Throughout this book so far we have talked about fields as functions of position only. In most physical applications of this theory the fields are also functions of time. To physicists fields are dynamical things that both experience and effect interactions with other systems, so the study of their development in time is of great importance.

As the time dependence of any physical quantity is usually governed by some basic dynamical law and the laws are different in different branches of physics, it might be thought that the ideas of vector analysis would be too general and all embracing to have anything to say about the time dependence of fields. To some extent this is true and we cannot here go far into dynamical problems. But there are two aspects of time dependence that can be very profitably discussed as aspects of vector analysis. It is fitting to conclude any study with a chapter on these topics, particularly because they provide an excellent illustration of how our general ideas can actually be used in a real physical context.

In this chapter we shall talk of scalar and vector fields ø(t, r) and f(t, r) and, in addition to the differentiation symbolised by ▽, we shall frequently form derivatives of the fields with respect to time: ∂ø/∂t and ∂f/∂t. We assume that differentiation with respect to time does not alter the vector character of a field: like ø itself, ∂ø/∂t is a scalar and, like f, ∂f/∂t is a vector.

The material of chapters 7–9 is so interrelated that it would not have been practical to provide separate exercises for each, or even to arrange the exercises in an order that relates strictly to the topics of the individual chapters. The student should, however, find little difficulty in identifying questions that require the understanding of earlier material only. An attempt at logical progress has been made except that the first three questions clearly relate to chapters 7, 8 and 9 separately. Sections 2 and 3 of chapter 10 do not call for illustration by exercises, but the last few questions in the following relate in part to section 1 of chapter 10.

At this stage in the book the idea behind the quasi-square and quasi-cube technique can be taken for granted. The technique is still relevant to some of the exercises, but many are now independent of it – and particular fields that are of importance in physics are now introduced when possible. However, some of the exercises still deal with quite artificial constructs. As before, these serve to illustrate general ideas by non-trivial particular examples. In some case the work is saved from being intolerably cumbersome only by the symmetries built in. Artificial though this may be, training in the exploitation of symmetries is of value in itself.

Scalar fields

Many physical quantities may be suitably characterised by scalar functions of position in space. Given a system of cartesian axes a scalar field ø can be represented as ø = ø(r), where r is the position vector defined in chapter 2. There is no real difference between this way of referring to a scalar field and the alternative statement ø = ø(x, y, z), except that in this latter form one is definitely committed to a particular set of cartesian coordinates for r, while the form ø(r) can be taken to refer to any coordinate axes – or indeed to any other equivalent way of defining the position vector r of a point.

In dealing with functions of a single variable, x say, the universal standby that helps one visualise the function is the simple graph, fig. 1. This notion is familiar enough to need no explanation and is so closely associated with the function itself that it is sometimes difficult to remember that the graph is not the function. But when we come to functions of more than one variable, things become rather different. Let us go one step at a time and first consider a function of two variables f(x, y). The values over which x and y vary cover all or part of the xy-plane – a plane where before we had a line – and the function f(x, y) may then be thought of as plotted in a third dimension which we may call z; z = f(x, y).

The justification for adding one more to the many available texts on vector analysis cannot be novelty of content. In this book I have included a few topics that are more frequently encountered as part of the discussion of a specific physical topic – in the main in fluid dynamics or electrodynamics – but this in itself does not justify telling the whole story over again, perhaps least of all in the classical way – in bold-type vectors, excluding suffixes, tensors, n-dimensional space etc. However, after giving a course in the standard shape for many years (to Scottish students in their second year – roughly comparable in level to an English first-year University course) I gradually developed a way of presenting the subject which gave the old tale a new look and seemed to me to make a more coherent whole of it. Once my course had taken this shape the students were left without a suitable text to work from. More and more ad hoc handouts became necessary and finally the skeleton of this book emerged.

In preparation for publication I spent more time on devising the sections of exercises than on the main text which was largely in existence from the start. The exercises were indeed devised rather than compiled and I am well aware that they reveal my idiosyncrasies. I see them as a necessary part of learning a language.