Today I'm going to take a look at a pure mathematics exam
from University. Now pure mathematics deals with
mathematics that is more abstract in concepts rather than applied. That doesn't mean that pure mathematics doesn't have
applications, but it means you're dealing with things like the theory behind
numbers and functions and the abstract nature of pure mathematics makes it quite difficult.
I will say that I took pure mathematics throughout my physics education,
not only because I did a double major in math, but because it is useful for physics.
Now the exam I have here is a real exam
that would be taken I think your second or third year of
university study, that's undergrad.
It would take you three hours to complete and in this booklet there are eight questions.
Four of them deal with real analysis and four of them deal with complex analysis.
The difference is that complex analysis
deals with
thinking about the theory behind functions in the complex pl ane.
So we've got a bunch of i (the square root of negative one).
We've got a bunch of is in the working. This particular exam paper
I have is from the University of Manchester because they uploaded
all of their exams for the public to see so shout out to them and
let's have a look at it. Even though there are eight questions in this exam
you're only going to have to answer five of them, so
it wants you to pick two questions from real analysis, two questions from complex analysis
and then the last question you choose to do is going to be up to you.
Because of that this exam is really quite long,
so it will take you much longer than three hours if you were going to try and attempt all of the questions.
So let's look at the first section, which is the real analysis section.
So real analysis deals with the properties of real numbers and functions. You want to
understand the theory behind mathematics and
ideas like sequences, convergence, limits
continuity and smoothness. So the first question we have is a question about limits. You're asked to find
essentially or show that the limit of a certain function is a certain number or a certain limit and
you do that with various techniques.
I won't go into exactly the details of how you would do it, but using the epsilon-delta definition
you want to show the limit. So essentially you just want to take
x to be really really close to a certain value and
see what the entire function
essentially is bounded by or what the limit is. Next we have the intermediate value theorem, now the intermediate value theorem
talks about if you have two points connected by a continuous curve. You can actually draw what we're working with here...
you've got an axes, you've got a point A
and you've got a point B.
If that's a continuous curve
and you've got say a certain line
You've got one point below the line and one point above the line, then at least one place on
the curve is going to cross that line.
So
essentially you need to cross a certain line to get from point A to point B.
That's what this theorem is talking about and then you're asked to prove some things kind of related to it.
Question
part 2 down here is
talking about what's probably quite a familiar idea in calculus.
Assume that a function f is differentiable on a certain range [a,b] with a local maximum at c.
Prove that the derivative of the function at point c is 0.
So really you're proving that if a function has a local maximum,
that the derivative at that point is 0 and that's an idea that underpins a lot of
calculus and optimization problems, but you're not often asked to prove it.
We've got another question here that kind of deals with calculus ideas. We've got
being asked for a proof of the product rule for differentiation
and a few other things about
differentiation, functions and limits. Our last question in the real analysis section
is this one here. I probably won't go into this one very much.
You can look at the kind of language that these questions use it's quite abstract
and when I took this for my own real analysis course
it was definitely my least favorite math course and it's because there's so much new language
involved that it's just really hard to talk about or to think about, well that's what I found.
It's really hard to describe to other people as well so when I was always complaining about real analysis
I didn't quite know how to tell people
what it was about. it's the theory of
mathematics essentially.
But yeah, it's also very hard to Google questions like this because there are so many symbols,
not a lot of language. Moving on to Section B. This is the complex analysis stuff. Now
I actually really like complex analysis in comparison to real analysis
because complex analysis seems to have
more applications.
Complex analysis I have actually seen used in physics and thermodynamics,
and fluid mechanics and
essentially like I said before, we're extending the real functions like
exponentials, logarithms and trigonometry
into the complex domain and range. So we've now got
functions that can be separated into real and complex parts. This first question
deals with something called the cauchy-riemann theorem and
essentially
it's a set of partial derivatives that can be used to test if a function is
complex differentiable. That means that the derivative
exists, it's also called holomorphic if that is true.
These questions deal with
being asked to show that certain functions are holomorphic and why,
proofs surrounding that really.
Number two in the complex
analysis part,
this is dealing with mainly
definitions of trig
functions in the complex plane so you can see the definition of sine there
using complex numbers.
You can see it's being used
with a definition that includes hyperbolic sine and cosine so some of these
definitions of trig functions you might not have seen before if you haven't done much work in the complex plane.
These questions are actually I think the most
straightforward in this exam paper,
but they still look pretty alien if you're not used to
definitions like this.
Now we're getting into my favorite part of complex analysis, and why I like it.
It's because there are some really cool applications to integrals using complex analysis and they don't
sort of strike you as being obvious, but once you start to learn about them, I think they're pretty cool.
So essentially here, we have this big crazy figure and
what we're doing is taking integrals
around certain paths. Now usually
or often I guess certain integrals are quite hard to do normally, but
there's this thing that you learn about in complex analysis where you can take an integral
that's usually quite difficult to do, you can
describe it as a function,
take that function and transfer it into the complex plane,
do some crazy maths with things like the residue theorem, Cauchy theorem,
work out the integral in that complex domain and then convert it back into the real space
where you end up with the answer that you wanted all along.
And the last one is a similar vein this is
using residue theorem
which again allows you to carry out integrations in the complex space that otherwise would have been quite difficult
just in real space.
I have included this entire exam in the description so you can
have a good read of it there. I know it's probably quite hard to see the details here, so have a look,
I haven't provided solutions,
that's because this is actually a real exam from the University of Manchester website, but hopefully that gave you a good insight into
what some pure mathematics looks like at university.
This by no means is the most advanced you're going to do, like I said, this is probably
second to third-year work. It might not necessarily be combined into one paper I
personally did courses
where real analysis and complex analysis were separated into two different courses.
But yes second to third-year levels probably about right. There is definitely more advanced work than this.
Hope it didn't scare you off. Thanks for watching.