We all continue to feel a frustration because of our inability to foresee the soul's ultimate fate. Although we do not speak about it, we all know that the objectives of our science are, from a general point of view, much more modest than the objectives of, say, the Greek sciences were; that our science is more successful in giving us power than in giving us knowledge of truly human interest. [E. P. Wigner, 1972, The place of consciousness in modern physics, in Consciousness and Reality, eds. C. Muses and A. M. Young, New York, Outerbridge and Lizard, pp. 132–141].

In this book we have explored many methods of mathematics as used in the physical sciences. Mathematics plays a central role in the physical sciences because it is the only language we have for expressing quantitative relations in the world around us. In fact, mathematics not only allows us to express phenomena in a quantitative way, it also has a remarkable predictive power in the sense that it allows us to deduce the consequences of natural laws in terms of measurable quantities. In fact, we do not quite understand why mathematics gives such an accurate description of the world around us [120].

It is truly stunning how accurate some of the predictions in (mathematical) physics have been. The orbits of the planetary bodies can now be computed with an extreme accuracy. Morrison and Stephenson [72] compared the path of a solar eclipse at 181 BC with historic descriptions made in a city in eastern China that was located in the path of the solar eclipse.

Many problems in mathematical physics exhibit a spherical or cylindrical symmetry. For example, the gravity field of the Earth is to first order spherically symmetric. Waves excited by a stone thrown into water are usually cylindrically symmetric. Although there is no reason why problems with such a symmetry cannot be analyzed using Cartesian coordinates (i.e. (x, y, z)-coordinates), it is usually not very convenient to use such a coordinate system. The reason for this is that the theory is usually much simpler when one selects a coordinate system with symmetry properties that are the same as the symmetry properties of the physical system that one wants to study. It is for this reason that spherical coordinates and cylindrical coordinates are introduced in this section. It takes a certain effort to become acquainted with these coordinate systems, but this effort is well spent because it makes solving a large class of problems much easier.

Introducing spherical coordinates

In Figure 4.1 a Cartesian coordinate system with its x-, y-, and z-axes is shown as well as the location of a point r. This point can be described either by its x-, y-, and z-components or by the radius r and the angles θ and ϕ shown in Figure 4.1. In the latter case one uses spherical coordinates. Comparing the angles θ and ϕ with the geographical coordinates that define a point on the globe one sees that ϕ can be compared with longitude and θ can be compared with colatitude, which is defined as (latitude – 90 degrees).

In most situations, the equations that we would like to solve in mathematical physics are too complicated to solve analytically. One of the reasons for this is often that an equation contains many different terms which make the problem simply too complex to be manageable. However, many of these terms may in practice be very small. Ignoring these small terms can simplify the problem to such an extent that it can be solved in closed form. Moreover, by deleting terms that are small one is able to focus on the terms that are significant and that contain the relevant physics. In this sense, ignoring small terms can actually give a better physical insight into the processes that really do matter.

Scale analysis is a technique in which one estimates the different terms in an equation by considering the scale over which the relevant parameters vary. This is an extremely powerful tool for simplifying problems. A comprehensive overview of this technique with many applications is given by Kline and in Chapter 6 of Lin et al. Interesting examples of the application of scaling arguments to biology are given by Vogel.

With the application of scale analysis one caveat must be made. One of the major surprises of classical physics of the twentieth century was the discovery of chaos in dynamical systems. In a chaotic system small changes in the initial conditions lead to a change in the time evolution of the system that grows exponentially with time.

The updates and changes from the earlier version of the book have to a large extent been driven by the comments of readers and reviewers. The second edition has been extended with Chapters 2, 24, and 25 that cover dimensional analysis, the asymptotic evaluation of integrals, and variational calculus, respectively. In a few places, new material has been inserted, such as Section 19.5 that covers the reciprocity of wave propagation. A number of teachers and students remarked that the level of difficulty of the problems in the first edition was highly variable. The problems in the second edition contain more hints and advice to make these problems more tractable.

The topic of this book is the application of mathematics to physical problems. Mathematics and physics are often taught separately. Despite the fact that education in physics relies on mathematics, it turns out that students consider mathematics to be disjoint from physics. Although this point of view may strictly be correct, it reflects an erroneous opinion when it concerns an education in the sciences. The reason for this is that mathematics is the only language at our disposal for quantifying physical processes. One cannot learn a language by just studying a textbook. In order to truly learn how to use a language one has to go abroad and start using that language. By the same token one cannot learn how to use mathematics in the physical sciences by just studying textbooks or attending lectures; the only way to achieve this is to venture into the unknown and apply mathematics to physical problems.

It is the goal of this book to do exactly that; problems are presented in order to apply mathematical techniques and knowledge to physical concepts. These examples are not presented as well-developed theory. Instead, they are presented as a number of problems that elucidate the issues that are at stake. In this sense this book offers a guided tour: material for learning is presented but true learning will only take place by active exploration. In this process, the interplay of mathematics and physics is essential; mathematics is the natural language for physics while physical insight allows for a better understanding of the mathematics that is presented.

The material of this chapter is usually not covered in a book on mathematics. The field of mathematics deals with numbers and numerical relationships. It does not matter what these numbers are; they may account for physical properties of a system, but they may equally well be numbers that are not related to anything physical. Consider the expression g = df/dt. From a mathematical point of view these functions can be anything, as long as g is the derivative of f. The situation is different in physics. When f(t) is the position of a particle, and t denotes time, then g(t) is a velocity. This relation fixes the physical dimension of g(t). In mathematical physics, the physical dimension of variables imposes constraints on the relation between these variables. In this chapter we explore these constraints. In Section 2.2 we show that this provides a powerful technique for spotting errors in equations. In the remainder of this chapter we show how the physical dimensions of the variables that govern a problem can be used to find physical laws. Surprisingly, while most engineers learn about dimensional analysis, this topic is not covered explicitly in many science curricula.

Two rules for physical dimensions

In physics every physical parameter is associated with a physical dimension. The value of each parameter is measured with a certain physical unit. For example, when I measure how long a table is, the result of this measurement has dimension “length”.

Many physical systems have the property that they can carry out oscillations at certain specific frequencies only. As a child (and hopefully also as an adult) you will have discovered that a swing in a playground will move only with a specific natural period, and that the force that pushes the swing is only effective when the period of the force matches the period of the swing. The patterns of motion at which a system oscillates are called the normal modes of the system. A swing has one normal mode, but you have seen in Section 13.6 that a simple model of a tri-atomic molecule has three normal modes. An example of a normal mode of a system is given in Figure 20.1 which shows the pattern of oscillation of a metal plate which is driven by an oscillator at a fixed frequency. The screw in the middle of the plate shows the point at which the force on the plate is applied. Sand is sprinkled on the plate. When the frequency of the external force is equal to the frequency of a normal mode of the plate, the motion of the plate is given by the motion that corresponds to that specific normal mode. Such a pattern of oscillation has nodal lines where the motion vanishes. These nodal lines are visible because the sand on the plate collects at these lines.