The present edition leaves the original essentially unchanged. However, several misprints and minor slips have been corrected, details of the presentation improved, parts of subsections 1.2.6, 3.1.3, 3.4.3, 5.1.1, and 5.1.3 have been rewritten, and a final section “What is a Differential Equation?” has been added.

In the light of classroom experience (my own experience, that of Professor Bowden and several colleagues) it was advisable not to deviate from the original general conception of the work. The preface to the original edition mentioned the difficulty of reconciling the pedagogically appropriate with the historically correct. Also this somewhat delicate balance was left undisturbed.

Professor Bowden and I are deeply grateful to Professor Anneli Lax for incorporating this revised version into the New Mathematical Library, for numerous improvements, both major and minor, of the text, and for editorial expertise which greatly facilitated the toil and trouble of revision.

Mechanics is the study of the action of forces on bodies. That part in which the bodies are at rest and, consequently, the forces are in equilibrium, is called statics in contrast to the other part, dynamics, in which the forces are not in equilibrium and, consequently, the bodies not at rest. Here we shall be concerned with the simpler and firstdeveloped branch, statics, which is conveniently introduced by consideration of the contributions of Stevinus and Archimedes. Although the first real achievements are due to Archimedes and preceded Stevinus' by many centuries, I prefer to discuss the latter first.

STEVINUS AND ARCHIMEDES

Stevinus, a Dutchman, lived in the 16th Century, contemporary with Descartes, a century or so before Newton, Leibniz, and the invention of the differential calculus. He was a brilliant applied mathematician who was fascinated by the usefulness of mathematics: for Stevinus, mathematics to be good had to be good for something. He was one of the first to use decimal fractions and showed their usefulness for everyday affairs, he invented the first horseless carriage, and he constructed dykes, which still serve Holland to this day. His achievements are commemorated by his statue in his native city, Brügge. If you ever go there, look him up. Meanwhile we shall consider his derivation of the Law of the Inclined Plane.

Inclined Plane

Even crude, casual, unavoidable everyday experience presents the curious with questions. Indeed, the simpler the experience the more difficult to avoid meeting pertinent questions head-on.

The reader should be somewhat familiar with the concepts and the techniques of integral and differential calculus; yet knowledge of the theory of differential equations is not a prerequisite. What such equations are and how they must be treated will be explained (roughly but sufficiently for our purpose) later, when they naturally emerge from physical problems. It will turn out that differential equations are useful in science. We cannot understand how and why they are useful before we have used them.

SECTION 1. FIRST EXAMPLES

Rotating Fluid

One lump of sugar, or two? Cream? We have all observed a lady taking tea. What happens? The faster she stirs, the higher up the side of her cup the tea climbs. If she stirs too fast she spills it and ruins an afternoon. Her teacup contains a problem for her and a problem for us. Our problem is amenable to mathematical treatment: What is the surface shape of the rotating tea?

First consider a motionless liquid. We have all seen a glass of water when no one is kicking the table. Its surface looks flat, yet closer examination shows its surface to be not entirely horizontal; it curls up ever so slightly at the edges, due to surface tension. For water substitute mercury, and surface tension causes precisely the opposite effect, a curling down at the edges. A phenomenon distinctly visible in a mercury barometer.

SECTION 1. MEASUREMENT

The Tunnel

Astronomers have measured the distance of the Sun from the Earth; even the distance of the fixed stars. How did they do this? Not by strolling through outer space with a measuring rod. The distance of places that cannot be reached is calculated from the distance of places that can be reached. To measure the stars we get down to Earth; cosmological survey has a terrestial base.

We begin with a terrestial problem. Due to increasing population a certain city of ancient Greece found its water supply insufficient, so that water had to be channeled in from a source in the nearby mountains. And since, unfortunately, a large hill intervened, there was no alternative to tunneling. Working from both sides of the hill, the tunnelers met in the middle as planned. See Figure 1.1.

How did the planners determine the correct direction to ensure that the two crews would meet? How would you have planned the job? Remember that the Greeks could not use radio signal or telescope, for they had neither. Nevertheless they devised a method and actually succeeded in making their tunnels from both sides meet somewhere inside the hill. Think about it.

