How to find stationary values of functions of a single variable f(x), of several variables f(x, y, …) and of constrained variables, where x, y, … are subject to the n constraints gi(x, y, …) = 0, i = 1, 2, …, n will be known to the reader and is summarized in Sections A.3 and A.7 of Appendix A. In all those cases the forms of the functions f and gi were known, and the problem was one of finding the appropriate values of the variables x, y, etc.

We now turn to a different kind of problem in which we are interested in bringing about a particular condition for a given expression (usually maximizing or minimizing it) by varying the functions on which the expression depends. For instance, we might want to know in what shape a fixed length of rope should be arranged so as to enclose the largest possible area, or in what shape it will hang when suspended under gravity from two fixed points. In each case we are concerned with a general maximization or minimization criterion by which the function y(x) that satisfies the given problem may be found.

The calculus of variations provides a method for finding the function y(x). The problem must first be expressed in a mathematical form, and the form most commonly applicable to such problems is an integral.

Compared to communication in other species, human speech is a highly differentiated faculty, enabling us to communicate complex information in an efficient way. There are many aspects to the psychological study of language, including its production and understanding (listening, articulation, memorization), and the use of indirect means of communication through writing and reading. In all of these aspects cross-cultural differences can be observed. In this chapter we will deal with the main theme of cross-cultural psycholinguistic research, namely the extent to which underneath different words and rules of grammar there is commonality between languages.

In the first section research on linguistic relativity is presented, addressing the question of to what extent speaking a particular language influences one's thinking. We look at two topics on which much of the discussion about linguistic relativity has been focussed, namely perception and categorization of colors, and orientation in space. We present the case of relativism and counterarguments based on empirical cross-cultural studies. The second section is on universalist approaches, especially the notion of universal grammar. Again, not only the evidence in favor, but also challenges are presented.

On the Internet we present some additional information. There is an entry on language development (Additional Topics, Chapter 8), pointing out some of the complexities a child has to master in order to acquire a language.

The notion of development comes into this book at three levels. First, there is phylogenetic development. It deals with variation across species, and the emergence of new species over long periods of time. This form of development will be discussed in Chapter 11. Second, the term “development” can refer to cultural changes in societies. Development in this sense will be touched upon in Chapter 10 (where we discuss the anthropological tradition of cultural evolution), and in Chapter 18 (where we focus on national development). In the present and the following chapter we are mainly concerned with the course of development of the individual through the life span, or ontogenetic development. In this chapter, we will focus on cultural similarities and differences in developmental patterns in infancy and early childhood; the next chapter will deal with late childhood, adolescence and adulthood.

Culture as context for development

Individual development can be considered as the outcome of interactions between a biological organism and environmental influences. Although we consider the separation of “nature” and “nurture” to be largely an outdated distinction (see Chapter 11, and the last section of Chapter 12), the relative importance of the biological and the cultural (environmental-experiential) components of behavior has formed the major dimension underlying the differences between various schools of thinking on ontogenetic development in the psychological literature.

In Chapter 14, we developed the basic theory of the functions of a complex variable, z = x + iy, studied their analyticity (differentiability) properties and derived a number of results concerned with values of contour integrals in the complex plane. In this current chapter we will show how some of those results and properties can be exploited to tackle problems arising directly from physical situations or from apparently unrelated parts of mathematics.

In the former category will be the use of the differential properties of the real and imaginary parts of a function of a complex variable to solve problems involving Laplace's equation in two dimensions, whilst an example of the latter might be the summation of certain types of infinite series. Other applications, such as the Bromwich inversion formula for Laplace transforms, appear as mathematical problems that have their origins in physical applications; the Bromwich inversion enables us to extract the spatial or temporal response of a system to an initial input from the representation of that response in “frequency space” – or, more correctly, imaginary frequency space.

