Calculus: Concepts and Connections is a readable introduction to the
basic topics of single and multivariable calculus from the basic idea of
limits up through Stokes' Theorem. It is a textbook clearly aimed at
the "average" student. The main difference between this and other
comparable textbooks such as Stewart's Calculus is that
explanations of concepts and examples are more extensive and more explicit
attempts are made to establish the connections between concepts. The
standard topics are treated in more or less the usual order. The authors have obviously taken pains to develop good exercise sets. Typically, each chapter has writing exercises, a collection of fairly routine problems, exercises designed to use a graphing calculator or computer algebra system, and exploratory problems. The latter are intended to be more challenging and to provoke a deeper level of understanding. A nice feature in the text is the use of an icon to identify pitfalls arising from injudicious use of calculators or computer algebra systems.
There are well over two hundred explicit examples of applications in the text or in exercises; these fall into the general categories of biology, chemistry, economics, physiology, engineering, physics and sports. While these are not especially deep applications, they generally have enough meat on them to be interesting.
Intuitive explanations and arguments of plausibility are usually favored over proofs, though there are a variety of proofs of varying levels of generality and rigor. In addition, there is an appendix with more rigorous proofs of some results.
This would be an appealing self-study text as well as a good choice for a class of average ability. It would probably seem too slow moving for strong students. [William J. Satzer]
For roughly the last thirty-five years, practitioners of control
theory have been aware that complex exponentials play a significant
role in the solution of control problems governed by partial
differential equations. The resulting "non-harmonic" analysis
techniques have been described in a variety of journal articles, but
Fourier Series in Control Theory represents the first
appearance of a synthesis of the relevant literature as well as a
collection of new results.
The story begins with the solution of a problem with a vibrating string having two free endpoints. If one can observe only oscillations of the left endpoint of the string of length l over a finite time interval T, what if anything can be said about the unknown initial data? A fairly direct approach using Fourier series and Parseval's equality establishes bounds on the L2 norm of the speed of the left endpoint as long as T is at least twice l. However, even a slight modification to the assumptions makes the method unworkable.
The book proceeds to generalize and expand the approach from this simpler problem to solve various controllability problems for vibrating strings, beams, membranes, plates, shells or systems of such. The authors' methods have applications broader than control theory: they offer, for example, a new proof of Bernstein's generalization of Pólya's theorem on singularities of the Dirichlet series.
The prerequisites for the reader are pretty formidable. Knowledge of the basics of linear partial differential equations and familiarity with Lebesgue and Sobelev spaces is assumed, as is a fair piece of introductory functional analysis. From the control theory side, the concepts of controllability, observability and stabilizability are reviewed in the context of systems governed by partial differential equations. Experience with these concepts in the simpler setting of ordinary differential equations would be useful background.
This is primarily a monograph aimed at experts and researchers in the field, although it would be accessible to a graduate student willing to fill in the background. The exposition is clear but terse.[William J. Satzer]
Instabilities, Chaos and Turbulence is an exploration
of instability and turbulence in fluid dynamics. This text is not a
litany of techniques but rather introduces non-linear dynamics and
chaos in complex systems by telling the story of a fluid's transition
to turbulence in various settings.
After an introductory chapter and a review of two-dimensional systems of linear differential equations, Manneville introduces some classical examples of two-dimensional non-linear systems (the Duffing oscillator, the van der Pol oscillator) and discusses several techniques for solving these systems. The remainder of the text is devoted to modeling instability in fluids.
A fundamental example is: apply heat to the bottom of pot containing water and model how the water transitions from conduction to convection to turbulence. Manneville describes how the parameters of the system (the depth of the water, the viscosity of the water, the dimensions of the pot) establish thresholds (bifurcation points) which determine these transition points. He then turns his attention to the appearance of instability and turbulence in open flows such as occurs when an obstacle is placed in a flowing river. The models presented are meant to be practical representations of real-world behavior. As such, Manneville makes frequent references to experimental results, comparing the theoretical results to the empirical data and discussing hypotheses about why discrepancies exist.
