1 00:00:11,970 --> 00:00:17,670 My name is Dominic Vella, and I am professor of applied maths here in the Mathematical Institute, 2 00:00:17,670 --> 00:00:23,730 and I'm also a tutorial fellow at Lincoln College, which is one of the colleges in the centre of town. 3 00:00:23,730 --> 00:00:29,160 I've got this photo here in the front, like just on the off chance that it's raining, which fortunately is not today, 4 00:00:29,160 --> 00:00:35,040 but also to show you that some of the buildings in the centre of Oxford particularly beautiful and well worth visiting. 5 00:00:35,040 --> 00:00:42,390 This is a photo of Lincoln's library. So my job today is to tell you a little bit about what applied maths is. 6 00:00:42,390 --> 00:00:50,400 And historically, it's something that's really been viewed as sort of developing mathematical models of real world phenomena. 7 00:00:50,400 --> 00:00:53,770 And historically, maybe it's been focussed on physical objects. 8 00:00:53,770 --> 00:01:02,010 So particular some of the things that people in the department here think about how brains fold and trying to understand 9 00:01:02,010 --> 00:01:09,990 the very convoluted folds that you get in human brains and are actually observed only in relatively large mammals, 10 00:01:09,990 --> 00:01:15,840 mammals larger than the ferret or the rat. And the question is what gives rise to this very fine pattern? 11 00:01:15,840 --> 00:01:20,280 And can we understand how it helps with cognitive ability? 12 00:01:20,280 --> 00:01:25,950 So that's a problem in solid mechanics. I'm interested in some mechanics, but also how fluids flow. 13 00:01:25,950 --> 00:01:31,290 And one of the things that I've thought about a little bit is what happens when you pour yourself a cup of juice. 14 00:01:31,290 --> 00:01:34,980 So this is a movie where you see orange juice being poured into a cup. 15 00:01:34,980 --> 00:01:39,420 And if you look at the bubbles that rise to the surface, they sort of clump together. 16 00:01:39,420 --> 00:01:41,640 And then over the period, over a period of time, 17 00:01:41,640 --> 00:01:50,970 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. 18 00:01:50,970 --> 00:01:57,360 But increasingly applied maths is not just about the physical world, but also about the digital world and how to deal with data. 19 00:01:57,360 --> 00:02:04,020 So an example may be that some of you might be too young to know about photos on paper, 20 00:02:04,020 --> 00:02:08,730 but actually a big problem with photos on paper is that they can get creased and damaged. 21 00:02:08,730 --> 00:02:11,310 And the question is, if that's your only copy of a photo, 22 00:02:11,310 --> 00:02:16,740 what do you do to try and reconstruct the original image and actually mathematical algorithms, 23 00:02:16,740 --> 00:02:21,840 the people in the study that try and even paint the image so that you go from 24 00:02:21,840 --> 00:02:27,030 this damage photo the top to something closer to the original down bottom. 25 00:02:27,030 --> 00:02:33,070 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. 26 00:02:33,070 --> 00:02:39,990 Let's give you a taste of how applied mathematicians do that and also to describe a little bit about how 27 00:02:39,990 --> 00:02:45,750 what techniques we teach you in the undergraduate course here at Oxford that will allow you to do that. 28 00:02:45,750 --> 00:02:50,820 So the basic idea is that there will be some fundamental principle that we then try it out, 29 00:02:50,820 --> 00:02:56,550 try to write down mathematically and then from that mathematical expression of the fundamental principle, 30 00:02:56,550 --> 00:03:00,270 we then try and see what the logical consequences are. 31 00:03:00,270 --> 00:03:08,280 So a simple example of a fundamental principle is Newton's second law is something you hopefully would have seen at school in science, 32 00:03:08,280 --> 00:03:13,680 GCSE or A-level physics, for example. And Newton's second law tells you that F equals M. 33 00:03:13,680 --> 00:03:20,190 It says that the force are not on a body. F is equal to the mass times the acceleration. 34 00:03:20,190 --> 00:03:25,980 And if you know that the acceleration is the rate of change of lost in velocity is the rate of change of position, 35 00:03:25,980 --> 00:03:32,580 then you know that that acceleration can be written as a second derivative with respect to time of the position. 36 00:03:32,580 --> 00:03:40,230 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. 37 00:03:40,230 --> 00:03:47,730 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. 38 00:03:47,730 --> 00:03:56,460 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. 39 00:03:56,460 --> 00:04:01,980 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? 40 00:04:01,980 --> 00:04:08,890 If I think about the forces tangential to the motion, then I can write down that the component of gravity is minus MJG times. 41 00:04:08,890 --> 00:04:13,450 The sign of the angle. Theatre features that angle up there and that is the force. 42 00:04:13,450 --> 00:04:22,500 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. 43 00:04:22,500 --> 00:04:28,860 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, 44 00:04:28,860 --> 00:04:37,110 which is that for very small angles, sign theatre is approximately theatre and say this equation becomes relatively simple. 45 00:04:37,110 --> 00:04:43,740 It says that the second derivative of a function is minus some constant squared times the function itself. 46 00:04:43,740 --> 00:04:46,140 And if you've done the calculus of trigonometric functions, 47 00:04:46,140 --> 00:04:52,620 you might recognise that the solution of that equation is going to be a trigonometric function, sign or cause. 