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

by Abbena; Elsa

• ISBN13: 9781584884484
• ISBN10: 1584884487
• Edition: 3rd
• Format: Hardcover
• Copyright: 2006-06-21
• Publisher: Chapman & Hall/

Summary

Presenting theory while using Mathematicain 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 Mathematicafor 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 Mathematicacode and added a Mathematicanotebook 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 Mathematicathat 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 Mathematicato 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	p. 1
Famous plane curves	p. 39
Alternative ways of plotting curves	p. 73
New curves from old	p. 99
Determining a plane curve from its curvature	p. 127
Global properties of plane curves	p. 153
Curves in space	p. 191
Construction of space curves	p. 229
Calculus on Euclidean space	p. 263
Surfaces in Euclidean space	p. 287
Nonorientable surfaces	p. 331
Metrics on surfaces	p. 361
Shape and curvature	p. 385
Ruled surfaces	p. 431
Surfaces of revolution and constant curvature	p. 461
A selection of minimal surfaces	p. 501
Intrinsic surface geometry	p. 531
Asymptotic curves and geodesics on surfaces	p. 557
Principal curves and umbilic points	p. 593
Canal surfaces and cyclides of dupin	p. 639
The theory of surfaces of constant negative curvature	p. 683
Minimal surfaces via complex variables	p. 719
Rotation and animation using quaternions	p. 767
Differentiable manifolds	p. 809
Riemannian manifolds	p. 847
Abstract surfaces and their geodesics	p. 871
The Gauss-Bonnet theorem	p. 901
Table of Contents provided by Blackwell. All Rights Reserved.