Version: November 17, 2015
Material:
Test 2: Wednesday, 2015 November 18
Chapter 7: Fourier Series and Transforms; essentially the entire chapter with
emphases on Fourier series as solutions to the Laplace equation,
the Dirchlet conditions, partial sums and Bessel's inequality,
sine and cosine series, exponential
series, translation of coordinate axes and linearity,
even and odd functions, Parseval's Theorem,
different intervals, applications of Fourier series, plucked strings and
other waveforms such as square waves, triangle waves, sawtooths, ... ,
driven systems, summation of series, solutions of definite integrals.
Fourier transforms. Differences between Fourier series and transforms and
problems for which each is useful. Show that Fourier transforms are limits
(in a sense) of Fourier series. Many essential features of Fourier series
have analogues in Fourier transforms. Important properties of Fourier
transforms, linearity, translations, axis stretching,
transforms of spatial derivatives, and convolutions. Applications of
Fourier transforms, definite integreals, solutions to differential equations,
representations of pulses and other trasnsients.
Chapter 13: Partial Differential Equations, 13.1-13.5 (Cartesian
coordinates), 13.9 (Fourier transform
solutions). Types of problems with partial differential equations,
diffusion type problems (heat flow, B-field, Schroedinger equation, ...),
wave equations (free, damped, driven),
Seperation of variables as a technique to turn a partial
differential equation into a system of ordinary differential equations,
d'Alembert's solution for the wave equation and its generalization to
general second order partial differential equations, characeristics and
their significance, classes of second order differential equations (
hyperbolic, parabolic, elliptic) and the the nature of each as a propagation
equation or equilibrium equation, Fourier transform solution of partial l
differential equations.
Chapter 8: Ordinary Differentail Equations (general skills assumed
known by students).