# Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition

CRC Press, Jun 21, 2006 - Mathematics - 1016 pages
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.

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### Contents

 Curves in the Plane 5 Famous Plane Curves 39 Alternative Ways of Plotting Curves 73 New Curves from Old 99 Determining a Plane Curve from its Curvature 127 Determining a Plane Curve from Its Curvature 127 5 Determining a Plane Curve from Its Curvature 128 Notebook 5 146 Global Properties of Plane Curves 153
 Notebook 14 452 Surfaces of Revolution and Constant Curvature 461 Notebook 15 488 A Selection of Minimal Surfaces 501 Intrinsic Surface Geometry 531 Notebook 17 548 Asymptotic Curves and Geodesics on Surfaces 557 Notebook 18 582

 Notebook 6 181 Curves in Space 191 Notebook 7 217 Construction of Space Curves 229 Notebook 8 254 Calculus on Euclidean Space 263 Notebook 9 283 Surfaces in Euclidean Space 287 Notebook 10 320 Nonorientable Surfaces 331 Notebook 11 352 Metrics on Surfaces 361 Notebook 12 379 Shape and Curvature 385 Notebook 13 420 Ruled Surfaces 431
 Principal Curves and Umbilic Points 593 Canal Surfaces and Cyclides of Dupin 639 The Theory of Surfaces of Constant Negative Curvature 683 Notebook 21 712 Minimal Surfaces via Complex Variables 719 Rotation and Animation Using Quaternions 767 Differentiable Manifolds 809 Riemannian Manifolds 847 Notebook 25 868 Abstract Surfaces and Their Geodesics 871 The GaussBonnet Theorem 901 Bibliography 931 Name Index 953 Notebook Index 977 Copyright

### References to this book

 Elementary Differential Geometry, Revised 2nd EditionBarrett O'NeillLimited preview - 2006
 Riemannian Geometry: A Modern IntroductionIsaac ChavelLimited preview - 2006
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