








Theory of Curves and Surfaces






Organization:  Tokyo Denki University 
Department:  Natural Sciences 






College






This course gives an introduction to classical differential geometry of curves and surfaces. Although the basic structure is traditional, the numerical and experimental approach is adopted in various examples, making use of Mathematica. Therefore this is also a Mathematica literacy through basic differential geometry.






Theory of Curves and Surfaces: An Introduction to Classical Differential Geometry by Mathematica (in Japanese) by Tazawa Pearson Education, 309 pages with CDROM






The theory of curves and surfaces was established long ago. Yet applying the general theory to individual objects is not easy. For instance, integrating the curvature over a curve or constructing a curve with assigned curvature can be very difficult even in the simplest cases. This is because it is not possible in general to solve differential equations explicitly. Hence, the examples appearing in this field have been confined to a small group of calculable objects. But, numerically, Mathematica does these calculations easily, and this makes it possible for us to deal with a wide range of examples, as the reader shall recognize throughout this book. The existing best book on differential geometry by Mathematica was written by Alfred Gray. But the approach of the present course is a little different, and I believe it is sufficiently unique, at least at this moment. In short, we use Mathematica not only to calculate geometric quantities or to draw geometric objects, but also to visualize the basic notions of differential geometry and to perform experiments in geometry making use of Mathematica graphics, animations in particular. Topics: Preliminaries:  Vector functions
 Mathematica Literacy
Plane Curves:  Plane Curves
 Well Known Plane Curves
 Generic Plane Curves
 Arclength and the Moving Frame
 Curvature
 The Fundamental Thereom of Plane Curves
 Global Properties of Plane Curves
 Plane Curves by Mathematica
Space Curves:  Space Curves
 The Moving Frame, Curvature, and Torsion
 Geometric Meanings of Curvature and Torsion
 The Fundamental Theorem of Space Curves
 Global Properties of Space Curves
 Space Curves by Mathematica
Surfaces:  Preliminary Observations
 Surfaces
 The Tangent Plane and the Unit Normal Vector
 The First Fundamental Quantities
 The Second Fundamental Quantities
 Mean Curvature and Gaussian Curvature
 The Structure Equations
 The Gauss Map
 Minimal Surfaces
 Geodesics
 Surfaces by Mathematica
Visualizing a Flat Torus












http://math.kn.dendai.ac.jp/tazawa/Mathematica.html







   
 
