Modern Differential Geometry of Curves and Surfaces with Mathematica, Second EditionThe Second Edition combines a traditional approach with the symbolic manipulation abilities of Mathematica to explain and develop the classical theory of curves and surfaces. You will learn to reproduce and study interesting curves and surfaces - many more than are included in typical texts - using computer methods. By plotting geometric objects and studying the printed result, teachers and students can understand concepts geometrically and see the effect of changes in parameters. Modern Differential Geometry of Curves and Surfaces with Mathematica explains how to define and compute standard geometric functions, for example the curvature of curves, and presents a dialect of Mathematica for constructing new curves and surfaces from old. The book also explores how to apply techniques from analysis. Although the book makes extensive use of Mathematica, readers without access to that program can perform the calculations in the text by hand. While single- and multi-variable calculus, some linear algebra, and a few concepts of point set topology are needed to understand the theory, no computer or Mathematica skills are required to understand the concepts presented in the text. In fact, it serves as an excellent introduction to Mathematica, and includes fully documented programs written for use with Mathematica. Ideal for both classroom use and self-study, Modern Differential Geometry of Curves and Surfaces with Mathematica has been tested extensively in the classroom and used in professional short courses throughout the world. |
Contents
Curves in the Plane | 1 |
Studying Curves in the Plane with Mathematica | 25 |
Famous Plane Curves | 49 |
Alternate Methods for Plotting Plane Curves | 75 |
New Curves from Old | 97 |
Determining a Plane Curve from its Curvature | 127 |
Global Properties of Plane Curves | 153 |
Curves in Space | 181 |
Surfaces in 3Dimensional Space | 359 |
Surfaces in 3Dimensional Space via Mathematica | 391 |
Asymptotic Curves on Surfaces | 417 |
Ruled Surfaces | 431 |
Surfaces of Revolution | 457 |
Surfaces of Constant Gaussian Curvature | 481 |
Intrinsic Surface Geometry | 501 |
Differentiable Manifolds | 521 |
Tubes and Knots | 207 |
Construction of Space Curves | 217 |
Calculus on Euclidean Space | 245 |
Surfaces in Euclidean Space | 269 |
Examples of Surfaces | 295 |
Nonorientable Surfaces | 317 |
Metrics on Surfaces | 341 |
Riemannian Manifolds | 557 |
Abstract Surfaces | 573 |
Geodesics on Surfaces | 595 |
The GaussBonnet Theorem | 627 |
Principal Curves and Umbilic Points | 641 |
Common terms and phrases
a_,b_ a*Cos abstract surface alpha tt asymptotic curves Axes->None Boxed->False catenoid Christoffel command compute constant coordinates Corollary Cosh curvature and torsion curve alpha cycloid D[alpha tt defined Definition denote differentiable function differentiable manifold differential equation ellipse Evaluate example formula gamma Gauss map Gaussian curvature geodesic given helicoid Hence hyperboloid implies inversion involute k₁ Lemma Mathematica mean curvature metric minimal curve minimal surface Möbius strip monkey saddle nonparametric form nonzero open subset parabola paraboloid ParametricPlot patch x:U plane curve plot principal curvatures principal curves Proof regular patch regular surface second fundamental form Section shape operator signed curvature Simplify Sin[t space curve sphere Sqrt surface in R3 surface of revolution tangent vector Theorem theta torus u_,v_ unit normal unit-speed curve vector field Weierstrass ди მა