Some other topics that could have been considered, had space permitted, are the location of the (complex) zeros of a polynomial, the approximate evaluation of certain types of contour integrals using the methods of steepest descent and stationary phase, and the so-called “phase-integral” solutions to some differential equations. However, for these and many more, the brief outlines given in the final section of this chapter will have to suffice.

The reader will be familiar with how, through Taylor series (see Section A.6 of Appendix A), complicated functions may be expressed as power series. However, this is not the only way in which a function may be represented as a series, and the subject of this chapter is the expression of functions as a sum of sine and cosine terms. Such a representation is called a Fourier series. Unlike Taylor series, a Fourier series can describe functions that are not everywhere continuous and/or differentiable. There are also other advantages in using trigonometric terms. They are easy to differentiate and integrate, their moduli are easily taken and each term contains only one characteristic frequency. This last point is important because, as we shall see later, Fourier series are often used to represent the response of a system to a periodic input, and this response often depends directly on the frequency content of the input. Fourier series are used in a wide variety of such physical situations, including the vibrations of a finite string, the scattering of light by a diffraction grating and the transmission of an input signal by an electronic circuit.

The Dirichlet conditions

We have already mentioned that Fourier series may be used to represent some functions for which a Taylor series expansion is not possible.

Since Mathematical Methods for Physics and Engineering (Cambridge: Cambridge University Press, 1998) by Riley, Hobson and Bence, hereafter denoted by MMPE, was first published, the range of material it covers has increased with each subsequent edition (2002 and 2006). Most of the additions have been in the form of introductory material covering polynomial equations, partial fractions, binomial expansions, coordinate geometry and a variety of basic methods of proof, though the third edition of MMPE also extended the range, but not the general level, of the areas to which the methods developed in the book could be applied. Recent feedback suggests that still further adjustments would be beneficial. In so far as content is concerned, the inclusion of some additional introductory material such as powers, logarithms, the sinusoidal and exponential functions, inequalities and the handling of physical dimensions, would make the starting level of the book better match that of some of its readers.

To incorporate these changes, and others to increase the user-friendliness of the text, into the current third edition of MMPE would inevitably produce a text that would be too ponderous for many students, to say nothing of the problems the physical production and transportation of such a large volume would entail.

In this chapter, we shift our focus from behavior that is primarily social to behavior that is cognitive. Social cognition was discussed in Chapter 4, where the phenomena of attribution, conformity and self-construal were presented as social psychological manifestations of the cultural context. In Chapter 8, we shall return to a consideration of cultural aspects of cognition, where links between language and culture are explored. In this chapter we focus on the more traditional cognitive phenomena that deal with knowing and interpreting the world, using such notions as intelligence, abilities and styles. We begin with a brief overview of the historical legacy of thinking about how human populations are similar and different in their cognitive lives. In each of the subsequent sections we present four perspectives on relationships between cognition and culture, beginning with a set of conceptualizations that involve a unitary view of cognition (captured in the notion of general intelligence). Thereafter we present cognitive styles, which are general preferences to deal with the world in a particular way. The third perspective is one that focusses on the East–West contrasts in cognition, where there has been much recent research on differences in the cognitive life of western and East Asian populations. Finally, the fourth perspective is contextualized cognition, in which cognitions are seen as task-specific and embedded in sociocultural contexts.

In the previous two chapters, we introduced the most important second-order linear ODEs in physics and engineering, listing their regular and irregular singular points in Table 7.1 and their Sturm–Liouville forms in Table 8.1. These equations occur with such frequency that solutions to them, which obey particular commonly occurring boundary conditions, have been extensively studied and given special names.

In this chapter, we discuss these so-called “special functions” and their properties. Inevitably, for each set of functions in turn, the discussion has to cover the differential equation they satisfy, their polynomial or power series form with some particular examples, their orthogonality and normalization properties, and their recurrence relations. In addition, as first introduced in this chapter, most sets possess a Rodrigues' formula and a generating function.

