Issue 28 / October - December 1999
Cybermath : Using The Internet As A Math Tool
Over the last decade, the Internet has impacted every aspect of our lives. It is now easy to perform very complicated tasks from your computer desktop by clicking buttons on the appropriate Web pages. For example, you can serve as your own travel agent by arranging your flight, car, or hotel reservations, and by searching for the lowest price by choosing the FareBeater option.1 You can do your shopping via computer, there by saving yourself a trip to a store or a mall.2 Since these Internet features make our lives easier and faster, they will continue to be a hot topic in the years to come.
While almost everybody who has access to a computer is somehow involved with the Internet for a variety of personal reasons, scientists and the academic community also use it for their own purposes. Examples are sharing data and information, browsing technical papers, searching for related documents, and posting their research activities to colleagues and other interested parties. Recent developments have proven to researchers and academia, both of which publish large amounts of technical literature, that the day of electronic publishing is close at hand.
The easiest way to publish electronically is to post documents on the Internet by generating Web pages. In this case, however, the user remains a passive recipient of information, for the Internet's interactive communication ability is not being used. This particular capability of the Web allows a scientist or researcher to generate Web pages with which the user can interact. One example of such interaction is performing mathematical operations over the Internet. This is very useful and powerful, for it allows the user to become an active receiver by gathering the needed information from that particular home page.
For instance, if you visit Rice University's home page for its Department of Mathematics, you will find several interactive math tools designed to help students taking the Ordinary Differential Equations course.3 These tools are activated by using Java or MATLAB programming language. For example, Figure 1 shows a tool called PPLANE, which draws the magnitude and direction of a differential vector in an x-y plane. PPLANE also graphs the linearization about equilibrium points, and displays eigenvalues, eigenvectors, nullclines, and stable and unstable orbits. The user can change the differential equation's variables and then run the associated MATLAB code over the Internet. A licensed MATLAB copy in the user's personal computer is not required, for the MATLAB routine is run on the server and displays the output on the user's browser. This makes it easy for the student to understand how the changes made alter the equation's features.
This technique is very efficient and powerful for a student who is still in the learning process. It also suggests that the Internet's interactive feature will affect the education system and the way courses are taught in the future.
Another home page that contains a wide variety of interactive math tools is found at the Web site for Dartmouth College's mathematics department.4 Professor Richard Williamson has written about 30 interactive math programs for common scientific problems. These vary from differential equations to heat equation solvers, from Newton's method of calculating the root of an equation to simulating a swing's motion. All of these programs are activated by the user's input parameters, and display the answer in the same manner.
Another interesting and very useful Web site is http://www.integrals.com (see Figure 2). This interactive site allows the user to take any symbolic integral over the Internet. It is provided by Wolfram Research, which also produces the well-known and widely used Mathematics tool MATHEMATICA. After the user enters an expression, the integrator automatically runs MATHEMATICA on the server, integrates the expression, and sends the result back to the user's browser.5 This site is already helping many calculus students with their homework, and is quite handy for researchers who deal with complex integrals in their everyday research.
Another useful interactive math Web site can be found at the Geometry Center Web page of the University of Minnesota, Science and Technology Center.6 This site offers such interactive math tools as hyperbolic triangles, Lorenz simulation, and interactive proofs of popular theorems. There is also a tool for taking numeric integrals. If you get tried of doing mathematics, you can take a break and play some Tetris games at the same site. In fact, interactive games on the Internet are also a particular type of interactive math tool. A better graphical version of Tetris can be found at http://www.geocities.com/SiliconValley/Pines/522 7/tetris.htm.
All of the above Web pages show that cybermath has found its way onto the Internet, thanks to the Web's interactive communication capability. It is not difficult to imagine that students in other majors will apply this useful and efficient tool to their own field, thus making the Web even more interactive
1 http://www.flifo.com/and http://www.reservations.com/, respectively.
2 http://mallblvd.net/and http://www.internet.net/index.html, respectively.