In the mathematical description of a physical system the basic laws are usually expressed as equations connecting local quantities. The equations consist of products and sums of the quantities and their derivatives, the actual combination of forms depending upon which variables are chosen to describe the situation.

The effect of external factors on the system is represented by the tying of the values of some of the quantities to particular values of others. The latter quantities are usually called the independent variables, and the former the dependent ones. A complete description (solution) of the physical system is obtained when the values of the dependent variables are known for all values of (i.e. as a function of) the independent ones.

In cases where derivatives appear in the expression of the physical laws it is thus necessary to convert a differential equation into a standard, tabulated, or computable function. This may take the form of a series or an integral which can be evaluated numerically. Methods by which the conversion can be effected form the content of this and the next two chapters and also (for equations containing more than one independent variable) of chapters 9 and 10.

Equations of the types considered in the first three of these chapters – those containing only one independent variable – are called ordinary differential equations. The two other chapters deal with some methods of solving partial differential equations, i.e. ones containing derivatives of dependent variables with respect to more than one independent variable.

This book is intended for students of physical science, applied science and engineering, who, for the understanding and practice of their principal subjects, need a working knowledge of applicable mathematics.

Since it is not possible in a single text to cater for all degrees of mathematical facility, nor for all tastes in abstraction, a broad middle course has been adopted, set at the level of what, at the risk of being misunderstood, I describe as the ‘average student’. It is hoped, however, that what is presented will also be of value to those who fall on either side of this central band, either as a less than rigorous introduction to the subject for the one group, or as an explanatory and illustrative text for the other.

The ground covered is roughly those areas of applied mathematics usually met by students of the physical sciences in their first and second years at university or technical college. Naturally much of it also forms parts of courses for mathematics students.

In any book of modest size it is impossible to cover all topics fully, and any one of the areas mentioned in this book can be, and has been, the subject of larger and more thorough works. My aim has been to take a ‘horizontal slice’ through the subject centred on the level of an average second-year student.

The preliminary knowledge assumed is that generally acquired by any student prior to entering university or college.

Although the major part of this book is concerned with mathematics of direct value in describing situations arising in physical science and engineering, this opening chapter, although directed to the same end, is of a less obviously ‘applied’ nature. It is concerned with those techniques of mathematics, principally in the field of calculus, which are the nuts and bolts of the more particularly orientated methods presented in later chapters.

Two particular factors have to be taken into account in its presentation; firstly the various levels of previous knowledge which different readers will possess, and secondly the fact that the subjects to be treated in this chapter form a less coherent whole than do those in any other.

The first of these has been approached at the ‘highest common factor’ level, namely, knowledge has been presumed only of those topics which will normally be familiar to a student who, in his previous studies, has taken mathematics in conjunction with other science subjects, rather than as his main or only subject. As a result, although several parts of this chapter will almost certainly be unfamiliar to him, the reader with more than this presumed level of knowledge may in some sections find it sufficient to make sure he can solve the corresponding exercises, marked by the symbol ▸, and then pass on to the next section.

As a result of the rather diverse nature of the topics considered, the degree of difficulty of the material does not vary ‘monotonically’ throughout the chapter.

In the preceding chapter the solution of ordinary differential equations in terms of standard functions or numerical integrals was discussed, and methods for obtaining such solutions explained and illustrated. The present chapter is concerned with a further method of obtaining solutions of ordinary differential equations, but this time in the form of a convergent series which can be evaluated numerically [and if sufficiently commonly occurring, named and tabulated]. As previously, we will be principally concerned with second-order linear equations.

There is no distinct borderline between this and the previous chapter; for consider the equation already solved many times in that chapter

The solution in terms of standard functions is of course

but an equally valid solution can be obtained as a series. Exactly as in ▸1 of chapter 5 we could try a solution

and arrive at the conclusion that two of the an are arbitrary [a0 and a1] and that the others are given in terms of them by

Hence the solution is

It hardly needs pointing out that the series in the brackets are exactly those known as cos x and sin x and that the solution is precisely that of (6.2); it is simply that the cosine and sine functions are so familiar that they have a special name which is adequate to identify the corresponding series without further explanation.

It will also be true of most of our examples that they have a name (although their properties will be slightly less well known), but the methods we will develop can be applied to a variety of equations, both named and un-named.

This chapter is of a preliminary nature and is designed to indicate the level of knowledge assumed in the development of the third and subsequent chapters. It deals with those elementary properties of vectors and their algebra which will be used later. The results and properties are usually stated without proof, but with illustrations, and a set of exercises is included at the end in section 2.10 to enable the student to decide whether or not further preliminary study is needed. It is suggested that the reader who already has some working familiarity with vector algebra might first attempt the exercises and return to this chapter only if he has difficulty with them.

