11. Math for research: motivation and review of prerequisites
11.1. Motivation for “math for research”
Approximating functions and data with sums of carefully chosen functions (like polynomials or trigonometric functions) is an almost daily task for working scientists. And yet I know of no curriculum that puts these topics together before students have reached an advanced calculus level.
Our goal here is to explore series approximations to various functions, and in the process to look at other places in which series come up.
The most important take-home is probably that polynomials can be used to to approximate reasonably well-behaved functions locally, and Fourier series can be used to approximate reasonably well-behaved periodic functions globally.
In doing so we will also get a tour of many interesting mathematical functions, as well as physical problems that can be solved thanks to these kinds of approximations.
We will not assume a knowledge of calculus at the start, but we will occasionally step out to learn some speficic topics in differentition and integration when we need it.
We will start with some mathematical experiments in which we start from known series and see how well they work. This will give us a dissatisfaction which can be expressed as “sure, I see that it works, but how do you get there?”
Then we will move on to learning how to generate the kinds of series we have been experimenting with. This will allow us to work such approximations for many different types of situations.
11.2. Rambling introduction
First of all: this is a no-anxiety working group. We get together and do serious work, but you are always encouraged to say “hey, I did not get get that at all - can we step back?” You will find that everyone is happy when we slow down or step back! Also: if you ever feel that you are not as good as others, you’re wrong. Everyone steps in and out of doing well and being confused, and you will certainly be at both ends, even if you don’t notice. That’s what makes it OK to say “please, let’s step back”.
As we study how to approximate things with polynomials (and other series), we find that the traditional school model does not completely work. In algebra and trigonometry they have text books which have codified the sequence in which you are supposed to study it, and they have tons of exercises. The books at https://openstax.org/subjects/math for example have tons of very good exercises.
But real work bridges these codified separations, and the textbooks are not as easy and mature. Sometimes one has to patch together many different sources of information, and different sources make different assumptions on what you already know.
This means that you are taken away from that comfort zone of everything being mapped out. In exchange for that discomfort you get to bridge areas, and your own thoughts can get profound.
And as we leave the comfort of the schoolkid approach, we replace that with the comfort that in our working group there is no shame in stepping back and re-tackling things from the same or different angles.
One challenge for us will be to find canned exercises for us to work from. Those are helpful, but harder to find as we progress. Our exercises might sometimes involve figuring out how to plot something, or how to write a couple of lines of code about something. We will move slowly on that.
Finally: the progression I have in mind is something like this:
sums of numbers
numerical solutions to differential equations
but we will happily meander and step back to basic math when we need more material from polymials, or trig, or exponentials, or even an occasional flash-forward to a calculus idea.
11.3. Review of prerequisites
The main prerequisite is what we covered in the “Visualizing algebra” working group.
We will introduce other ideas (such as derivatives and integrals) as the need comes up.
Here are some videos and other preparation tips which give you an interesting review, or just some attractive demonstrations.
“3 blue 1 brown” video on Taylor series. Note that we are not going to do it the way they do: they assume you know calculus, and we don’t. But it is visually quite lovely, so if you have a few minutes take a look.
Watch the “map of mathematics” video on youtube: https://www.youtube.com/watch?v=OmJ-4B-mS-Y revel in the British accent, and see how all areas of math fits together.
Watch the “Professor Dave” video on sequences and sums: https://www.youtube.com/watch?v=-DPkqpmm1sI&list=PLybg94GvOJ9FoGQeUMFZ4SWZsr30jlUYK&index=84
Have ready a way of calculating and plotting. A calculator is OK, as is a computer calculator program. For calculating I personally use “M-x calc”, a reverse polish notation (RPN) calculator built in to emacs. If you find RPN too nerdy then the fastest (i.e. no mouse) is probably to start a python3 interpreter and type expressions in to it, but anything you’re comfortable with… For plotting I recommend having gnuplot handy because it will allow us to manipulate the expressions we plot.