This book, a volume in SIAM's Classics in
Applied Mathematics series, lives up to its title and fits somewhere
between Ordinary Differential Equations by Carrier and Pearson and
the second edition of Ordinary Differential Equations by Hartman,
also "classics" in the series. This no-nonsense book is less heuristic
than the Carrier/Pearson text, yet not as theoretical as Hartman's
treatise. Originally published in 1996 by Wiley, the book is a successful
effort to present analytical and numerical techniques as two sides of the
same coin, as well as to provide an introduction to mathematical modeling.
The text's main source of examples is classical mechanics.Typos and errors in the original edition have been corrected, with no major changes in the text itself. In the prefaces to the original and current editions, the authors have provided Internet addresses for numerical programs. The URLs are no longer valid, but a graduate student or researcher should be able to locate appropriate software. There are some strange typographical mutations on pp. 25-26, where norms are introduced. The references have not been updated for this edition. Also, there are some inaccuracies in the new Index (mostly, it seems, with respect to entries appearing toward the end of the book). For example, 'inverse resonance' appears on p. 320, not p. 321; a mention of the 'Maxwell equations' occurs on p. 303, not p. 305.
The book is intended for both undergraduate and graduate courses, but I don't believe it would be palatable to American undergraduates unless they have had multivariable calculus, linear algebra, and an introductory ODEs course. Laplace transforms and series solutions are missing, although there is a great emphasis on one-step and multistep numerical methods, as well as treatments of stability, chaotic systems, singular perturbations, and boundary value problems. There are many good examples (in smaller print), but they are stated briefly, with few details. There are well-chosen exercises, but relatively few and only at the end of each chapter, with no answers at the back of the book (as is common in European texts). Overall, this is a fine text for a graduate course in applied differential equations and a valuable reference that should be on the shelves of researchers and those teaching differential equations.[Henry Ricardo]
AK Peters has recently published the first
translation into English of Elon Lages Lima's book Fundamental Groups
and Covering Spaces, which was originally published in Portuguese as
part of IMPA's "Project Euclides" series. The book introduces the reader
to some of the key ideas of algebraic topology, and it does so with clear,
easy-to-read exposition and many examples. I found the book an enjoyable
read, even knowing most of the punchlines ahead of time.
The book is divided into two parts. The first part is about homotopy, fundamental groups, and winding numbers. This reviewer found the chapter on the classical matrix groups, in which Lima looks at the groups SO(n), SU(n), and Sp(n) and computes what their fundamental groups are, to be especially interesting and novel to a book at this level. The second part of the book deals with covering spaces, starting with local homeomorphisms, liftings, and covering maps and working through a classification of the fundamental groups of compact surfaces. He concludes with a discussion of orientability in general, oriented double coverings in particular, and a nice discussion of the fact that the fundamental group of a nonorientable manifold has a subgroup of index two.
In addition to the lucid writing — for which translator Jonas Gomes certainly deserves a piece of the credit — and plentiful exercises, another nice feature of Lima's book is the number of references to the history of the ideas which he is presenting. While the book is by no means a history book, it does contain more references to the literature than one typically find in undergraduate textbook, which I found quite refreshing. All in all, I would not hesitate to recommend this book to an undergraduate wanting to learn the subject. [Darren Glass]
In the glad to have you back department,
I'm delighted that Springer has decided to reprint the two volumes of
B. L. van der Waerden's Algebra. Based in part on lectures by Emmy
Noether and Emil Artin, this is the book that brought "abstract algebra" to
the mathematical world. It is not for the faint of heart: van der Waerden
is terse and precise and does not spend much time on easy details. On the
other hand, the book reflects the excitement that accompanied the birth of
axiomatic algebra. It was written when the power of the new methods was
still news for most mathematicians, and a guide to the new ideas was still
needed. This is that guide. It covers all the usual topics and then
some. Not, perhaps, a book to give to an undergraduate (though who knows?
for some undergraduates, it might just work), but still a book to
treasure. I'm glad it's back. My only hesitation is to wonder whether it
would not have been better to bring it back as one fat volume instead of
two thin ones.
Also worth noting is the arrival in paperback of Fibonacci's Liber Abaci, whose hardcover edition we hailed a year ago. I said then that this was a book worth having, and I still think that's true, and even more so now that we have a less expensive (but still quite handsome) paperback edition.
I'm also pleased to see a reprint of
Calculus Problems and Solutions, by A. Ginzburg, just out from
Dover. It's not that I remember this specific book, but rather that I
remember (and value) this kind of book: page after page of hard, grungy,
technical calculus problems. (When I was a student, we used to use one by a
fellow called Demidovich, one of those amazingly cheap Soviet editions that
were very popular in the third world. I've never seen or heard of it
since.) No one would want a diet of this kind of problem, but they have
their place, and lately they have been hard to find, so I'm glad to have
this join the ranks of valuable reprints from Dover.
Finally, even though it's not a reprint and doesn't really fit this
section, I would like to go on record that I am delighted at the very
possibility of a book title such as The Structure of Spherical
Buildings. Yes, I know they aren't real buildings (they do have
apartments in them, however), but isn't the mental image one worth
treasuring? Kudos to whoever came up with that title. I hope they did it on
purpose. [Fernando Q. Gouvêa]
Ordinary Differential Equations in Theory and Practice, by Robert Mattheij and Jaap Molenaar. SIAM, 2002. Paperback, 405pp., $45.00 ($31.50 to SIAM members). ISBN 0-89871-531-8.
Fundamental Groups and Covering Spaces, by Elon Lages Lima, translated by Jonas Gomes. AK Peters, 2003. Hardcover, 230 pp., $49.00. ISBN 1568811314.
Algebra, volumes I and II, by B. L. van der Waerden. Springer-Verlag, 2003. Paperback, 265 pp. + 284 pp., $34.95 each. ISBN 0-387-40624-7 and 0-387-40625-5.
Calculus: Problems and Solutions, by A. Ginzburg. Dover, 2003. Paperback, 455 pp., $24.95. ISBN 0-486-43277-7.
The Structure of Spherical Buildings, by Richard M. Weiss. Princeton, 2004. Hardcover, 135 pp., $45.00. ISBN 0-691-11733-0.
Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.
Darren Glass is a VIGRE Assistant Professor of Mathematics at Columbia University. His research interests include number theory, algebraic geometry, and cryptography. He can be reached at glass@math.columbia.edu.
Fernando Q. Gouvêa is Professor of Mathematics at Colby College in Waterville, ME. He is interested in number theory, the history of mathematics, Christian theology, poetry, science fiction, comic books, politics, classics, and football (the real thing, not the American version).
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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 Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.
MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu). Last modified: Sat Apr 10 11:55:58 Eastern Standard Time 2004