Yoshihiko Tazawa

Tokyo Denki University

e-mail: tazawa@cck.dendai.ac.jp

**Abstract**

The purpose of this article is to show the advantage of using Mathematica
in the theory of surfaces. The examples in the classical differential geometry,
namely the theory of curves and surfaces, have been confined to a small
group of calculable objects, because of the difficulty in evaluating geometrical
quantities and solving differential equations explicitly. But Mathematica
has made it possible to deal with wide range of objects and to perform
experimental treatment of them, based on the power of numerical calculation,
the NDSolve and NIntegrate commands in particular, and graphical visualization
by the ParametricPlot command. I would like to introduce here several animations
and figures that I use in my differential geometry class for undergraduate
students. First of all, these graphics help students understand the basic
notions of differential geometry. Secondly, we can experiment on geometry
through these graphics. This article is one of the serial talks given by
the author at Developers Conference 95, IMS 97 in Rovaniemi, and WMC 98
in Chicago. They are all targeted for the experimental usage of Mathematica
on differential geometry. This time the topics are focused on the surfaces
in the 3-dimensional Euclidean space. The commands are contained in six
notebooks. If the commands in other notebooks cause any trouble, please
restart Mathematica. See also the explanation of the notebooks.