- Students must labor through four hard years before they begin to see
current research applications in their formal course work. Many students
drop out along the way.

- The standard curriculum solidified before the advent of the
computer, so
this most practical of physics tools is largely ignored. At Oberlin, for
example, physics majors are urged to learn a computer language, yet most of
those who do so apply this knowledge only to one or two relaxation problems
that might or might not be assigned in the electrodynamics course.

- The curriculum tends to produce graduates who manipulate symbols rather than understand physics. For example, any physics senior will be able to express a periodic function as a Fourier series with frequencies going from +\infinity to -\infinity, but few will be able to explain the meaning of the negative frequencies.

This proposal is an attempt in the direction of rectifying the faults while retaining the manifold and obvious benefits of the standard curriculum.

Our proposed computational physics course takes advantage of such mutually advantageous situations to teach algorithms as aids for understanding physics rather than as (somewhat distasteful) mechanisms for getting answers. In order to develop the connection between formal approaches and informal, physical (and largely pictorial) approaches, the course will use computer graphics in an essential way. The course will have only one prerequisite, namely the introductory "Mechanics and Relativity" course (which in turn has a prerequisite of one semester of calculus). Thus students will be ready for the course as early as the second semester of the freshman year (if they arrive at college with calculus advanced placement, as many do). In this position, we will use the course to inject excitement, research problems, and physical insight early on in a physics student's career. Also in this position, it will to useful to students in allied sciences such as chemistry, computer science, and geology. We will select examples to make the course attractive to such students.

The *ordinary differential equation* section will build directly
upon the
student's background in the "Mechanics and Relativity" course. The numerical
technique taught will be Runge-Kutta and examples will be drawn from the
field of chaos. This field will contribute excitement, of course, but is
also valuable in building a student's physical insight. Chaos relies heavily
on visualization techniques, such as the Poincare section, whose entire
purpose is to generate a good *qualitative* picture of the motion. These
techniques are also valuable for building qualitative understanding in
situations where analytic solutions are available. They are not commonly
used as such simply because the analytic solution is used as a crutch to
*avoid* a deep qualitative understanding of the situation.

Every professional physicist has a qualitative understanding
of *spectral
analysis*. Students usually do not, because Fourier techniques are introduced
accompanied by such a cacophony of difficult integrals and convergence
theorems that the student misses the fundamental qualitative point. These
topics are essential for a second encounter with the subject, but for our
introduction we will stress examples and qualitative pictures. This section
will give the course a much-needed experimental component.

The last section would normally be called "solutions of Laplace's
equation," but our students will not even know what a partial derivative is!
We will instead speak of *harmonic functions*, defined at those
which satisfy
the mean value relation. It is no more mysterious to say that the
temperature of a plate must be a harmonic function than it is to say that the
temperature obeys Laplace's equation. The mean value relation leads directly
to relaxation techniques and then to random walk techniques. Multigrid
methods (which apply relaxation to grids of varying size) will be introduced
both to speed the computations and to reinforce the qualitative understanding
of wavelength components developed in the spectral analysis section.