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Experiments in Differential Geometry with Mathematica

Yoshihiko Tazawa
Tokyo Denki University

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The classical theory of curves and surfaces was established long ago. However, there has always been difficulty in applying the general theory to individual curves or surfaces because calculating integrals or solving differential equations explicitly is not possible in general. So the examples appearing in this field have been confined to a small group of calculable objects.  Mathematica has made it possible to deal with a more generic group of objects. In this talk I will introduce some experimental visualizations that I have produced with Mathematica and used in my differential geometry class. They help students to understand basic notions of differential geometry without difficulty. Two examples are: 1) to generate a random closed differentiable plane curve and observe the change of its curvature, Gauss map, and total absolute curvature under the homotopy between the unit circle, and 2) to approximate a space curve with pieces of circles joined together twisted at each joining point according to the torsion.