Teaching and computer algebra

Today, microcomputers have developed to the point where they can run programs once restricted to mainframes. Personal computers are more widely available than mainframe systems, which facilitates the employment of algebraic systems as aids to teaching Mathematics on a broader scale in Physics and Engineering courses. The main systems, like Axiom , Maple , Mathematica , MuPad and Derive in the new generation group and Macsyma and Reduce in the old group, allow the student to perform not only algebraic calculations, which are the main purpose, but also graphic manipulation, numerical analysis and programming. Two aspects of teaching Mathematics can change due to the availability of these tools.

First, the student need not be submitted to exhaustive training in the use of calculation techniques like integration by parts, by substitution, and so on. These techniques are implemented in the computer algebra systems; in the case of integration, the powerful Risch-Norman algorithm[Geddes] allows the user to perform integral calculation which are impracticable and unwieldy whit pencil and paper. The computer carries out this long algorithm without complaining. These techniques are available to any student who has learned one of the computer algebra systems.

Second, use of these systems allows the instructor to give priority to the teaching of concepts, giving them the attention they deserve; teaching some of calculation methods becomes unnecessary which frees up time for discussing the concepts. Computer algebra systems may also help with procedures like plotting and animation. Imagine a teacher explaining the non-axially symmetrical normal modes of vibration of a circular membrane. Even with a lot of hand gesticulations and expert chalkboard drawing, only the student with excellent visualization skills will understand. A computer plot or an animation can greatly assist students in understanding such concepts.

The necessity of changing the traditional curriculum of exact areas due to the acessibility of computers and the advent of structured languages like C, Pascal and Basic has been already pointed out[Redish]. Computer algebra reinforces this necessity by order of magnitudes. Students who learn computer algebra systems in undergraduate courses are in a much more favorable position than professionals who only start to learn them later. Even today there are researchers who need to use computers in their work, but who are not familiar with the windows environment or even with a computer keyboard.

The vibrating membrane is a good example of how computer algebra systems can provide pedagogical tools for the analysis. The main aspects of these systems can be used to reach the desired goal. In this work we study the oscillation of membranes of different shapes under the new viewpoint of computer algebra. We concentrate on the pedagogical aspects and on the part of the theory in which these systems can help more. This is a classical problem of Mathematical Physics which was solved years ago. We skip some details which are easily obtained in the references on the subject[Butkov][Courant][Sagan][Morse].

In literature there are some references which use Maple to analyze the membrane problem[Rybowicz][Kreyszig endf], but a systematic approach like the one presented in this paper has not been undertaken yet, as far as we know. Maple[Heal][cite Monagan][Heck][Portugal] is a proper environment to analyze the membrane problem. The main aspects of the Maple system (algebraic, graphical, numerical and programming) have been used together to solve the problem. The algebraic aspect has been used to develop the problem step-by-step from the wave equation. This has been presented for the circular membrane with axially symmetric movement. The graphical aspect has been used to animate the normal modes and the movement due to initial conditions. The numerical aspect has been used to find the zeros of the Bessel functions and numerical integration when needed, and of course to plot. The programming has allowed us to make packages which provides tools to analyze the problem. With these tools, one can change physical parameters and choose many kinds of initial conditions. All material is available in the membrane web site.[In the membrane web site one can find the worksheets, text files and the packages used in this work. The address is http://daisy.uwaterloo.ca/~rportuga/membrane.html . There is a README file which contains installation instructions.]

We have used Maple V release 5 to perform the calculations. This release has procedures to calculate zeros of the Bessel functions. These procedures are missing in all earlier releases. Nevertheless, the worksheets[The worksheet is the main Maple user interface. It is a virtual notebook in which one can peform calculations, take notes and store plots and animations. It can be saved in a file (the default extension is .mws ) to be recovered later on.] are available for release 4, and the text files are available for release 3. The procedures for finding the required zeros of Bessel functions have been added for these releases.