In mechanical drawings the orientation of the design is chosen purely for convenience. So far as the builder is concerned, it is only the relative orientation of the elements of the design that matters. Indeed, some details may be more clearly understood if the print is turned through 90°, or even read upside down.

Mathematical description of the geometrical aspect of physical phenomena is rather like this. The vector notation which uses a single symbol is handy, precisely for the reason that it is independent of reference directions. Of course at some stage in a practical calculation it usually becomes necessary to specify actual components of velocity, force etc., and these will depend upon the choice of axes. Even here, however, some geometrical features are the same whatever axes are used. Thus the distance between two points and the angle between two directions are unaffected by rotations or shifts of axes. Such fixed quantities are called invariants. Another important example of an invariant is the scalar product of two vectors.

In the i,j,k notation of chapter 4, a given vector (displacement, velocity, force, etc.) can be written a = axi + ayj + azk. The coefficients ax, ay, az, depend upon the axes chosen, and are referred to as the components of a, resolved along the directions i,j,k. Indeed the vector a may be represented by (ax,ay,az). For a position vector, this is just the Cartesian system introduced in chapter 3.

Confronted with an expanse of still water in a pond, who can resist the temptation to do the obvious experiment, to fling a stone into its centre? The disturbance thus created travels outwards, is reflected at the pond's edge and eventually disappears. Equilibrium is restored.

Such a disturbance is called wave motion. It is similar to the oscillatory motion that we have studied in chapters 21 and 23, except that it describes the behaviour of an extended medium (the surface of the pond) whose equilibrium is disturbed. The appropriate mathematical description must involve a function of several variables, since the height of the pond's surface is a function of position (x, y) as well as time (t). Already we can expect that a partial differential equation may be the key to understanding it.

Physicists are always striving for unified descriptions of the physical world. In the nineteenth century a variety of apparently different phenomena came to be understood in terms of waves – elastic waves in solids, sound waves in air, and, most importantly, electromagnetic waves. These include light, x-rays, radio waves, infrared radiation and so on, and constitute in themselves another important “unification” of physics. All of these types of waves and many others have a common mathematical description, at least up to a point.

One of the great surprises of modern physics in this century was the realisation that matter itself also has wave-like properties, described by quantum mechanics.

Differential equations are the very essence of much of mathematical physics. In the nineteenth century there was a tendency to replace them with equivalent ‘variational principles’. They have since reasserted themselves as the most convenient mathematical expression of many physical laws, at least in their application. Let us see how they arise in some simple examples.

An object, initially at temperature T0, is heated in the open air. How does its temperature T vary? That is, what is the function T(t)? As usual, we must first give a little thought to the physics of this situation. It may be assumed that there is a fixed ratio between the gain (or loss) of heat and the rise (or fall) of temperature, the ratio being the ‘heat capacity’ C of the object. Given this, the problem is attacked simply by identifying the mechanisms of heat gain/loss. Suppose heat is supplied by the heater at the rate Q(t) (not necessarily constant). From this we must subtract the rate of heat loss to the surrounding atmosphere, which clearly depends on the temperature T of the object – the hotter it gets, the greater the loss. A common assumption is the simple proportionality of this rate of heat loss and the excess of T over the temperature Ta of the surroundings.

An invaluable accessory to a computer or experimental system is a curveplotter. The stony columns of data in the digital print-out come to life as plateaux, peaks, dips, shoulders and so on. Indeed, the pictorial language of numerical trends is in every sense, graphic. What business is complete without its sales chart? Nevertheless the presentation of experimental data in the form of a graph is a surprisingly modern convention. It did not become popular until the late nineteenth century. Before that, the general form of the results of experiments was described only roughly, in words. For example, Michael Faraday, who is credited (among many other things) with first recognising the characteristic property of semiconductors in 1833 simply said, ‘The conducting power increases with the heat’. At that time, whole textbooks were written on ‘Natural Philosophy’ without graphs, even though Descartes had long since provided the conceptual framework for them. Today, such a statement would rarely be made without an accompanying graph of the data or sketch of the inferred functional form.

A graph can present experimental data or a theoretical prediction or the comparison of both in a clear and compact way. To maximise its effectiveness in your own work, you should give a little thought to the choice of origin, scales and functional forms which you use.

First, there is no necessity whatever that the meeting of the horizontal and vertical axes should coincide with the zero of either variable.

Wherever there is a function, theory finds a star role for its ‘rate of change’. Indeed, relations (differential equations) which involve this quantity are the simplest means the human mind has found to express the principles of such sciences as dynamics, electromagnetism, thermodynamics, and even the statistical laws of chance.

Such an important and practical notion can hardly be abstruse. Although its inventors Newton and Leibniz were innovators of genius, common mortals can easily pick up its use. First of all comes the functional relation: speed versus time; heating-oil consumption versus outside temperature (a steepening curve!); mean river speed versus river width. Sometimes ‘continuous’ is interpreted loosely, as in the frequency of road repairs versus the mean traffic flow, but the idea remains the same. One variable is selected as ‘independent’ (time, perhaps), and the other as ‘dependent’. The choice is not always compelling – is traffic flow controlled by road repairs or are repairs caused by traffic? The mathematician, passionate for abstraction, denotes the independent variable by plain x, the dependent variable by y and expresses the relation as y = f(x). Actually, the mathematician probably has in mind an algebraic relation of y to x, and might object that the above examples do not constitute ‘differentiable’ relationships. With a graph or tabulation of y = f(x) there is usually no difficulty in numerically estimating the differentiated function. This is just the local slope plotted (or tabulated) against x.

