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Modern differential geometry of curves and surfaces with Mathematica. 3rd ed. (English) Zbl 1123.53001
Studies in Advanced Mathematics. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-448-7/hbk). 984 p. (2006).
There appeared two earlier editions of this book, namely [Modern differential geometry of curves and surfaces. Boca Raton, MA: CRC Press (1993; Zbl 0795.53001)] and an extended and improved version [Modern differential geometry of curves and surfaces with Mathematica. 2nd ed., Boca Raton, FL: CRC Press (1998; Zbl 0855.53001)]. Its ambition is to give the student an easier and deeper insight into classical differential geometry with the help of computer graphics and computer algebra tools of Mathematica. Mathematica is also used to visualize and understand topological ideas, for instance nonorientable surfaces such as Möbius strip, projective plane and Klein’s bottle. The program code can be downloaded from the publisher’s site http://www.crcpress.com.
Here, after the untimely death of Alfred Gray, the editors Elsa Abbena and Simon Salamon, present a version of the second edition containing some additional material and a new presentation differing from the former ones the way in which mathematical themes and associated Mathematica programs are organized. While Gray scattered them in the text and added a library of programs at the end of his book, here the relevant programs are placed at the ends of the chapters. Many of them have been enhanced. This separation of computer code from mathematical content is intended to make it easier to see their mutual dependences and designing them in a different computer language as for instance Maple.
Some of the additional material was already prepared by Alfred Gray himself. This concerns mainly chapter 21. It treates the theory of surfaces of constant negative curvature in Euclidean 3-space including the classical Bäcklund transformation. Plots of 1- and 2-soliton surfaces are shown, especially the Bianchi transform of the pseudosphere, named Kueun’s surface. We mention that a picture of a hand-made model of this surface was already shown in the early textbook of [L. Bianchi, “Lezioni di Geometria Differenziale” (see German translation, B. G. Teubner, Leipzig, Berlin) (1910; JFM 41.0676.01)] about 100 years ago. The use of quaternions for presenting orthogonal transformations of Euclidean 3-space is the other new feature added by the editors. It is the content of chapter 23. In comparision with the second edition various sections are expanded by additional text and graphics.

##### MSC:
 53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 68W30 Symbolic computation and algebraic computation 53A05 Surfaces in Euclidean and related spaces 53A04 Curves in Euclidean and related spaces 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 53-04 Software, source code, etc. for problems pertaining to differential geometry
##### Software:
CandS; Mathematica; Maple