My name is Dominic Vella, and I am professor of applied maths here in the Mathematical Institute, and I'm also a tutorial fellow at Lincoln College, which is one of the colleges in the centre of town. I've got this photo here in the front, like just on the off chance that it's raining, which fortunately is not today, but also to show you that some of the buildings in the centre of Oxford particularly beautiful and well worth visiting.
This is a photo of Lincoln's library. So my job today is to tell you a little bit about what applied maths is. And historically, it's something that's really been viewed as sort of developing mathematical models of real world phenomena. And historically, maybe it's been focussed on physical objects.
So particular some of the things that people in the department here think about how brains fold and trying to understand the very convoluted folds that you get in human brains and are actually observed only in relatively large mammals, mammals larger than the ferret or the rat. And the question is what gives rise to this very fine pattern? And can we understand how it helps with cognitive ability? So that's a problem in solid mechanics. I'm interested in some mechanics, but also how fluids flow.
And one of the things that I've thought about a little bit is what happens when you pour yourself a cup of juice. So this is a movie where you see orange juice being poured into a cup. And if you look at the bubbles that rise to the surface, they sort of clump together. And then over the period, over a period of time, they sort of move towards the edge of the glass that we're interested in understanding how that happens, why it happens and how fast it happens.
But increasingly applied maths is not just about the physical world, but also about the digital world and how to deal with data. So an example may be that some of you might be too young to know about photos on paper, but actually a big problem with photos on paper is that they can get creased and damaged.
And the question is, if that's your only copy of a photo, what do you do to try and reconstruct the original image and actually mathematical algorithms, the people in the study that try and even paint the image so that you go from this damage photo the top to something closer to the original down bottom. So what I'd like to do, having described a bit about what applied maths is really trying to do, which is more or less everything.
Let's give you a taste of how applied mathematicians do that and also to describe a little bit about how what techniques we teach you in the undergraduate course here at Oxford that will allow you to do that. So the basic idea is that there will be some fundamental principle that we then try it out, try to write down mathematically and then from that mathematical expression of the fundamental principle, we then try and see what the logical consequences are.
So a simple example of a fundamental principle is Newton's second law is something you hopefully would have seen at school in science, GCSE or A-level physics, for example. And Newton's second law tells you that F equals M. It says that the force are not on a body. F is equal to the mass times the acceleration.
And if you know that the acceleration is the rate of change of lost in velocity is the rate of change of position, then you know that that acceleration can be written as a second derivative with respect to time of the position. So if I know the force in an object and I know its mass, then I can work out what its position is going to be sometime later on.
And as an example of this that many of you would have seen in your A-level, I would think a little bit about what happens with a simple pendulum. So pendulum is a length of string of length L with a mass m hanging on the end of it and we pull it to one side and let go. And the question is, well, I want to write down Newton's second. Also, the question is what are the forces that are acting on this body?
If I think about the forces tangential to the motion, then I can write down that the component of gravity is minus MJG times. The sign of the angle. Theatre features that angle up there and that is the force. So it's got to be equal to the mass times. The acceleration and the acceleration is L Times the second derivative of the theatre with respect to time.
So from this equation, we can cancel em and then we can think about something else that you have seen, hopefully in your A-level, which is that for very small angles, sign theatre is approximately theatre and say this equation becomes relatively simple. It says that the second derivative of a function is minus some constant squared times the function itself.
And if you've done the calculus of trigonometric functions, you might recognise that the solution of that equation is going to be a trigonometric function, sign or cause. So we can write down the general solution as being some amplitude times the cause of omega, the some frequency times time plus a Faceshift say omega here is given as the square root of the acceleration due to gravity g divided by the length of the pendulum. OK, so what can we say, given this solution?
Well, the first thing we can say is that we know that cost goes backwards and forwards, and it oscillates. And that's just what we expect the pendulum to do as well. So that's good. What else can we say? Well, we've got this expression for what the frequency of the pendulum is. It's the square root of over L. And I think the first time that you see this formula is sort of a bit surprised. You're used to things how fast things move, maybe depending on the mass of an object.
