practical cases, and it is a method frequently adopted when solving problems by mechanical or electronic computers.