CHOICE CURRENT REVIEWS FOR ACADEMIC UBRARIES
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February 2011 Vol. 48 No. 06 SCIENCE & TECHNOLOGY Mathematics
Susannah Sieper Marketing Director A K Peters, Ltd. 5 Commonwealth Road Suite 2C Natick, MA 01760-1526
Thejollowing review appeared in the February 2011 issue ojCHOICE:
48-3330 QA641 2009-48247 CIP Banchoff, Thomas. Differential geometry of curves and surfaces, by Thomas Banchoff and Stephen Lovett. A K Peters, 2010. 331p bibl index afp ISBN 9781568814568, $49.00 This title has been reviewed jointly with "Differential Geometry of Manifolds," by Stephen Lovett. Banchoff (Brown Univ.) and Lovett (Wheaton College) have produced two connected works on modern differential geometry, assuming only minimal prerequisites. The first volume, Differential Geometry of Curves and Spaces, is a rich, concrete introduction to the theory of plane and space curves and surfaces in R1\0]3, full of classical topics that undergraduate students all too rarely see. Curves includes local notions such as curvature, torsion, and the Frenet frame, as well as global results such as the four-vertex theorem for plane curves and the Fary-Milnor theorem about knottedness of space curves, The study of surfaces requires development of more machinery and, accordingly, the authors begin with local definitions using coordinate patches, and then introduce the Gauss map, fundamental forms, geodesics, varieties of curvature, and treatments of ruled and minimal surfaces. There is a lucid discussion of tensors, phrased in local coordinate terms, leading to structural equations for surfaces and the Theorema Egregium. The volume culminates with the Gauss-Bonnet theorem and some of its consequences. What makes this book so special, however, are the dozens of illuminating online applets that accompany it. Readers can readily use these to visualize and interact with examples from the text, or to develop their own.<p>Differential Geometry of Manifolds, by Lovett alone, continues the first title, although it can be read independently. Intended to provide a working understanding of the differential geometry of n-dimensional manifolds, it does a good deal more, offering treatments of analysis on manifolds (including the generalized Stokes's theorem) in addition to Riemannian geometry. An especially interesting chapter on applications to physics includes some general relativity, string theory, symplectic geometry, and Hamiltonian mechanics. Appendixes give background in general topology, the calculus of variations, and multilinear algebra. (One small awkwardness: the Banchoff and Lovett volume frequently refers to the appendix on topology in Lovett's work.)<p>Both books are very carefully constructed and written with a deft touch and an enticing, friendly tone. The stated prerequisites of multivariable calculus and linear algebra are perhaps ambitious for undergraduates in places, but these works certainly would motivate readers to make the necessary investment of time and thought. These two books are valuable library acquisitions. Summing Up: Highly recommended, Upper-division undergraduates and graduate students. -- S. J. Colley, Oberlin College
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