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. 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.