This is what a pure mathematics exam looks like at university
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.