In the last chapter we saw how experimental data could be organised and how measures of centre and scatter could be worked out to give a collective description of the data. This description was useful because it helped tell us what to expect from future experiments. However, when making predictions about future experiments, we have to bear in mind that the predictions we make are going to be based on only a sample of results and will necessarily have a measure of uncertainty associated with them. Being able to express numerically the inevitable uncertainties in the results of statistical analyses is very important in science, but it requires first of all an understanding of probability and of probability distributions. In this chapter we cover the basics of probability and consider some of the theoretical probability distributions that are important in handling scientific data.

Scientific context

Example 1: radiation decay data revisited. To provide us with some data for discussing the basic ideas of probability, let us return to the radiation decay data that we considered in Chapter 14. For convenience we present the 500 data values again in Table 15.1. Suppose we want to know what the chances are of observing fewer than 4 counts in another 10-s exposure of the geiger counter to the radioactive source. Since the words ‘chance’ and ‘probability’ mean the same thing, then what we are really wanting to work out here is nothing more than a probability – the probability of observing fewer than 4 counts.

In Chapter 9 we defined what we meant by a definite and an indefinite integral and used the Fundamental Theorem of Calculus to obtain the integrals of basic functions. We also introduced numerical methods of integration because there are many functions for which analytic integrals cannot be found.

In this chapter we concentrate on techniques of integration that can be used to reduce complicated looking integrands to a basic form that can be evaluated analytically. We should stress at the outset that there are no standard rules of integration that will always work, unlike the standard rules for differentiation. There is a certain amount of guesswork and experience involved with the evaluation of integrals. In 10.2 we introduce various substitutions that often help and in 10.4 we consider a method for tackling integrals of products of functions.

Scientific context

Example 1: the centre of mass of a lamina. One of the important properties of a body is the position of its centre of mass; this is the point at which all the mass can be thought to act when the body is moving in a straight line. For such translational motion the body can be modelled as a particle with its mass concentrated at this point. Finding the centre of mass is thus important and makes use of the technique of integration. Without any loss of generality let us consider how integration is used in finding the position of the centre of mass of a uniform lamina; this is often called the centroid.

The contents of this book consist of the basic mathematics taught in nearly all first-year service courses in mathematics for science and engineering students. It is based on the experience of the three authors of teaching such material for many years.

There is a growing awareness that we must not teach mathematics in isolation from its applications. In teaching mathematics to scientists, technologists and engineers, there is plenty of opportunity to provide applications as part of the syllabus and teaching approach. There are few textbooks which recognise this. One of the aims of writing this text has been to encourage the teaching of mathematics through its applications in science.

The importance of teaching mathematics through its applications is reflected in the format of this book. Each chapter starts with two or three examples setting the new techniques to be introduced in the context of the scientific world. The aim here is to answer the often posed question from science students: ‘why do we need to learn this mathematics?’

Sections 2 and 4 of each chapter contain the basic mathematical techniques of each chapter, and sections 3 and 5 give worked examples showing the use of the new techniques introduced in the chapter.

Some of the material in the book will have been acquired by most students before entering higher education. For example, in the United Kingdom about half of the contents of the book appears listed in the syllabus of many GCE Advanced level and corresponding BTEC level 3 examinations.

We now begin a study of statistics. As well as being familiar with the mathematics that underpins the theoretical side of science we must also know how to analyse experimental data; for carrying out experiments, making measurements and collecting data are just as important in science as mathematical modelling.

Analysing experimental data is often fraught with difficulties, because one very important property, possessed by virtually any set of data values, is that different measurements of a given quantity will almost certainly show variation. This variation usually exists for one of two reasons. Firstly, variation could be present in what is being measured. For example, heights of people vary, daily hours of sunshine vary, the number of radioactive particles entering a geiger counter in a given time interval varies, and so on. Or, secondly, when measuring some fixed quantity, variation could arise due to errors of one kind or another creeping into the measuring process. We might, for example, take a sample of blood and analyse the blood for alcohol content. Even though we might use the same analytical technique, repeated determinations of the fixed concentration of alcohol would vary due to experimental error.

This variability not only affects the sort of conclusions we can draw from experimental data, but inevitably it means that in the first place we have to make repeated measurements of a quantity before we can say anything sensible about its value.

The need for series which converge for all values of the time; Poincaré's series.

We have already observed (§ 32) that the differential equations of motion of a dynamical system can be solved in terms of series of ascending powers of the time measured from some fixed epoch; these series converge in general for values of t within some definite circle of convergence in the t-plane, and consequently will not furnish the values of the coordinates except for a limited interval of time. By means of the process of analytic continuation it would be possible to derive from these series successive sets of other power-series, which would converge for values of the time outside this interval; but the process of continuation is too cumbrous to be of much use in practice, and the series thus derived give no insight into the general character of the motion, or indication of the remote future of the system. The efforts of investigators have therefore been directed to the problem of expressing the coordinates of a dynamical system by means of expansions which converge for all values of the time. One method of achieving this result is to apply a transformation to the t-plane.