So you might think, well, why doesn't the frequency depend on the mass? Well, we saw that it's because of the cancellation that happens in Second Law. But just to convince you of this, I take my daughters to the park and one of them is roughly 20 kilos and one of them is 10 kilos, and I pull them up on the swing and let go of them. And you see that they they fall with or they oscillate with the same frequency. OK, so we've seen that mass doesn't change the frequency of oscillation.
We could change G by going to Mars or the Moon and change a frequency that way. But a much simpler way to change the frequency is to play with the length of the pendulum. And this video I'm going to show you in a second here is from the Harvard Natural Sciences demo. What they've done is they set up a whole series of pendulum with different lengths, and then they let them go all at the same time. OK, so you can see that as we expect from our formula, the length is influencing the frequency.
The short one is going much faster than the low one or much is oscillating with a lower period. OK. And they've chosen the lengths of these pages in a very nice way so that you get these really mesmerising patterns. And if you go onto YouTube and watch the rest of the video, you'll see that actually comes full circle and goes back to the beginning and then carries on.
The reason I have included this video here is because it gives you the sense that it's not necessarily enough to just think about how things vary in time. It might also be interesting to think about how things vary in space as well. And if we want to be very crude, then actually applied, Massey University is thinking about some of the things that you've thought about at school as functions of one variable, but thinking about how we deal with functions of two variables.
So as in the pendulum example, we might be that something varies in space and time, or it might be that things vary in two spatial directions x y. So why do we need to think a little bit about this? Well, one thing that you need to think a little bit about is how we should talk about derivatives. How do things change? We have to introduce this curly debate. That's something that comes up a lot in two core first year courses.
So an introductory calculus, we worry about what difference between this curly divide and the straight divide it is. And then in the course called multivariable calculus, we sort of talk about doing integration as the reverse of differentiation, but with multiple variables. OK, so with that mutation, then it turns out that there are essentially three different kinds of differential equations that come up with three different equations that come up.
One is called the wave equation, and if you look at it, you can sort of say I've got a function f that varies with x and time. And if you look on the left hand side, you see the second derivative of F with respect to time. That's sort of coming from Newton's second law. It's a bit like the acceleration time in Newton's second law. And on the right hand side, I have a second derivative with respect to spatial code in the X that turns out to come from the forces.
The second equation that comes up is what's called the diffusion equation or the heat equation. Again, this looks similar, except on the left hand side you only have one time derivative. And I'll give you an example of or try and outline how this equation is derived so that you get a sense of a different physical principle that's useful in applied maths.
The final example equation that comes up a lot is what's called the +s equation, and that two looks a lot like the wave equation, except that the two times are now on the same side of the equation. So it's as if time was sort of an imaginary space coordinate, if you like.
OK, so these are the three equations that come up a lot. And again, a cool first year course called very serious and partial differential equations is really focussed on trying to understand how you would solve the different techniques you would use to solve each of these three equations. Obviously, we can't talk about that today, but what I want to give you instead is a sense of how the solutions of these equations behave.
And some examples of where you might have seen these in your everyday life, where it. OK, so what about the wave equation? As I said already, the physical principle behind the wave equation is often Newton's second law, but it turns out that in one dimension, you can write down the general solution of this equation as some function big f of x minus plus another function big g of X plus c t. And then the question is just what are the big f g?
Now I can't show you why that works, but hopefully you can see that this f of x minus c t is a way of moving to the right at some speed. C, You know that when you take a function of X minus A, that's like just translating the whole graph to the right. And if you do a function of X plus a constant, that's like moving the graph to the left.
So what we're doing is we're moving the graph Big F to the right with an amount that depends on time with this constant C and we move in the graph Big D to the left again at some speed C. OK, well, what can we do with this? Well, one example that you might think about in the third year course is actually basically to try and understand what this transmission or this this sorry troubling waves do when you're playing your music.
I'm sure that it's a familiar experience to parents and to children alike that you often get complaints about your music. And the question is, why is that what's going on there? So what we have is we have our music in our bedroom, which we think we are playing at a reasonable volume, OK? And it's hitting the wall and we hear it all being reflected by our parents hear stuff being transmitted. And the question is what bits are being transmitted and what bits are being reflected.
And again, for a calculation that you can do after the third day course, you find that the ratio of what's transmitted to what's reflected is inversely proportional to the frequency omega. OK. So what does that mean? Well, it means that if you have very low frequencies, that transmission to reflection is very large. And if you have very high frequency, the transmission to reflection is very small.
