In 1973, when I joined the staff of the University of Bristol as a lecturer in the Department of Engineering Mathematics, I found my new colleagues in the preliminary stages of planning a novel degree course to be known as Engineering Mathematics. One of the primary objectives of this degree course was to produce graduates with not only a sound mathematical education but also the ability to apply their mathematical knowledge to the solution of the problems of commerce and manufacturing industry. As a contribution to the design of this course I proposed to my colleagues that we should include, in the overall course structure, a course of practical mathematical activities aimed at introducing students to a range of industrially relevant mathematical and para-mathematical activities. Thus was born the Case Study course.

The concept of the course called for the collection of resource material from a wide cross-section of industry and commerce. To support this phase of the work a grant was sought from the Nuffield Foundation under what was then known as their Small Grants Scheme for Undergraduate Teaching. The grant was approved and the task of gaining the co-operation of industry and commerce in the creation of the resource material for the course proceeded throughout 1976 and 1977. The course itself was first used with undergraduate students in early 1978 and has been run annually since then.

Scientific context

Many scientific laws are often expressed as relations between two or more physical quantities. In general these laws are obtained in one of two ways. Either the results of experiment are used directly to formulate empirical laws, or existing scientific knowledge is used, often together with mathematics, to arrive at new theories which can then be validated later by experiment. In formulating scientific laws we attempt to find a formula between the symbols representing the physical quantities of interest. Sometimes this is not possible and the relationship has to be expressed in the form of a table of values or a graph, for example.

If two quantities are related so that the value of one of them is uniquely determined when the other is known, then we say that there is a functional relationship between the variables. In these opening chapters we consider the basic mathematical functions which occur in science.

Example 1: loading a steel wire. Table 1.1 shows the results of an experiment to investigate how the length of a piece of mild steel wire changes when weights are attached to it. The unextended length of the wire is 2 m and its mean diameter is 1 mm.

It is clear that l increases with W but the nature of the relationship between the quantities W and l is more easily seen if we plot the points on a graph.

Scientific context

In the first four chapters we have introduced various important functions and investigated their properties. In the last chapter we presented some of the more formal aspects of the properties of a function, such as its domain and image, and considered how to build up composite functions by combining basic functions. In this chapter we continue to investigate the properties of functions and introduce one more important function namely the function f(x) = 1/x.

Example 1: equations of state. Pure substances can exist in various phases, namely gas, liquid and solid, and whether the phase is a gas, a liquid or a solid is determined essentially by the values of the pressure, volume and temperature of the substance. Relationships between p, V and T lead to different models called equations of state. To investigate these models fully we obviously need to understand the properties of functions of more than one variable. Functions of several variables are discussed in Chapter 17, but to simplify matters, suppose we look at the changes that can take place in a substance when the temperature remains constant. (Such changes are called isothermal changes.)

Imagine then that an experiment is carried out where the volume of gas is measured at different pressures, the temperature remaining constant. Suppose the results of the experiment are those shown in Table 5.1. In a first attempt to produce a model we might plot a graph of the pressure against the volume, as shown in Fig. 5.1.

We now begin three chapters about integration.

As the word implies, integration is about bringing things together. In mathematics the process of integration involves bringing together function values in a special way to form a summation. This sum of function values is called an integral and we shall see that there are many circumstances in which we calculate the value of some quantity by the process of integration. At first sight it will appear that integration is just the opposite of differentiation, i.e. in a sense what we appear to be doing is ‘undoing’ the differential of a function. This process is called anti-differentiation. For example, cos (x) is the derivative of sin (x), and sin (x) is the anti-derivative of cos (x).

However, integration and antidifferentiation are conceptually different. The fact that the two processes give the same answer is a consequence of an important theorem called the Fundamental Theorem of Calculus. In this first chapter we define what we mean by the integral of a function and investigate the results of integrating the basic functions we have met before.

Scientific context

Example 1: A swinging pendulum. The photograph in Fig. 3.1 shows the motion of a swinging pendulum. The multiple images of the pendulum, obtained using stroboscopic lighting, represent its position at the times when the stroboscope flashed which was every tenth of a second. In Fig. 3.2 we have drawn a graph of the horizontal displacement of the pendulum bob from its central position (i.e. when the string is vertical).

None of the functions we have considered so far provides a model for the way the displacement changes with time and this is obvious if we draw the graph of the displacement over a much longer time interval. This graph is shown in Fig. 3.3. As the bob swings backwards and forwards its displacement varies from – 0.076 m to + 0.076 m in a periodic way.

Polynomials and exponential functions are not periodic and so it is clear that a new class of functions is going to be needed if we are to describe the kind of motion shown. This wave-type behaviour is a very common feature of many periodic systems in science.

Scientific context

In the first three chapters, by looking at appropriate examples from science, we have been able to motivate the study of polynomials, exponential functions and trigonometric functions. For example, by looking at the growth of a certain kind of bacterium and asking how large the colony would be after time t, we arrived quite naturally at the exponential function. However, had we looked at this growth problem from a different standpoint and tried to discover how long it would be before the colony reached a certain size, we would have been led to a quite different but not unrelated function. In our study of the growth process the independent variable was the time t and the dependent variable was the number N of the bacterium. Had we adopted this different standpoint, the role of the variables would have been reversed. N would now be the independent variable and t the dependent one. The function we would have been led to would have been the so-called inverse function of the exponential function. Many functions have inverses and these inverse functions often play an important role in mathematical modelling. In this chapter we look at those inverse functions that are of importance in science.

Example 1: bacterial growth revisited. The models of growth and decay that we have produced so far have all been geared to telling us how large something might be after a certain period, or what the temperature of a cooling body is after a certain time interval, and so on.

In Chapter 15 we looked at the basics of probability and at some of the important probability distributions that arise in the handling of scientific data. We did this because, as we stated, a study of probability is first needed if we are properly to address the problem of expressing numerically the inevitable uncertainties that exist in the results of statistical analyses. In this final chapter on statistics we consider the problem of quantifying the uncertainty associated with the results of statistical analyses and look specifically at sampling and at just what can be deduced about a population from only a finite, and often small, sample of results.

Scientific context

Example 1: estimating the value of the acceleration due to gravity. The time period t of a simple pendulum of length l is given by the well-known expression where g is the acceleration due to gravity. By observing the time for, say, 50 oscillations of the pendulum, for each of a set of given l values, a plot of t2 versus l can be drawn up which should be a straight line of slope 4π2/g. By measuring the gradient it is thus possible to obtain a reasonably accurate estimate of g in a fairly straightforward way.

Scientific context

In earlier chapters we made a study of many of the functions commonly used in science and we showed how to obtain their derivatives. The functions we have considered so far have been functions of a single variable. In science there are many situations where some quantity of interest is related to not just a single variable but to several. In such situations these relationships would be modelled mathematically by functions of several variables.

In order to understand the properties of functions of one variable, we needed to differentiate and sketch out the graph of the functions. Similarly, we need to carry out the same operations in the several variable case. To differentiate functions of several variables involves knowing how to work out what are called partial derivatives and obtaining partial derivatives is the central theme of this first chapter on partial differentiation.

As we shall see, the partial derivative of a function of several variables is a straightforward extension of ordinary differentiation just as functions of several variables are a straight forward extension of the single variable case.

Example 1: van der Waal's equation of state. Pure substances can exist in various phases, namely gas, liquid and solid, and whether the phase is a gas, a liquid or a solid is determined essentially by the values of pressure, volume and temperature of the substance.