Completed discussion of damped, driven harmonic oscillators, then began discussing dynamics in higher dimensional spaces.
On damped, driven harmonic oscillators: cited the
famous result of Fourier that any periodic driving force can be
written as a superposition of sinusoidal driving forces with
frequencies 
(with the summation running over all
integers n). This allows you to find the particular solution for any
periodic driving force, by superposition of the solutions for each
harmonic driving force n. Then discussed the full solution for the
oscillator's motion: a sum of the general homogeneous solution (which
can be underdamped, critically damped, or overdamped, as discussed
before) and the particular solution obtained above. Because the
homogeneous solutions all eventually decay (even if they're not
optimally damped), they are called "transients". Eventually the motion
becomes dominated by the particular solution, which is thus labeled
the "steady state" solution.
We then began discussing the generalization of our methods to dynamical systems with higher spatial dimension. Reviewed the following facts about vectors: definition of a vector, vector addition, vector multiplication by a scalar, components, dot products, cross products, coordinate invariance of both products, their form in components, and Newton's second law as a vector equation. We then began to define derivatives of vectors. We first considered the simple case of a vector dependent on a scalar variable, like time. After expressing the vector explicitly as a sum of its component vectors in each coordinate direction, we were able to differentiate with respect to time using the product rule. We found that, if the coordinate axes are time-independent, we can just differentiate wrt time component by component. Otherwise, we must include terms describing the time derivatives of our coordinate axes as well.
-- KB