Definitions

The simplest kind of physical quantity is one which can be completely specified by its magnitude, a single number together with the units in which it is measured. Such a quantity is called a scalar and examples include temperature, time, work, and [scalar] potential.

Quantities which require both a magnitude (≥ 0) and a direction in space| to specify them are known (with a few exceptions, such as finite rotations, discussed below) as vectors; familiar examples include position with respect to a fixed origin, force, linear momentum and electric field. Using an arbitrary but generally accepted convention, vectors can be used to represent angular velocities and momenta, the axis of rotation being taken as the direction of the vector and the sense being such that the rotation appears clockwise when viewed parallel [as opposed to antiparallel] to the vector.

In a previous chapter it was shown how to find stationary values of functions of a single variable f(x), of several variables f(x, y,…) and of constrained variables f(x, y,…) subject to gi(x, y,…) = 0, (i = 1, 2, …, m). In all these cases the forms of the functions f and gi were known and the problem was one of finding suitable values of the variables x, y,…

We now turn to a different kind of problem, one in which there are not free variables which must be chosen in order to bring about a particular condition for a given function, but in which the functions are free and must be chosen to bring about a particular condition for a given expression which depends upon these functions.

To give a more concrete example of the type of question to be answered, we may ask the following. ‘Why does a uniform rope suspended between two points take up the shape it does? Why doesn't it hang in an arc of a circle or in the form of three sides of a rectangle? Is it possible to predict the shape in which it will hang, that is to find a function, y = y(x), that gives the vertical height of the rope as a function of horizontal position?’

It frequently happens that the end product of a calculation or piece of analysis is one or more equations, algebraic or differential (or an integral), which cannot be evaluated in closed form or in terms of available tabulated functions. From the point of view of the physical scientist or engineer, who needs numerical values for prediction or comparison with experiment, the calculation or analysis is thus incomplete. The present chapter on numerical methods indicates (at the very simplest levels) some of the ways in which further progress towards extracting numerical values might be made.

In the restricted space available in a book of this nature it is clearly not possible to give anything like a full discussion, even of the elementary points that will be made in this chapter. The limited objective adopted is that of explaining and illustrating by very simple examples some of the basic principles involved. The examples used can in many cases be solved in closed form anyway, but this ‘obviousness’ of the answer should not detract from their illustrative usefulness, and it is hoped that their transparency will help the reader to appreciate some of the inner workings of the methods described.

The student who proposes to study complicated sets of equations or make repeated use of the same procedures by, for example, writing computer programmes to carry out the computations, will find it essential to acquire a good understanding of topics hardly mentioned here.

In this and the following chapter the solution of differential equations of the types typically encountered in physical science and engineering is extended to situations involving more than one independent variable. Only linear equations will be considered.

The most commonly occurring independent variables are those describing position and time, and we will couch our discussion and examples in notations which suggest these variables; of course the methods are not restricted to such cases. To this end we will, in discussing partial differential equations (p.d.e.), use the symbols u, v, w, for the dependent variables and reserve x, y, z, t for independent ones. For reasons explained in the preface, the partial equations will be written out in full rather than employ a suffix notation, but unless they have a particular significance at any point in the development, the arguments of the dependent variables, once established, will be generally omitted.

As in other chapters we will concentrate most of our attention on second-order equations since these are the ones which arise most often in physical situations. The solution of first-order p.d.e. will necessarily be involved in treating these, and some of the methods discussed can be extended without difficulty to third- and higher-order equations.

The method of ‘separation of variables’ has been placed in a separate chapter, but the division is rather arbitrary and has really only been made because of the general usefulness of that method.

The remarkable coloured reflection from certain crystals of chlorate of potash described by Stokes, the colours of old decomposed glass, and probably those of some beetles and butterflies, lend interest to the calculation of reflection from a regular stratification, in which the alternate strata, each uniform and of constant thickness, differ in refractivity. The higher the number of strata, supposed perfectly regular, the nearer is the approach to homogeneity in the light of the favoured wave-lengths. In a crystal of chlorate described by R. W. Wood, the purity observed would require some 700 alternations combined with a very high degree of regularity. A general idea of what is to be expected may be arrived at by considering the case where a single reflection is very feeble, but when the component reflections are more vigorous, or when the number of alternations is very great, a more detailed examination is required. Such is the aim of the present communication.

The calculation of the aggregate reflection and transmission by a single parallel plate of transparent material has long been known, but it may be convenient to recapitulate it. At each reflection or refraction the amplitude of the incident wave is supposed to be altered by a certain factor. When the light proceeds at A from the surrounding medium to the plate, the factor for reflection will be supposed to be b′, and for refraction c′; the corresponding quantities when the progress at B is from the plate to the surrounding medium may be denoted by e′, f′.