OK, so as we maybe what you see in everyday life that high frequencies tend to bounce off the wall, low frequencies are transmitted. And that's why when you hear music or the traffic light next door's car, it's really the bass that you hear rather than the tune. OK, so there are lots of other solution types of solution of the wave equation. For example, if you think about a drum, the solutions are irrelevant when you hit a drummer.
What are called normal modes and normal modes depend on the shape of the boundary, the shape of your drum, but they also depend on the way that you hit them. So if you hit the drum perfect in 70, you might expect to get this dome shape and that has a particular frequency associated with it. If you hit it in slightly other ways, you might get combinations of different moods. And these are the modes of a circular drum, so you get this sort of this hat shape.
And again, each shape has a different frequency associated with it. And again, the techniques are used to calculate these are covered in this optional third vehicles called waves and compressible flight. What's really important is that the shape of the object is important in determining the frequencies of which it likes to oscillate, and that the and that that frequency, this kind of frequency depends on the shape and then the shape of the object gives you a particular frequency.
So you can see that in the violin, where you can put powder on the surface of violin as it vibrates and you get certain nodes of the oscillation patterns. OK. And just as a simple demonstration of this, you could imagine taking a simple mug, OK, and what I'm going to do is I'm just going to tap it with a spoon. OK, so this is an ordinary mug. But the key thing is, if I tap it here, you can hear one note, OK, and if I tap it slightly off off that angle, can you give us a different name?
OK. So what's happening there is that by tapping the mug in different places, you're exciting, different modes of oscillation, and each of those modes of oscillation has a different frequency. What's special about the mug? Well, it's certainly symmetric. So if I just tap in a different place, it shouldn't make any difference. But of course, there's a handle. OK? And say the handle, the mass of the handle effect, which modes vibrate depending on where I hit it and what those frequencies are.
So I've talked about these different modes, what are called eigen modes as being the shape that a drum will have if you hit it and actually you've already walked past an example of that today. So if you go outside and look up at the ceiling, you'll see there's this big glass roof, which is precisely the eigen mode. Actually, the second I have made of a drum with this strange boundary shake.
OK. So I've talked about why the frequency of the oscillation depends on the shape, and then you might if you're a mathematician, you might start to think, Well, if I listened very carefully to the sound the drum makes and I pick out all of the different frequencies that are there in that sound, maybe I can also tell what shape the drum is going to have.
And so that's a question that was posed in the 60s and it wasn't solved until the early 90s when these three mathematicians down here say that that actually you can have drums with very different shapes or apparently different shapes that have exactly the same frequencies. Okay, so there's a slightly odd shaped drum, but to actually prove that the spectra, the frequencies of these two drums is identical is a little bit involved.
OK, so what about other examples of the wave equation in that you may have come across? Well, you may remember a few years ago there was a lot of excitement about. Sorry about the discovery of gravitational waves. So this is a simulation of two black holes colliding. OK, and what you're seeing is the sort of background of space time. As they get closer and closer, the time left until the collision is shown in the top left. And when we get to zero, it will rapidly slow down.
But what you might be able to see or just start to see is that actually the colour is changing in space, and that's really because gravitational waves are being emitted and that's what was detected at the end of 2015. So in the second movie, we're just as email. And then you'll be able to see these waves propagating across space time in a very similar way to the way in which ripples move on the surface of the pond after you drop a stain into it.
OK, so these these are the gravitational waves, another example that you may have actually seen already today is what's called the phantom traffic jam, and that's the situation in which you're travelling on the motorway or on any road and everything's going fine. And then all of a sudden the traffic stops and you think, OK, maybe there's been an accident or someone's broken down.
And then after a few minutes, equally, suddenly everyone starts moving again and you never see any sign of an accident or a breakdown. So the question is what's going on there? And the answer is that people are imperfect drivers and as a demonstration of this. The BBC One show a few years ago did sort of experiment where they asked a series of drivers to just drive around this nice circle and told them to all stick to 10 miles an hour,
OK? They, even though they've all been given instructions to stick to a particular speed, there are little variations in the speed at which they go. And, of course, because they're trying to avoid banging into each other as well, what you see is that actually there's a wave sort of ends up with a stop wave that kind of propagates backwards all the way around the circle. OK, now of course, the mathematics behind this is slightly different to the mathematics of the wave equation.