Of course, had not the source been on a higher level than the city, there would not have been gravity to make the water flow through this aqueduct. But, to better concentrate on the crux of the matter, let us neglect the complication due to difference of levels.

In these lectures we will discuss:

1. (1) Very simple physical or pre-physical problems; problems that could be discussed at the high school level.

2. (2) The relation of mathematics to science and of science to mathematics. This relation is a two-way street. Though more usual, it is not always the case that mathematics is applied to science; also there is traffic in the opposite direction. Good driving takes note of the oncoming traffic.

3. (3) Elementary calculus, for without some calculus one's idea of how mathematics is applied to science is necessarily inadequate.

Also, as their title indicates, these lectures will deal with my ideas about methods. First, let me say that there is no one teaching method which is the method; there are as many good methods as there are good teachers. To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important; he must develop his personality.

In my presentation I shall, by and large, follow the genetic method. The essential idea of this method is that the order in which knowledge has been acquired by the human race will be a good order for its acquisition by the individual.

Whereas statics, we recall, is that part of mechanics which is concerned with the equilibrium of bodies, dynamics is that part which is concerned with the motion of bodies. The former, as we have had occasion to note, goes back to the Greeks; to Archimedes' discovery of the Law of the Lever and his application of it to the integral calculus. The latter is relatively new; it starts with Galileo.

SECTION 1. GALILEO

Galileo is known by his first name; his family name is Galilei. He was born in 1564 and died in 1642. To believers in the transmigration of souls the date of his death is important. Not only did he die in the year in which Newton was born, conveniently for their speculations, he died shortly before Newton was born. A much more important date is 1636, the year in which he completed the book on which his fame so securely rests, the Dialogue Concerning Two New Sciences. Although many of his brilliant predecessors, beginning with Aristotle, and including that most versatile of versatile geniuses, Leonardo da Vinci, had been interested in the free fall of heavy bodies, Galileo was incomparably the greatest dynamicist of them all. He inherited a dogma and bequeathed a science.

His tomb is to be found in Florence, in the Church of Santa Croce, among those of Leonardo and Michelangelo the artists, Dante the poet, and Machiavelli the politician.

To date, what have we done? First we discussed measurement, especially in astronomy; then simple but pervasive topics culled from the history of statics, and finally, great discoveries from the history of dynamics—so many of which hark back to the stars. We have seen something of the role played by mathematics in the development of science; that the aim of physics is to condense its knowledge into mathematical formulae; that, as Galileo so delightfully expressed it, the book of Nature is written in mathematical characters.

Yet this view, although undeniable, is one-sided—or should I say unidirectional? Of course mathematics helps physics. But you must not suppose that help always flows downstream from mathematics to physics; the river of thought is tidal. My object in this chapter is to navigate an incoming tide, to show how help flows also from physics to mathematics.

My lecture-room navigation will not be reproduced here as my upstream voyage is already carefully charted in my Mathematics and Plausible Reasoning, Vol. 1, pp. 142–167, to which the interested mariner is directed.

The following treatment of integral transforms in applied mathematics is directed primarily toward senior and graduate students in engineering and applied science. It assumes a basic knowledge of complex variables and contour integration, gamma and Bessel functions, partial differential equations, and continuum mechanics. Examples and exercises are drawn from the fields of electric circuits, mechanical vibration and wave motion, heat conduction, and fluid mechanics. It is not essential that the student have a detailed familiarity with all of these fields, but knowledge of at least some of them is important for motivation (terms that may be unfamiliar to the student are listed in the Glossary, p. 89). The unstarred exercises, including those posed parenthetically in the text, form an integral part of the treatment; the starred exercises and sections are rather more difficult than those that are unstarred.

I have found that all of the material, plus supplementary material on asymptotic methods, can be covered in a single quarter by first-year graduate students (the minimum preparation of these students includes the equivalent of one-quarter courses on each of complex variables and partial differential equations); a semester allows either a separate treatment of contour integration or a more thorough treatment of asymptotic methods. The material in Chapter 4 and Sections 5.5 through 5.7 could be omitted in an undergraduate course for students with an inadequate knowledge of Bessel functions.

The exercises and, with a few exceptions, the examples require only those transform pairs listed in the Tables in Appendix 2.