From the fifteenth century onwards, Southeast Asia’s widening global connections stimulated commercial activity and fostered cultural interaction along well-traveled seaborne and overland trade routes. Emblematic of this invigorated maritime activity was the prosperous town of Melaka, on the west coast of the Malay Peninsula, which became renowned as a center for regional and international commerce. First mentioned in Chinese sources in 1403, Melaka’s emergence on the historical scene serves as a marker for the beginning of Southeast Asia’s early modern period. It also signals a time of transition from the period dominated by the “classical” states to one in which new polities and rulers, though always conscious of their illustrious predecessors, were intent on marking out their own spheres of authority. Across the region flourishing maritime commerce brought not merely increased wealth, but new religious ideas and technologies that enabled many polities to expand their territory and for some the opportunity to develop more elaborate bureaucratic structures.

Because of its location at the crossroads of international trade, Southeast Asia was inevitably affected by the ripple effects of world developments, most notably the consolidation of the Ming dynasty in China. In 1369, a decade after his accession in 1358, the first Ming emperor made clear his policies towards the Nanyang, the “Southern Ocean” – a term generally used by the Chinese to refer to Southeast Asia – by sending a series of missions to announce his victory over the Mongols and to proclaim the establishment of a new regime in China. Less directly felt, but no less significant, was the rise of powerful Islamic empires in Central Asia and India. The prestige of Islam steadily rose as the Ottoman Turks extended their conquests into the Balkans, Greece, and Europe. In 1453 Ottoman armies captured Constantinople, the heavily fortified capital of “Rum” (the name Arabs gave to the Byzantine Empire). The fall of Constantinople also meant that the Ottoman and the Mongols controlled the land and maritime trade routes to Asia.

This text is written for those who have studied calculus in the sixth form at school, and are now ready to review that mathematics rigorously and to seek precision in its formulation. The question sequence given here tackles the key concepts and ideas one by one, and invites a self-imposed precision in each area. At the successful conclusion of the course, a student will have a view of the calculus which is in accord with modern standards of rigour, and a sound springboard from which to study metric spaces and point set topology, or multi-dimensional calculus.

Generations of students have found the study of the foundations of the calculus an uncomfortable business. The reasons for this discomfort are manifold.

(1) The student coming from the sixth form to university is already familiar with Newtonian calculus and has developed confidence in the subject by using it, and experiencing its power. Its validity has been established for him/her by reasonable argument and confirmed by its effectiveness. It is not a source of student uncertainty and this means that an axiomatic and rigorous presentation seems to make heavy weather of something which is believed to be sound, and criticisms of Newtonian calculus seem to be an irritating piece of intellectual nit-picking.

(2) At an age when a student's critical capacity is at its height, an axiomatic presentation can have a take-it-or-leave-it quality which feels humiliating: axioms for the real numbers have none of the ‘let's-play-a-game’ character by which some simpler systems appeal to the widespread interest in puzzles. The bald statement of axioms for the real numbers covers up a significant process of decision-making in their choice, and the Axiom of Completeness, which lies at the heart of most of the main results in analysis, seems superfluous at first sight, in whatever form it is expressed.

The states of matter

The following is a thought experiment. Nothing written here should be construed as instructions for practical action. This is not a homework problem.

You go to your freezer, take out an ice cube, and put it in an experimental fluid of your choice. Some of the choicest fluids for this experiment are made along the banks of glens in Scotland. A scientist often feels compelled to repeat an experiment many times to be sure that the results are dependable. If you do so in this case, you might not be able to remember the results.

Here is what you observe. The fluid starts out at room temperature. The ice is initially at the temperature of the freezer, say, 260 K or -13°C. At first the ice warms up, cooling the fluid. But then, when its temperature reaches 273 K or 0°C something happens that is both extraordinary and dramatic. The ice refuses to get any warmer. Whereas just before it had no difficulty raising its temperature as it absorbed heat, it now absolutely refuses to warm the slightest bit more. Instead it undergoes a catastrophic change in its microscopic state. As it absorbs more heat from its surroundings, the molecules separate from their solid crystal structure. The ice melts into water. The whole profound transformation takes place without permitting any change in temperature until it is completely finished. If that were not the case, you wouldn’t cool your drinks with ice. You might as well throw cold rocks in them.

The thermodynamic variables

In Chapter 1 we made a truly remarkable simplification of nature. For a body of a given U, N and V it's not necessary to know what each of its atoms is doing. If we allow the body to come to equilibrium, all the microscopic information necessary to specify its macroscopic state is contained in the value of a single variable, the entropy. In this chapter we shall explore some of the consequences of that profound insight.

If we know explicitly the equation U = U(S, V, N), then we have enough information to find anything we want. For this reason S, V and N are said to be the proper variables of the energy U. We have already had some examples of how information can be extracted from this equation, and we shall have some more (Problems 2.5 and 2.7). In reality the equation is hardly ever known explicitly for any real system, but the fact that it exists in principle – which was the point of the first chapter – is what we wish to exploit.

There are seven thermodynamic variables, and these can be grouped into classes in various ways. One way is to assemble them into conjugate pairs. Each pair, when multiplied together, has the units of energy. They can also be classified as either extensive or intensive. Both are shown in Table 2.1.