I presented a few slides ago, but the features of this wave propagation and various other things are really generic features of the wave equation. What I want to do now is to move on to the diffusion equation. It's often also called the heat equation because it's used to describe the mathematics of how heat flows. And I want to go through the derivation of the heat equation a little bit just because it gives you an idea of a different physical principles and Newton's second law.
So what we do is we think about a metal bar, OK, that's got there's no heat loss through the sites. I'm just going to imagine putting some heat in at one end, and I want to think about how it flows along the length of the bar.
OK. So if I think about the temperature, which is going to vary with space and time and I think about how does the temperature in this little cylinder, this little cylinder with the dust curves on either side vary between time T and time T plus Delta G. Well, if the temperature has changed and that means there must be a change in the internal energy. The change in internal energy is a change in temperature times the area times the length times,
the density times what's called the specific heat capacity. How can that energy have appeared a contest have appeared from nowhere. It must have either flown in from the left or not flown out from the right. So really it's a difference between how much heat flows in from the left and how much heat flows in from the right that will determine how much the temperature changes in an instant of time.
So I've just introduced this Q of X, and I haven't told you anything about what it is, but Fourier told us that actually heat flows from hot to cold and in mathematics, we write that as Q the heat flux, the flow of heat is minus d t by the X case. Heat flows from high T again down to low T. So if I substitute this into the right hand side here and then 10 X do that, Delta X tends to zero. And Delta T 10 to zero, then what I find is exactly this diffusion equation.
Well, what can I say about how that or the way in which the solution to this behave? Well, the way in which you will have seen the consequences of the heat equation in everyday life is when cooking, and it's typically important in our house because I really like this brownie recipe. OK. And so what you see when you look at the recipe for brownies is, as you expect, it might tell you the ingredients.
It tells you what you need to make this brownie. But then somewhat surprisingly, I think the first instruction is that you should lightly grease a 20 centimetre shallow square cake tin and line the base. So that wasn't what I was expecting. I was expecting to do some mixing. So the question then is, well, why does it tell us what size tin we should use? And the real particular problem in our house is that we don't have a 20 centimetre cake tin.
We only have a 15 centimetre cake tin. And so the question is what should we do if we only have a 15 cents make sense to me, the cake tim. Obviously, one solution would just be to make two sort of scale the ingredients, but that would mean having less brownie. So that's not really a solution. We want to think about how we should change the cooking of the brownie to accommodate this.
OK, so to a mathematician, cooking is really just heating up the browning, you take a thin slab of brownie mix, you put it in the oven, the oven hot, the top and bottom are getting heated that he has to diffuse through to the centre and say that the centre gets sufficiently hot to cook the eggs and whatever else. OK. So basically, what you're waiting for is you're waiting for the temperature to diffuse through the thickness.
That means we're going to use a diffusion equation, and the diffusion equation tells us that the temperature changes with time according to the derivative with respect to the thickness of a y. Well, second derivative with respect to the thickness.
Now, as I said, we've not don't have time to talk about how to solve this, but I can say that basically in terms of the solution, what matters is the differentiating is a little bit like dividing by time and differentiating in respect to why it's a little bit like dividing by the thickness h. So I differentiate respect to time once I divide by time, once I differentiate with respect to y twice. I divide by the thickness twice and the important things.
So that tells me that the time it takes to cook is going to be proportional to the square of the thickness of the brownie mix. OK, so that's the sort of conclusion from this very simple mathematical model. And of course, the recipe is already hinted fixes a particular volume of mix. So if I change my area of my tin, then I'm going to change the thickness of my brownie mix. And that's going to mean that I'm going to change the cooking time in proportion to the thickness squared.
So in particular, if I use my 15 centimetre cake tin, it's going to mean that I need to cook three times longer. OK, just because of the way the area scales and so on. Now, of course, there are lots more serious applications of the diffusion equation and one that's particularly important to the people doing research here is what's called by interaction between multiple chemical species. So what they do is they write down a different diffusion equation for the diffusion of a chemical one.