The final chapter of the text is a generic discussion of why it is appropriate to use non-linear dynamics to model the Earth's climate; it does not introduce any additional mathematics or present any specific models of the Earth's climate (except for a very simple one as an exercise). The first appendix reviews the linear algebra used in the text. (It contains much more than is typically taught in a first course in linear algebra.) In a second appendix, the author provides a survey of numerical techniques for approximating solutions to initial value problems. Each of the eight chapters ends with some quite interesting exercises. They are often very involved, sometimes running for 2 or 3 pages each.
I like Manneville's decision to tell a story about instabilities in fluid dynamics. Unfortunately, it takes a while for this story to become apparent — the book would be much stronger if the path of the story were more clearly laid out in the introductory chapter. The drawback of his approach is that it is more difficult to lift specific techniques from such a text. However, I much prefer a narrative approach to a more traditional technique-example style. The pace of the text is probably too fast for most undergraduates. It requires the reader to be adept at the techniques of linear algebra, and ordinary and partial differential equations, and to have some knowledge of basic thermodynamics and Newtonian mechanics. For someone interested in seeing non-linear dynamics in a specific context, Instabilities, Chaos and Turbulence is a nice book.[Stephen T. Ahearn]
Biostatistics: A
Foundation for Analysis in the Health Sciences (8e) by Wayne Daniel is
a textbook appropriate for advanced undergraduate and beginning graduate
students, as well as a useful reference for researchers in the health and
biological fields. While a number of texts exist for these audiences, this
has been written at a more accessible level for those without any
mathematical exposure beyond algebra. This accessibility should be
particularly advantageous for those looking to use it as a text for
advanced undergraduates.
The book begins laying the groundwork for statistical inference as well as providing some background in sampling in the first chapter. While oftentimes this is not done until after probability has been presented, Daniel uses it as motivation for the development of the theory that is needed for formal inference procedures later in the text. This also provides the student with no statistical background some insight into the notational differences seen in the following chapter on descriptive statistics.
All the standard topics that are generally taught in an introductory statistics course are covered. Specialized topics such as diagnostic testing, logistic regression, survival analysis, and life tables make it especially appealing for those in a biostatistics course, or for those wishing to use it as a basic reference in a medical or biological research setting. In addition, the text has impressive coverage of nonparametric and distribution-free statistical methods for hypothesis testing, analysis of variance, and regression analysis.
The text makes use of statistical software packages throughout to demonstrate many of statistical procedures it presents. MINITAB is the primary package used and many of the necessary commands are given for readers who wish to take advantage of software to assist them in analyzing data. In addition to MINITAB, SAS output is given for a few topics such as contingency table analysis and analysis of variance. For survival analysis, SPSS is the primary package used with details given on how to implement the methods covered in the text.
While some texts that incorporate statistical software commands and output often are difficult to use for those who prefer an alternate package (or no package at all), this is not true of Daniel's book. Readers not using the same packages should not find the information on software distracting. This adds to its broad appeal and reinforces its appropriateness for a variety of audiences seeking a comprehensive summary of basic methods used to analyze data — particularly in the biological and health sciences. [Liam O'Brien]
Publication Data
Calculus: Concepts & Connections, by Robert T. Smith and Roland B. Minton. McGraw-Hill, 2006. Hardcover, 1104 pp., $132.80. ISBN 0-07-282623-1.
Fourier Series in Control Theory, by Vilmos Komornik and Paola Loreti. Springer-Verlag, 2005 Hardcover, ix + 226 pp., $89.95. ISBN 0-387-22383-5.
Instabilities, Chaos, and Turbulence: An Introduction to Nonlinear Dynamics and Complex Systems, by Paul Manneville. Imperial College Press, 2004. Hardcover, 391 pp., $84.00, ISBN 1-86094-483-3; Paperback, 391 pp., $38.00, ISBN 1-86094-491-4.
Biostatistics: A Foundation for Analysis in the Health Sciences, by Wayne W. Daniel. John Wiley, 2005. Hardcover, 782 pp., $109.95. ISBN 0-471-45654-3.
Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.
Stephen T. Ahearn teaches at the Department of Mathematics and Computer Science of Macalester College.
Liam O'Brien is assistant professor of statistics at Colby College in Waterville, ME.
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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: Wed Aug 24 10:53:00 Eastern Daylight Time 2005