In this chapter we shall first reflect upon the procedure for solving second-order differential equations numerically, just as we did with first-order equations. Indeed, the obvious method is closely similar. For simplicity, we shall focus on the oscillator equation (20.6). Again we shall temporarily disregard the fact that our chosen equation has a simple analytic solution. This is ‘simple harmonic motion’, x = A sin (ωt + B) or C sin ωt + D cos ωt. Instead we shall ask how we would set about solving it numerically as there are some general lessons to be learned from this. Let us try to find the analogue for second-order equations of Euler's method already studied for first-order equations. Again we shall choose convenient numerical values for purposes of illustration, so we take ω2 = 4 and will look for a solution in the range t = 0 to t = 4.

We shall try to find this by a step-by-step procedure, with a steplength Δt = 0.1. But again we need a ‘jumping-off point’ – some starting information. If we are given, say, x = 1 at t = 0 we can use the differential equation to get d2x/dt2 at t = 0, but this is not enough information to make the required step. We need to be given (or choose) a starting value for dx/dt, say 2, as well, and use two equations to get started.

For the last hundred years, much of the theory of physical science has been concerned with the analysis of solutions of partial differential equations – the Laplace Equation, the Poisson Equation, Maxwell's Equations, the Wave Equation, the Diffusion Equation, Schrödinger's Equation.…

We have touched upon some of these but have stopped at the very brink of a wider discussion of them, in their three-dimensional forms. We can already see or anticipate some of the lines of attack which might be used when the solution of a partial differential equation is not obvious.

First, the solution may be represented as a sum over some convenient set of functions and an attempt made to find the coefficients. This reduces the problem to one of matrix algebra, which is often a great relief. A further simplification of the problem is achieved if an awkward term can be discarded from the equation and the solution obtained without it. The missing term can then be put back in an approximate way by use of perturbation theory, a technique which is used everywhere in quantum mechanics but has its antecedents in early work on the dynamics of planetary orbits. Such methods have been given an elegant formal basis in the modern theory of linear operators.

Finally, the solution function may be represented by values on a discrete grid of points, whereupon the differential equation becomes a difference equation and we are once more in the realm of matrix algebra.

For the practical physicist and engineer complex numbers provide a handy tool for the analysis of AC circuits. A few basic mathematical rules enable reactive circuit elements like inductances and capacitances to be treated with nothing more difficult than Ohm's law. The first-order linear equation (18.6), dealing with the voltage/current relationship in a simple LR circuit can be used to illustrate the benefits of the complex approach. We shall really just be repeating the steps which led us to (19.13) but we shall see them in a different light.

For generality it is helpful to regard the driving voltage as having both cos and sin components, expressed as A cos ωt – B sin ωt (the choice of sign is to comply with later conventions). Then the responding current also has both components and, following chapter 19, is written a cos ωt – b sin ωt. The mathematical object is then to calculate the coefficients a, b in terms of A, B by equating the driving voltage to the voltage fall across the elements L, R, as before.

Now the voltage change across the resistance is simply proportional to the current so that the cosine and sine oscillations of the voltage are the same, multiplied by R. In engineering terms this is the ‘quadrature response’. On the other hand the inductance requires a time differentiation which effectively exchanges the components (with a change of sign) to give the voltage fall – ωLa sin ωt – ωLb cos ωt.

Musicians, whether they saw, hammer or blow, are all acquainted with overtones. The octave of a note simply doubles the frequency (or very nearly, for the well-tempered piano) and coincides with its first overtone. An octave plus a (major) fifth trebles the frequency and defines the second overtone, and so on. A single sustained note (unless produced by a good tuning fork) actually contains many such overtones, a fact musically exploited by the harmonies of Claude Debussy. The richness and quality of a musical note depends very much upon its tonal content.

The Greeks, first in the field as so often, developed the arithmetic of tonal or harmonic perfection, incorporating it even into architecture and their ideas concerning the motion of the planets. However, the mathematics of modern harmonic theory is largely due to Joseph Fourier (1768–1830) who used it in his treatise on the propagation of heat. Fourier's idea was to represent a function by a series of sine (and cosine) functions. Although wave motion (with its acoustic applications) and diffusion (with its thermal applications) are mathematically described by different equations (see chapters 25,26), both may be solved in a very general way by the method of Fourier.

As well as these two areas of physics, Fourier analysis turns up characteristically in the description of electrical signals.

As explained in the previous chapters, a physical vector such as a velocity, force or torque is described by its components referred to a fixed set of axes. Three dimensions would be usual but this may be reduced to two in some cases, or even (in relativistic physics) increased to four, reckoning time as a dimension. Also, the idea of a set of variables as constituting a ‘vector’ can be extended to non-geometrical systems whose ‘dimensions’ then refer to the number of degrees of freedom of the system, that is, the number of variables needed to identify what state it is in. Two such vectors may be linearly related through an array of coefficients or matrix. Here we discuss the use and basic properties of matrices, and in the following chapter we outline their more formal manipulation.

An illustration is provided by the analysis of electrical circuits. The standard approach is to regard the circuit as consisting of loops. This requires us to introduce as many currents (as variables) as there are loops. The mathematical connection between the loop voltages and currents can be represented by a square array or matrix of impedance coefficients, as follows.

Consider the 2-loop circuit shown in fig. 7.1. In the left-hand loop a current I1, driven by the battery potential V1, flows through the series resistances R,R1; in the right-hand loop, the current I2 driven by V2 flows through R and R2.