Today I started teaching my first real lecture course (as pointed out in the comments, the link is only accessible within the Imperal network).
I am teaching the second-year physics students mathematical techniques of Fourier Series and Fourier Transforms — this is the theorem that you can represent any function as a sum of so-called sinusoidal waves. That bit I think I explained all right. But then we had to start getting down into the mathematical details. Unfortunately, I think I lost them somewhere trying to make the analogy between vectors (i.e., arrows in space) and functions; you can describe a vector by giving its value in three perpendicular directions (x, y, z, for example), just like you can describe a function f(t) by giving its value at each value of t. A full set of these directions (x, y, z in the case of spatial vectors, or the individual values of t for the function) is called a basis. But we can rotate our vector to describe it any basis that is convenient.
The idea behind Fourier Series is that there is a specific basis made of sine and cosine waves — and expanding our function in this basis lets us understand things like sound and light in terms of frequency: light as a mixture of colors or sound as a mixture of pitches. For many problems in physics, these mixtures (with the somewhat more technical name of “linear superpositions”) are described by very simple formulae. Indeed, in addition to his laws of motion, Isaac Newton is famous for the first description of light this way (although he didn’t have the mathematical technology that Joseph Fourier would only develop in the 19th Century).
Indeed, there are some mathematical formulae behind all of this — not too complicated technically, but I’m not sure I was able to get the concepts behind them through to the students. It’s hard to calibrate to exactly what the students already know (which may not be the same as what they’ve already seen in their coursework!). Also, I worry that I may have drowned them in a sea of notation without actually explaining what I meant in quite enough words.
(In the unlikely event that any of Imperial’s second-year students are reading this, feel free to leave an anonymous comment and let me know what you thought!)