"By the year 2000, American colleges and universities will be lean and mean, service oriented and science minded, multicultural and increasingly diverse---if they intend to survive their fiscal agony" ("Campus of the Future," TIME, April 13, 1992).
If our educational institutions are to remain competitive, it is essential that our students be prepared to enter a society that is increasingly dependent on advanced technology. Therefore, students must be exposed to advanced computer tools in our science and mathematics courses--even in introductory courses--to prepare them for a world where computers play an important role.
In the 10 years since Mathematica was introduced, it has already had a significant impact on the way physics is taught and how research is performed. This presentation will review our experiences during the last five years to integrate Mathematica into the physics curriculum at SMU, starting from the introductory-level courses up to the graduate level. By examining canonical problems from the undergraduate curriculum and contrasting the standard solution "before Mathematica" with the modern standard solution "after Mathematica," we will also examine how Mathematica allows the student to better focus on the underlying concepts. To be specific, I will consider two examples.
The double pendulum is a topic that is generally treated as an "advanced" topic, but with Mathematica the solution is elementary. Additionally, using the animation capability of Mathematica, it is trivial to animate the final result, allowing students to discover results on their own by varying the amplitudes of the separate normal modes.
Boundary value problems in electrostatics is a topic that beginning students find overwhelming. With the power of Mathematica, it is easy to show these solutions are quite straightforward--especially with the help of the different coordinate systems built into Mathematica. When students finish the problem with pen and paper, they have only a set of formulas that may mean very little. With Mathematica's 2D and 3D graphics, we can plot the final solution to verify visually that the boundary conditions are satisfied. This technique encourages the student to think about the solution and not simply grind out the math.
For the lower-level courses, it was the user-friendly interface and the intuitive graphics capabilities of Mathematica that encouraged and tempted the student to experiment with different methods of solving problems. The tedium of the algebra or calculus no longer was an impediment. For example, in 1993 we added Mathematica as a component of our calculus-based introductory physics course. The students were given an initial tutorial at the beginning of the semester and then assigned problems from their text that were particularly well suited to the capabilities of Mathematica. With intermediate-level courses, the students were able to delve deeper into the capabilities of the program in solving more complex problems.
It is said that there are only two problems in physics that can be solved exactly: the Kepler problem of an orbiting mass and the simple harmonic oscillator--everything else is perturbation theory. With Mathematica we were not limited to analytic perturbation theory, but could easily devise numerical solutions to real-life problems that were not solvable using traditional analytic methods. Examples include projectile motion with air resistance, simple harmonic motion with a damping force, and chaotic double-well systems.