Abstract:

Abstract concepts in mathematics can be made concrete, visual, and interactive with advanced numerical facilities, not to mention symbolic computation, provided by Mathematica, a widely used computer algebra system (abbreviated as CAS). This paper presents patterns of CAS usage in class and a special toolkit with which classical differential geometry is introduced in “mathematics laboratory” sessions. Our attempt is one of good practices in this area that have already been in action. However, our attempt is new in that learners can be much more active as a “little researcher” arranging experiment apparatus and observing mathematical phenomena in the laboratory.
Recurring concerns of course designers of college mathematics are:
(1) Mathematics class should be more interactive,
(2) Appropriate choice of intervention points is the key to the successful course design,
(3) Learners should work through appropriate amount of mathematical experiments with tractable problems with pencil-and-paper. Concerning (3), we often encounter the situation in which tractable and non-trivial examples are rare and we are forced to give only theoretical explanation without a rich set of examples. The Gauss-Bonnet theorem is one such example, which we will study in detail in this paper.

These recurring concerns motivated us to pursue patterns of CAS usage in class to contribute to engineering education. Widely accepted facts are:
(1) CAS provides students and instructors with unrestricted opportunities of experiments with interactive graphics;
(2) CAS also provides intervention points in class at which the results are checked against intuition;
(3) even CAS cannot give meaningful results unless students give it a tractable instruction, which means, pedagogically best, they are forced to streamline their ideas beforehand;
(4) we can eliminate cumbersome lower level calculations of no concern, which makes the learners concentrate on the most important aspect of the topic.

Our research is based on these common observations.

To demonstrate our ideas concretely, we take up one of the most popular and impressive theorems in differential geometry, the Gauss-Bonnet theorem. The theorem claims that there is a certain link between metrical quantity and a topological invariant. We consider this topic relevant since there is an increasing demand on familiarity with topology among various fields of scientists and engineers.

Following our CAS usage patterns, learners write a small piece of code that calculates tangent, normal vector fields, and their derivatives in various directions. These vector fields are visualized as dynamic objects of Mathematica; the learners can move the vectors around on the surface through the GUI that they themselves produced. Next, the learners write code to calculate the curvature and carry out integration on the surfaces, and then observe that the values depend only on the topology of surfaces. Manifolds with boundaries can be treated further with advanced numerical facilities of Mathematica.

In most textbooks, the Gauss-Bonnet theorem is presented with a rigorous proof with a limited number of concrete examples. Adding more examples meets the difficulty of growing computation burdens when done only with pencil and paper. We can overcome this by full use of our patterns in class.

Keywords:

Classical differential geometry, visual, interactive, math class, Gauss-Bonnet theorem, numerical method, Mathematica, 3D graphics.