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

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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

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