jamesbond 2007-9-10 15:03
Mathematical Tools for Physicists【Trigg】
[Name] Mathematical.Tools.for.Physicists
[Author] George L. Trigg
[Publisher] WILEY-VCH Verlag GmbH & Co. KGaA
[Key Words] Mathematics, Physics
[Briefing]
Mathematics is a central structure in our knowledge. The rigor of mathematical proof places the subject in a very special position with enormous prestige. For the potential user of mathematics this has both advantages as well as disadvantages. On the one hand, one can use mathematics with confidence that in general the concepts, definitions, procedures, and theorems have been thoroughly examined and tested, but the sheer amount of mathematics is often very intimidating to the non-expert. Since the results of mathematics once proved stay in the structure forever, the subject just gets larger and larger, and we do not have the luxury of discarding older theories as obsolete.
So the quadratic formula and the Pythagorean theorem are still useful and valid even though they are thousands of years old. Euclid’s Elements is still used as a text in some classrooms today, and it continues to inspire readers as it did in the past although it treats the mathematics from the time of Plato over 2300 years ago.
Despite the prestige of mathematical proof, most mathematics that we use today arose without proof. The history of the development of calculus is a good example.
Neither Newton nor Leibniz gave definitions of limits, derivatives, or integrals that would meet current standards. Even the real number system was not rigorously treated until the second half of the nineteenth century. In the past, as in modern times, large parts of mathematics were initiated and developed by scientists and engineers. The distinction between mathematicians and scientists was often rather vague. Consider for example, Newton, Euler, Lagrange, Gauss, Fourier, and Riemann. Although these men did important work in mathematics, they were also deeply involved in the sciences of their times. Toward the end of the nineteenth century a splitting occurred between mathematics and the sciences. Some see it in the development of non-Euclidean geometry and especially axiomatic methods reminiscent of Euclid.
At this time mathematics appeared to be taking its own path independent of the sciences. Here are two cases that participated in this division. In the late nineteenth century Oliver Heaviside developed the Operational Calculus to solve problems in electrical engineering. Although this calculus gave solutions in agreement with experiment, the mathematicians of Heaviside’s time could not justify or condone his procedures. Physicists also found the Dirac delta function and Green’s functions extremely useful and developed an appropriate calculus for their use, but the underlying mathematical theory was not available. It was not until the early 1950’s that Laurent Schwartz was able to give a rigorous mathematical foundation for these methods with his Theory of Distributions. Also, early in the twentieth century the relatively new subject of Group Theory was seen as being of use in applications to chemistry and physics, but the few texts available at the time were written in a rather abstract and rigorous mathematical style that was not easily accessible to most non-mathematicians.
The subject was quickly labeled the ‘‘Gruppenpest’’ and ignored by many researchers.
Needless to say, today group theory with its applications to symmetry is a fundamental tool in science.
With the complexity of each field in science and engineering growing so rapidly, a researcher in these fields has little time to study mathematics for its own sake. Each field hasmorematerial than can possibly be covered in a typical undergraduate program, and even graduate students must quickly pick a sub-area of specialization. Often, however, there is a sense that if we just knew more mathematics of the right sort, we could get a better grasp of the subject at hand. So, if we are still in school, we may take a mathematics course, or if not in school, we may look at some mathematical texts. Here some questions arise: which course should we take, do we have the correct prerequisites, what if our mathematics instructor has no knowledge of our field or any applications that we are interested in, are we really in the right course? Furthermore, most texts in mathematics are intended for classroom use. They are generally very proof oriented, and although many now include some historical remarks and have a more user friendly tone, they may not get to the point fast enough for the reader outside of a classroom.
This book is intended to help students and researchers with this problem. The eighteen articles included here cover a very wide range of topics in mathematics in a compact, user oriented way. These articles originally appeared in the Encyclopedia of Applied Physics, a magnificent twenty-three volume set edited by George L. Trigg, with associate editors Eduardo S. Vera and Walter Greulich andmanaging editor EdmundH.
Immergut. The full Encyclopedia was published in the 1990’s by VCH, a subsidiary of John Wiley & Sons, New York. Each article in this volume covers a part of mathematics especially relevant to applications in science and engineering. The articles are designed to give a good overview of the subject in a relatively short space with indications of applications in applied physics. Suggestions for further reading are provided with extensive bibliographies and glossaries. Most importantly, these articles are accessible.
Each article seeks to give a quick review of a large area within mathematics without lapsing into vagueness or overspecialization.
Of course not all of mathematics can be covered in this volume: choicesmust bemade in order to keep the size of the work within bounds.We can only proceed based on those areas that have been most useful in the past. It is certainly possible that your favorite question is not discussed here, and certainly the future will bring newmathematics and applications to prominence, but we sincerely expect that the articles in this volume will be valuable to most readers.
