Modern Differential Geometry of Curves and Surfaces with Mathematica: (Textbooks in Mathematics)

Presenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library, where nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted. Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use Mathematica to plot many interesting curves and surfaces. Including as many topics of the classical differential geometry and surfaces as possible, it highlights important theorems with many examples. It includes 300 miniprograms for computing and plotting various geometric objects, alleviating the drudgery of computing things such as the curvature and torsion of a curve in space.

Table of Contents:
Curves in the Plane. Famous Plane Curves. Alternative Ways of Plotting Curves. New Curves from Old. Determining a Plane Curve from its Curvature. Global Properties of Plane Curves. Curves in Space. Construction of Space Curves. Calculus on Euclidean Space. Surfaces in Euclidean Space. Nonorientable Surfaces. Metrics on Surfaces. Shape and Curvature. Ruled Surfaces. Surfaces of Revolution and Constant Curvature. A Selection of Minimal Surfaces. Intrinsic Surface Geometry. Asymptotic Curves and Geodesics on Surfaces. Principal Curves and Umbilic Points. Canal Surfaces and Cyclides of Dupin. The Theory of Surfaces of Constant Negative Curvature. Minimal Surfaces via Complex Variables. Rotation and Animation using Quaternions. Differentiable Manifolds. Riemannian Manifolds. Abstract Surfaces and their Geodesics. The Gauss–Bonnet Theorem.

About the Author :
Abbena, Elsa; Salamon, Simon; Gray, Alfred

Review :
“This is a nicely readable textbook on differential geometry. It offers an outstanding, comprehensive presentation of both theoretical and computational aspects … There are hundreds of illustrations that help the reader visualize the concepts. … It is a nicely written book, strongly recommended to all with an interest in differential geometry, its computational aspects and related fields.” — In EMS Newsletter, June 2007