Showing a Fourier transform to a physics student generally produces the same reaction as showing a crucifix to Count Dracula. This may be because the subject tends to be taught by theorists who themselves use Fourier methods to solve otherwise intractable differential equations. The result is often a heavy load of mathematical analysis.

This need not be so. Engineers and practical physicists use Fourier theory in quite another way: to treat experimental data, to extract information from noisy signals, to design electrical filters, to ‘clean’ TV pictures and for many similar practical tasks. The transforms are done digitally and there is a minimum of mathematics involved.

The chief tools of the trade are the theorems in Chapter 2, and an easy familiarity with these is the way to mastery of the subject. In spite of the forest of integration signs throughout the book there is in fact very little integration done and most of that is at high-school level. There are one or two excursions in places to show the breadth of power that the method can give. These are not pursued to any length but are intended to whet the appetite of those who want to follow more theoretical paths.

The book is deliberately incomplete. Many topics are missing and there is no attempt to explain everything: but I have left, here and there, what I hope are tempting clues to stimulate the reader into looking further; and of course, there is a bibliography at the end.

In Chapters 3 and 4 we discussed functions f of only one variable x, which were usually written f(x). Certain constants and parameters may also have appeared in the definition of f, e.g. f(x) = ax + 2 contains the constant 2 and the parameter a, but only x was considered as a variable and only the derivatives f (n)(x) = dnf/dxn were defined.

However, we can equally well consider functions that depend on more than one variable, e.g. the function f(x, y) = x 2 + 3xy, which depends on the two variables x and y. For any pair of values x, y, the function f(x, y) has a well-defined value, e.g. f(2, 3) = 22. This notion can clearly be extended to functions dependent on more than two variables. For the n-variable case, we write f(x 1, x 2, …, xn ) for a function that depends on the variables x 1, x 2, …, xn . When n = 2, x 1 and x 2 correspond to the variables x and y used above.

Functions of one variable, like f(x), can be represented by a graph on a plane sheet of paper, and it is apparent that functions of two variables can, with a little effort, be represented by a surface in three-dimensional space.

In Chapter 9 we discussed the algebra of vectors and in Chapter 10 we considered how to transform one vector into another using a linear operator. In this chapter and the next we discuss the calculus of vectors, i.e. the differentiation and integration both of vectors describing particular bodies, such as the velocity of a particle, and of vector fields, in which a vector is defined as a function of the coordinates throughout some volume (one-, two- or three-dimensional). Since the aim of this chapter is to develop methods for handling multi-dimensional physical situations, we will assume throughout that the functions with which we have to deal have sufficiently amenable mathematical properties, in particular that they are continuous and differentiable.

Differentiation of vectors

Let us consider a vector a that is a function of a scalar variable u. By this we mean that with each value of u we associate a vector a(u). For example, in Cartesian coordinates a(u) = ax (u)i + ay (u)j + az (u)k, where ax (u), ay (u) and az (u) are scalar functions of u and are the components of the vector a(u) in the x-, y- and z-directions respectively. We note that if a(u) is continuous at some point u = u 0 then this implies that each of the Cartesian components ax (u), ay (u) and az (u) is also continuous there.

The first two chapters of this book review the basic arithmetic, algebra and geometry of which a working knowledge is presumed in the rest of the text; many students will have at least some familiarity with much, if not all, of it. However, the considerable choice now available in what is to be studied for secondary-education examination purposes means that none of it can be taken for granted. The reader may make a preliminary assessment of which areas need further study or revision by first attempting the problems at the ends of the chapters. Unlike the problems associated with all other chapters, those for the first two are divided into named sections and each problem deals almost exclusively with a single topic.

This opening chapter explains the basic definitions and uses associated with some of the most common mathematical procedures and tools; these are the components from which the mathematical methods developed in more advanced texts are built. So as to keep the explanations as free from detailed mathematical working as possible – and, in some cases, because results from later chapters have to be anticipated – some justifications and proofs have been placed in appendices. The reader who chooses to omit them on a first reading should return to them after the appropriate material has been studied.

For reasons that are explained in the preface to Foundation Mathematics for the Physical Sciences (FMPS), the text of the third edition of Mathematical Methods for Physics and Engineering (MMPE) (Cambridge: Cambridge University Press, 2006) by Riley, Hobson and Bence, after a number of additions and omissions, has been republished as two somewhat overlapping texts. Essential Mathematical Methods for the Physical Sciences contains most of the more advanced material, and specifically develops mathematical methods that can be applied throughout the physical sciences; FMPS is an augmented version of the more introductory material, principally concerned with mathematical tools rather than methods. The full text of MMPE, including all of the more specialised and advanced topics, is still available under its original title.

As in the third edition of MMPE, the penultimate subsection of each chapter of FMPS consists of a significant number of problems, nearly all of which are based on topics drawn from several sections of that chapter. Also as in the third edition, hints or outline answers are given in the final subsection, but only to the odd-numbered problems, leaving all even-numbered problems free to be set as unaided homework.