"Of the
making of new books there is no end," said the Preacher. He (she?) might
have added "and of the reprinting of old books there is not enough." All
too often great old books become unavailable and students are denied the
opportunity to learn from them. This is all the more serious when the books
in question are written by the original creators of the topics they
discuss.Here's one. Jürgen Moser's Stable and Random Motions in Dynamical Systems is motivated by the stability problem in celestial mechanics: can we prove that the solar system is stable? In the first chapter, Moser explains the historical roots of the question, makes it precise, and sets up the mathematical questions that the rest of the book will address. The other chapters center on two big theorems: the Kolmogorov-Arnold-Moser (KAM) theorem, dealing with quasi-periodic motions, and the Smale-Birkhoff theorem, connecting dynamical systems with Bernoulli processes. These are, respectively, the "stable" and "random" aspects mentioned in the title.
The book is part of Princeton University Press's Landmarks in
Mathematics series. The series consists of paperback reprints of old
mathematical classics, ranging from Milnor's Topology from the
Differentiable Viewpoint to Cartan and Eilenberg's Homological
Algebra. All in all, a worthy endeavor. [Fernando
Q. Gouvêa]
My first reaction to Modern Applied Mathematics for Engineers was that I wouldn't find it very interesting. After all, books with titles like this appear all the time. Then I ran into this:
All those who deal with engineering education are well aware of the precarious state of basic mathematical skills and conceptual comprehension which is typical of modern students. Most beginning engineering graduate students haven't had a good experience with mathematics. (...) The mathematical methods encountered in each course do not evolve into unified patterns which the future engineer would be able to recognize and use universally.(...)What the authors are saying is that we mathematicians aren't doing a good job of teaching mathematics. Specifically, we're not getting across concepts that have wide applicability. To counter this problem, their book packs a great deal of material into a small space, attempting a unifying point of view. And concepts are given priority over proofs:We strongly believe that this situation can and should be radically improved. We think that it is truly possible to offer a beginning graduate student a concise yet comprehensive course which summarizes, unifies, and completes his/her mathematical knowledge by constructing a comprehensive system of operating mathematics and setting up patterns that can be recalled for use in a wide range of engineering disciplines. This book is our response to this challenge.
With this in mind we have mostly forgone the theorem-proof format for a more informal style. We must confess that it was done with a certain relish; in this sense, as in many others, this is a book written by engineers for engineers. However, we have not eliminated all the theorems and have not presented the applied mathematics as merely a bag of tricks. The important theorems we retained are used as pivotal points in the exposition of particular concepts. They play an important role in summarizing concepts and making the student consciously realize that any established method or technique has well-defined limitations.Well, that's interesting. Here's a way to "sell" the importance of theorems: they tell us the conditions for something to be true. And the book is also interesting. It starts with set theory, then covers logic, algebraic structures, linear algebra, metrics and topology, Banach and Hilbert spaces, Fourier series, transforms, and partial differential equations. While most mathematicians know most of this stuff, I suspect they may find much to ponder in the approach and style chosen by the authors. This one's worth a look. [Fernando Q. Gouvêa]
Finally, here's a very different kind of applied
mathematics. Structure and Interpretation of Classical Mechanics is
basically an expository account of classical mechanics. But it takes a few
unusual stances. First of all, the authors use "correct" notation for
partial derivatives, avoiding the conceptual mess that the Leibnizian
notation often brings with it. They do this, they claim, not only because
it is right and the other notation is wrong (the authors quote a strong
argument to this effect from Spivak's Calculus on Manifolds), but
also because this notation avoids critical conceptual traps allowed by less
formal notation. Second, the authors use the Scheme programming
language to introduce a computational aspect to their exposition. Again,
they do this not just because it allows them to make pictures of their
dynamical systems, but because making the mathematics sufficiently precise
to be attacked computationally is itself a valuable exercise that will help
students understand it better. As they say, "Our requirement that our
mathematical notations be explicit and precise enough that they can be
interpreted automatically, as by a computer, is very effective in
uncovering puns and flaws in reasoning." Third, they focus on understanding
motion rather than on deriving equations, and finally, they include many
recent results into the text. The book, which comes out of a course taught
at MIT, is one that mathematicians may want to read, not just for the
mathematical content, but also for the pedagogical ideas it reflects. Maybe
we can learn something here. [Fernando Q. Gouvêa]
Stable and Random Motion in Dynamical Systems, by Jürgen Moser. Princeton University Press, 2001. Softcover, 200 pp, $14.95. ISBN 0-691-08910-8.
Modern Advanced Mathematics for Engineers, by Vladimir V. Mitin, Dmitri A. Romanov, and Michael P. Polis. Wiley, 2001. Hardcover, 308 pp., $89.95. ISBN 0-471-41770-X.
Structure and Interpretation of Classical Mechanics, by Gerald Jay Sussman and Jack Wisdom. MIT Press, 2001. Hardcover, 534pp., $60.00. ISBN 0-262-019455-4.
Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of FOCUS and MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.
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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouvˆa, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.
MAA Online is edited by Fernando Q. Gouvˆa (fqgouvea@colby.edu). Last modified: Wed May 30 13:57:38 -0500 2001