Geometry is relevant

What have the following situations in common?

Preliminaries

Today my particle accelerator in Geneva is not operating, and I have decided to have the day off. I take a bus ride to a nearby village, and get off at the bus-stop in the main square. The village is very pretty, and I wander around enjoying the chalets and the views. I also venture into the surrounding hillside, with its criss-crossing paths. After a couple of hours, I begin to think it is time to return to the village, to catch the bus back to Geneva. If I know that I have walked a total of 6 miles, how far am I from the bus-stop?

The answer of course is that we cannot tell – it could be anything from zero distance to 6 miles. It all depends on how straight or crooked my walk had been up till then. This is because the problem is basically one involving vectors, in which not only a magnitude is involved, but also a direction. The distance that I am from the bus-stop is given by the length of the vector formed by adding all the vectors corresponding to the various straight bits of path along which I had walked.

Vectors are thus very useful in any problem in which directions are implied.

This book originated from a one-semester course on introductory engineering mathematics taught at MIT over the past ten years primarily to first-year graduate students in engineering. While all students in my class have gone through standard calculus and ordinary differential equations in their undergraduate years, many still feel more awe than confidence and enthusiasm toward applied mathematics. Upon entering graduate school they need a quick and friendly exposure to the elementary techniques of partial differential equations for studying other advanced subjects and the existing literature, and for analyzing original problems. For them a popular first step is to take a course in advanced calculus, which is usually taught to large classes. To cater to a large audience with diverse backgrounds, an author or instructor tends to concentrate on mathematical principles and techniques. Applications to physics and engineering are often kept at an elementary level so that little effort is needed to set up the examples before, or interpret them after, finding the solutions. In some branches of engineering, students get further exposure to and practice in theoretical analysis in many other courses in their own fields. However, in other branches such reinforcements are less emphasized; all too often practical problems are dealt with by tentative arguments undeservingly called the Engineering Approach.

In engineering endeavors rooted in physical sciences, deep understanding and precise analysis cannot usually be achieved without the help of mathematics.

Introduction

In previous chapters we have only discussed techniques of getting exact solutions. Clearly, the problems must be sufficiently idealized for these techniques to be effective. For more practical problems either the boundary geometry or the governing equations are less simple, and one must often be content with approximate solutions. Among methods of approximation two are the most important. If the problem is close to one that is solvable exactly, perturbation methods are powerful tools for getting analytical answers. If, however, the problem is far from anything that can be solved exactly, strictly numerical methods via discretization must be employed. In general, analytical perturbation methods are much more effective in gaining qualitative insight, while numerical methods are good in producing quantitative information. Sometimes the two can be mixed for studying small departures from a basic state that must itself be solved numerically.

In this chapter we shall give an introductory account of the analytical approach of perturbation methods. To have a bird's-eye view of the subject, let us first outline the typical ideas and procedure of a perturbation analysis.

(i) Identify a small parameter. This is a very important first step which must be taken by recognizing the physical scales relevant to the problem. One then normalizes all variables with respect to these characteristic scales. In the normalized form, the governing equations will display certain dimensionless parameters, each of which represents the relative importance of certain physical mechanisms.

While programming languages such as FORTRAN are designed for numerical computations, a number of them, e.g., MACSYMA, MAPLE, MATHEMATICA, REDUCE and THEORIST, have been developed to handle symbolic manipulations. With these languages one can use the computer to manipulate algebraic and trigonometric operations, differentiation and integration, Taylor series expansions, Laplace transform, and the solution of simple ordinary differential equations. When repetitive and lengthy manipulations are needed, these symbolic languages are extremely useful for speed and accuracy. In this chapter we first introduce some basic commands of MACSYMA and then illustrate their use for perturbation analysis. Only the bare essentials are introduced to get the reader started. More proficient users of the computer may very well wish to consult the MACSYMA manuals in order to find more shortcuts. There are also a few books devoted exclusively to the basics and applications of MACSYMA. Computer Algebra in Applied Mathematics – An Introduction to MACSYMA by R.H. Rand (1984) and Variational Finite Element Methods – A Symbolic Computation Approach by A.I. Beltzer (1990) are both helpful.

Getting started

MACSYMA can be installed on a variety of computers. On each the entry protocol may be slightly different. On a personal computer (PC) you simply enter WINDOWS, point the icon at the MACSYMA logo and click the mouse. After a short wait a few lines of trademark information appear on the monitor screen, followed by the first command line, (c1)

All the command lines are preceded by the letter c and numbered consecutively.

Beyond elementary functions such as exponential, logarithmic, sinusoidal and hyperbolic functions, there is a host of so-called special functions that arise frequently in physical problems. Examples are Bessel functions, Legendre polynomials, Mathieu functions, hypergeometric functions, etc. Often these special functions emerge from the solution of partial differential equations when the boundary possesses a certain special geometry. For example, Bessel functions are associated with circular boundaries, while Legendre polynomials are associated with spherical boundaries, etc. In this chapter we choose to acquaint the readers only with the basic properties of the Bessel functions, and with applications in wave propagation and fluid flow. Certain essential facts such as series definitions, recursion formulas, orthogonality and asymptotic approximations will be discussed. Though far from exhaustive, these facts can already go a long way toward many applications, and can prepare the reader for further study of advanced aspects and other special functions. For quick access to further properties the reader should take advantage of some of the popular handbooks of special functions such as Erdelyi (1953a) and Abramowitz and Stegun (1964). For thorough theoretical expositions the reader must consult more advanced treatises such as Watson (1958).

Circular region and Bessel's equation

In this section we give a practical motivation for the need of Bessel functions by examining wave motion in a circular domain.

Thus far we have only dealt with real variables; the use of complex representation for a sinusoidal function of time is just a matter of convenience involving only the real variable t and no new principles. To an analytical engineer the techniques of complex variables are essential because of their wide range of applications. Many two-dimensional potential theories in classical hydrodynamics, static electricity, steady diffusion, etc., can be directly solved by complex functions. The inverse Fourier and Laplace transforms are often most efficiently evaluated in a complex plane. In contrast to most methods of real variables where the mathematical details are tailored to suit the geometry of the boundaries, conformal mapping is a radically different tool whose effectiveness lies in altering the boundaries themselves.

In the following four chapters, we give a guided tour of the basic principles of complex functions, together with applications that range from the elementary to the slightly advanced. In the present chapter the basics of analytic functions and the rules of differential and integration are explained. In Chapter 10 these basics are applied to the techniques of Laplace transform. In Chapter 11 elements of conformal mapping are introduced with examples from hydrodynamics. One of the most beautiful applications of complex functions in continuum mechanics is the formulation and solution of certain mixed boundary-value problems. In Chapter 12 two examples from hydrodynamics and elasticity are examined and the Riemann–Hilbert technique is explained.