And then for a chemical to and there's an interaction between these two chemicals. And then you can do various analyses on this. I'm going to show you a video of a numerical solution of this equation down here. And the key thing is that on the left is going to be a video of what solution looks like if you have a rectangle and on the right, if you have a square, OK? And what you see is that very quickly, because you've got these two chemicals interacting,
you get stripes of one chemical, followed by a stripe of another. If you solve it in a rectangular domain, what is it? Whereas if you solve it on a square domain, you tend to get spots. So as I say, this is something that's an area of active research. But it's believed that at least this this relationship between the domain size and what pattern you get is seen in the natural world. So if you look at big cats in particular, a lot of them tend to have spotty coats and stripy tails.
And it's thought that this is really a consequence of the change in aspect ratio as you go along the tail. OK, so this is covered in a various third and fourth year courses in mathematical biology. The final equation is what's called the +s equation, which is used to govern the potential around electric fields, how fluids flow past aircraft wing and then also wings, and then also things like the evacuation of paint.
So this is a time lapse movie of paint drying, and what you can see is it's more interesting than it's meant to be, right? So basically you you can see the sort of wet front propagating and so on. So the important characteristic of the plastics equation is that it really doesn't like curved regions.
So you know that you might have heard in science or physics that when you have a very sharp conductor, charges accumulate in that region of very high curvature, and that's really a consequence of the plastics equation. The consequence of that is that tool. Shop buildings tend to get struck by lightning. But in your everyday life, there are some other examples as well.
So if you spill a drop of coffee and then it turns out that the contact line the region around the edge has an evaporative singularity, an evaporation rate and that pulls liquid to the edge and leaves behind a very dark ring of coffee right at the edge. Another example of thinking about cooking again is that when you cook potato wedges, you see that they don't tend to get cooked evenly all across the surface. And instead, you get very but you get sort of browner or burnt edges.
And that's because these sharp edges are where the evaporation happens most. That means as the smallest water content and hence the highest temperature. And so those regions get burnt quickest. Again, that's just a consequence of the +s equation not liking sharp corners. OK, so of course, I've told you about some applications, but there are other aspects of applied maths,
including how we deal with problems that we can't solve analytically. And as you might expect, that involves using a computer. But the important thing is to use the computer intelligently and that's again crudely this the field of numerical analysis. And there are a series of courses on numerical analysis in the undergraduate course. At Oxford, for example, the second year numerical analysis course really asks questions about how should we deal with matrices efficiently?
How should we think about a function on a computer? We can't tell it the whole function. We have to tell it the functions, behaviour at certain points. So how do we do that in a clever way? And then in the third year, there are various options again, which are really about how to control the errors that you make by solving a problem on a computer.
And again, there's two different courses one that, again, loosely speaking, is focussed on solving the diffusion equation and then the second one that's loosely speaking focussed on solving the +s equation. OK, so I've told you some examples of problems of plague mass in the real world, and I hope I've given you a sense of at least why I study applied maths, which is that I think it really gives you a new perspective on the world around you.
So whoever that's thinking about why potato wedges are bent on the edges or why a boat travelling alone to sea has a particular wave pattern behind it? And I think when I was your age, I think I was very worried about whether I wanted to do science or whether I wanted to do maths and actually doing applied maths is kind of got the best of both worlds. You sort of get to learn about interesting scientific problems and to develop mathematical models of those problems and then to solve them.
And of course, because it's mathematics, the key thing is that you're allowed to make abstractions. Just think about that model of the brownie. Of course, the real brownie is not an infinitely thin, infinitely long, thin slab, but it tells me the essential mathematics that's going on in the cooking of a brownie. I realise that there are a lot of parents in the audience as well as I just wouldn't to say a few words about why the skills learnt and applied maths are useful in the job market,
essentially. And there are various reasons. For example, the way in which fluid flows are described mathematically is used a lot in finance. A key thing that we get students to do is really to learn how to code, how to deal with data. And again, that's a key skill. And crucially, a lot of our students go into a whole range of different careers, whether it's in finance, software engineering, cryptography, teaching and so on.
So I'm almost done, I just wanted to say that if you're interested in some of the applications I told you about today, there's a very nice short book called Applied Maths, a very short introduction by Alan Greeley, which is a very short book. And it's really in the same spirit as a talk that I gave you here. And that's the main reason that I mention it now. But I also mention it because I am sort of my boss, and it's good to keep him happy. Thank you very much.