1. Math for research: motivation and review of prerequisites
1.1. 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. 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.
1.2. 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:
sequences
sums of numbers
taylor polynomials
taylor series
fourier series
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.
1.3. Diving in - polynomials and their humps
To get into the spirit of the Taylor series we start out in a completely visual mannger. Students can use a local plotting progrma like gnuplot, or an oline plotting calculator like geogebra or desmos.
Start out by plotting
and ask two questions:
What is the degree of the polynomial?
How many “humps” does it have.
Then do the same for
and ask the same questions. Finally do the same for
The students will have made the connection that a polynomial of \(n^{\rm th}\) degree has \(n-1\) humps. When I recapitulate what they have noticed I make a point of saying “up to \(n-1\) humps”.
The plots we get should look something like:
Figure 1.3.1 A 1st degree polynomial (purple straight line) has 0 humps, a 2nd degree has one hump (green parabola), and a cubic has up to two humps (blue curvy cubic).
Now ask the question: look at a sin() wave and ask yourself: how many humps does it have?
Figure 1.3.2 A simple sin() wave.
The students will answer “infinitely many humps”, at which point I throw out this kind of a challenge:
Can you imagine yourself saying “maybe the sin() wave is an infinite degree polynomial”?
As I say this to the students I put on a disgusted face because there is no such thing as an infinite degree polynomial. I point out that one does not just bandy the term “infinity” - in math you can talk about infinity (and infinitesimal), but it these terms are always used carefully and with clear definitions, since there is no actual number that is infinity.
Still, I leave them thinking that maybe one can use higher and higher degree polynomials to get closer and closer to a sin() wave.
1.4. Some possible materials to peruse on the side
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.