Although each of these aspects needs to be treated in sufficient detail for the enquiring reader to be satisfied about the validity of the results stated, their serial presentation, for one set of functions after another, tends to become rather overwhelming. Consequently it is suggested that once the reader has become familiar with the general nature of each of the aspects, by studying, say, Sections 9.1 to 9.3 on Legendre functions, associated Legendre functions and spherical harmonics, he or she may treat other sets of functions more lightly, turning in the first instance to the summary beginning on p. 377, and only referring to the detailed derivations, proofs and worked examples in Sections 9.4 to 9.9 when specific needs arise.

In this chapter, we turn to the study of statistics, which is concerned with the analysis of experimental data. In a book of this nature we cannot hope to do justice to such a large subject; indeed, many would argue that statistics belongs to the realm of experimental science rather than in a mathematics textbook. Nevertheless, physical scientists and engineers are regularly called upon to perform a statistical analysis of their data and to present their results in a statistical context. This justifies the inclusion of the subject in a book such as this, but we will concentrate on those aspects of direct relevance to the presentation of experimental data.

Experiments, samples and populations

We may regard the product of any experiment as a set of N measurements of some quantity x or of some set of quantities x, y, …, z. This set of measurements constitutes the data. Each measurement (or data item) consists accordingly of a single number xi or a set of numbers (xi, yi, …, zi), where i = 1, …, N. For the moment, we will assume that each data item is a single number, although our discussion can be extended to the more general case.

As a result of inaccuracies in the measurement process, or because of intrinsic variability in the quantity x being measured, one would expect the N measured values xi, x2, …, xN to be different each time the experiment is performed. We may therefore consider the xi as a set of N random variables.

LIFE

Our evidence for St.'s life comes mainly from the Siluae, particularly 3.5, an epistle to his wife, and 5.3, the poem on his father's death. St. was a poet of two cities and two cultures. He was born around 50 ce in Naples, ‘practically a Greek city’ (quasi Graecam urbem, Tac. Ann. 15.33), where Hellenic culture was supported by Roman wealth and power. The city celebrated Greek-style games founded by Augustus; Greek and Latin, and probably Oscan, were spoken in the city, with Greek remaining the language of cultural prestige in many official contexts. St. was the son of an eminent grammaticus and Greek professional poet who, composing in a tradition of extemporaneous poetry, won prizes at the major Greek games and was honoured with a statue in the Athenian agora. St. was doubly privileged: born in a city with a rich cultural heritage, he was taught a demanding curriculum in Greek literature by his father in a period when such knowledge was crucial for social advancement.

At some point in St.'s youth father and son moved to Rome where St. senior had a successful career as grammaticus to the Roman elite and probably also to the imperial family. He enjoyed too the great honour of reciting his poem on the civil war of 68–9 in the temple of Jupiter Optimus Maximus on the Capitol (5.3.199–204). The father thus may have been closer to the imperial court than the son.

In this chapter and the next, the solution of differential equations of types typically encountered in the physical sciences and engineering is extended to situations involving more than one independent variable. A partial differential equation (PDE) is an equation relating an unknown function (the dependent variable) of two or more variables to its partial derivatives with respect to those variables. The most commonly occurring independent variables are those describing position and time, and so we will couch our discussion and examples in notation appropriate to them.

As in the rest of this book, we will focus our attention on the equations that arise most often in physical situations. We will restrict our discussion, therefore, to linear PDEs, i.e. those of first degree in the dependent variable. Furthermore, we will discuss primarily second-order equations. The solution of first-order PDEs will necessarily be involved in treating these, and some of the methods discussed can be extended without difficulty to third- and higher-order equations. We shall also see that many ideas developed for ODEs can be carried over directly into the study of PDEs.

Initially, in the current chapter, we will concentrate on general solutions of PDEs in terms of arbitrary functions of particular combinations of the independent variables, and on the solutions that may be derived from them in the presence of boundary conditions. We also discuss the existence and uniqueness of the solutions to PDEs under given boundary conditions.