Season 9 Episode 0
Beth is on the show to put the fun into fundamental theorems, with calculus and algebra and more!
Further Reading
Mirzakhani Society
Beth mentioned that she’s on the committee for the Mirzakhani Society. That’s the society for women and non-binary students studying Maths at the University of Oxford. They hold events and sometimes they make T-shirts, like the one they gave me which says “Girls just want to have fundamental theorems of calculus”. It's a nerdy T-shirt. It’s a wordy T-shirt. It’s a nerdy wordy T-shirt and I love it. If they work out how to sell these online then I will add a link here.
Fundamental Theorem of Calculus
During the livestream we jokingly mentioned “terms and conditions” several times. Here’s a precise statement of the first fundamental theorem of calculus, so that you get an idea of what those terms and conditions are.
If $f$ is a continuous real-valued function defined on a closed interval $[a,b]$ then let $F$ be the function defined for all $x$ in $[a,b]$ by $$F(x)=\int_a^x f(t)\,\mathrm{d}t.$$ Then $F$ is differentiable on the open interval $(a,b)$ and $F’(x)=f(x)$ for all $x$ in $(a,b)$.
The conditions of the theorem are the parts labelled “if”. The claim here asks that we consider a function that’s continuous and real-valued and defined on a closed interval $[a,b]$. Each of those is a restriction of the sort of function that we’re talking about for the rest of the statement, and each of them has a definition. Definitions are important!
Without those conditions; well, we’re on our own. For example, the theorem makes no claims about functions that aren't real-valued; possibly such functions do lead to $F(x)$ with $F’(x)=f(x)$, possibly not, or possibly that doesn’t even make sense to ask.
Some of the work that mathematics students do at university is to make sure they’ve understood why the lecturer has included the “terms and conditions”; perhaps these restrictions make the proof easier, or perhaps they’re needed because the claims just aren’t true for other functions. Sometimes the terms and conditions might not be necessary though, and something researchers think about is whether known results can be extended or strengthened by reducing the conditions needed for the proof.
As a metaphor, suppose you’ve worked out how to change the tyres on cars. You might boast “Given any red car, I can change the tyres for you”. Impressive. But that claim could be strengthened when you realise that the colour of the car doesn’t matter. And then the claim could be extended if you find out how to change the tyres on buses and other things too, until perhaps you’re going around saying “Given any four-wheeled vehicle, I can change the tyres for you”. Stronger claim!
To find out more about functions and differentiation and integration, see this previous OOMC episode with Aditi.
To find out more about the history of calculus, see this previous OOMC episode with Zakkai on YouTube.
Fundamental Theorem of Algebra
I’ve made an interactive version of the animation from the episode, on Desmos. You can move the action forwards or backwards, and you can change the polynomial to explore what happens.
Separately, Beth found a teaching resource that combines Sudoku with the fundamental theorem of algebra. It asks you to answer questions based on the fundamental theorem of algebra, put the answers into a Sudoko, and then solve the Sudoku. If you get a copy of this, note that this author uses “imaginary number” to mean “a complex number that isn’t real”, where other people might use “imaginary number” to mean “the product of a real number and the number $i$". Definitions are important!
A concept that we mentioned very briefly was “winding number”. This is an idea from topology, and it’s a count of the number of times that a curve goes around a particular point. The definition for the winding number of a contour in the complex plane is pretty involved (see the Wolfram Mathworld encyclopedia, for example). Just understanding the definition relies on complex numbers and integration!
The Wikipedia page for winding number has an intuitive description and a nice animation, which might be more helpful.
Being a mathematician needs some combination of intuition and rigour; enough intuition to have ideas about what’s true based on your understanding of complex information, and then enough rigour to prove it from the definitions.
If you want to get in touch with us about any of the mathematics in the video or the further reading, feel free to email us on oomc [at] maths.ox.ac.uk.