Fourier series. Given a function, you should be able to compute a Fourier series in either real or complex form, with any given period. You should also know how to compute Fourier sine series and Fourier cosine series, and you should understand the relation of these series to Fourier series in general. Be able to make use of any symmetry when computing coefficients for the various series mentioned above. (Tolstov, 1.1-1.3, 1.5-1.15)
Convergence. There are three types of convergence that we have discussed: point-wise, uniform, and mean. You should know the convergence results that we discussed in class, and be able to apply them to decide what function a given series converges to. (Tolstov, 1.10, 2.7)
Orthogonal functions. Know the definition of an orthogonal set of functions. Be able to calculate the best least-squares fit for a function relative to a given set of orthogonal functions. Know what completeness of an orthogonal set means, and how it relates to convergence in the mean and Bessel's inequality. (Tolstov, 2.1-2.7, 2.10)
Heat-flow problems. Be able to explain the physical origin of the the three ingredients of heat-flow problems---the heat equation itself, the boundary conditions, and the initial conditions. Be able to find and solve the steady-state heat flow problem, and be able to derive the transient problem as well. In addition, in the three problems that we studied in detail---the problem of the bar with temperatures fixed at the ends, with insulated ends, and with one end at fixed temperature and the other insulated. You should be able to separate variables up to the point of arriving at an eigenvalue problem. You will not be expected to solve the resulting eigenvalue problems. (Powers, 2.1-2.5)
Office hours. Instead of my usual 11-12 Wednesday office hours, I will have office hours from 9:30 to 11:15 am and 2:30-3:30 pm on Wednesday, February 24.