48 00:04:52,620 --> 00:04:59,040 So we can write down the general solution as being some amplitude times the cause of omega, 49 00:04:59,040 --> 00:05:03,960 the some frequency times time plus a Faceshift say omega here is given as the square 50 00:05:03,960 --> 00:05:09,450 root of the acceleration due to gravity g divided by the length of the pendulum. 51 00:05:09,450 --> 00:05:12,270 OK, so what can we say, given this solution? 52 00:05:12,270 --> 00:05:16,590 Well, the first thing we can say is that we know that cost goes backwards and forwards, and it oscillates. 53 00:05:16,590 --> 00:05:21,060 And that's just what we expect the pendulum to do as well. So that's good. 54 00:05:21,060 --> 00:05:25,770 What else can we say? Well, we've got this expression for what the frequency of the pendulum is. 55 00:05:25,770 --> 00:05:31,350 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. 56 00:05:31,350 --> 00:05:37,320 You're used to things how fast things move, maybe depending on the mass of an object. 57 00:05:37,320 --> 00:05:45,080 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. 58 00:05:45,080 --> 00:05:51,110 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, 59 00:05:51,110 --> 00:06:00,000 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. 60 00:06:00,000 --> 00:06:03,420 OK, so we've seen that mass doesn't change the frequency of oscillation. 61 00:06:03,420 --> 00:06:07,590 We could change G by going to Mars or the Moon and change a frequency that way. 62 00:06:07,590 --> 00:06:12,600 But a much simpler way to change the frequency is to play with the length of the pendulum. 63 00:06:12,600 --> 00:06:16,860 And this video I'm going to show you in a second here is from the Harvard Natural Sciences demo. 64 00:06:16,860 --> 00:06:28,920 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. 65 00:06:28,920 --> 00:06:34,500 OK, so you can see that as we expect from our formula, the length is influencing the frequency. 66 00:06:34,500 --> 00:06:40,880 The short one is going much faster than the low one or much is oscillating with a lower period. 67 00:06:40,880 --> 00:06:46,260 OK. And they've chosen the lengths of these pages in a very nice way so that you get these really mesmerising patterns. 68 00:06:46,260 --> 00:06:48,480 And if you go onto YouTube and watch the rest of the video, 69 00:06:48,480 --> 00:06:53,100 you'll see that actually comes full circle and goes back to the beginning and then carries on. 70 00:06:53,100 --> 00:06:56,700 The reason I have included this video here is because it gives you the sense 71 00:06:56,700 --> 00:07:00,900 that it's not necessarily enough to just think about how things vary in time. 72 00:07:00,900 --> 00:07:05,970 It might also be interesting to think about how things vary in space as well. 73 00:07:05,970 --> 00:07:08,520 And if we want to be very crude, then actually applied, 74 00:07:08,520 --> 00:07:14,370 Massey University is thinking about some of the things that you've thought about at school as functions of one variable, 75 00:07:14,370 --> 00:07:17,970 but thinking about how we deal with functions of two variables. 76 00:07:17,970 --> 00:07:27,570 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. 77 00:07:27,570 --> 00:07:35,160 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. 78 00:07:35,160 --> 00:07:43,920 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. 79 00:07:43,920 --> 00:07:50,700 So an introductory calculus, we worry about what difference between this curly divide and the straight divide it is. 80 00:07:50,700 --> 00:07:52,890 And then in the course called multivariable calculus, 81 00:07:52,890 --> 00:08:00,470 we sort of talk about doing integration as the reverse of differentiation, but with multiple variables. 82 00:08:00,470 --> 00:08:02,360 OK, so with that mutation, 83 00:08:02,360 --> 00:08:09,680 then it turns out that there are essentially three different kinds of differential equations that come up with three different equations that come up. 84 00:08:09,680 --> 00:08:17,210 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. 85 00:08:17,210 --> 00:08:22,040 And if you look on the left hand side, you see the second derivative of F with respect to time. 86 00:08:22,040 --> 00:08:27,380 That's sort of coming from Newton's second law. It's a bit like the acceleration time in Newton's second law. 87 00:08:27,380 --> 00:08:36,200 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. 88 00:08:36,200 --> 00:08:40,560 The second equation that comes up is what's called the diffusion equation or the heat equation. 89 00:08:40,560 --> 00:08:45,860 Again, this looks similar, except on the left hand side you only have one time derivative. 90 00:08:45,860 --> 00:08:50,570 And I'll give you an example of or try and outline how this equation is derived so that 91 00:08:50,570 --> 00:08:55,160 you get a sense of a different physical principle that's useful in applied maths. 92 00:08:55,160 --> 00:08:59,360 The final example equation that comes up a lot is what's called the +s equation, 93 00:08:59,360 --> 00:09:05,120 and that two looks a lot like the wave equation, except that the two times are now on the same side of the equation. 94 00:09:05,120 --> 00:09:09,850 So it's as if time was sort of an imaginary space coordinate, if you like. 95 00:09:09,850 --> 00:09:13,270 OK, so these are the three equations that come up a lot. And again, 96 00:09:13,270 --> 00:09:18,730 a cool first year course called very serious and partial differential equations is really focussed on trying to 97 00:09:18,730 --> 00:09:25,750 understand how you would solve the different techniques you would use to solve each of these three equations. 98 00:09:25,750 --> 00:09:33,010 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. 99 00:09:33,010 --> 00:09:37,950 And some examples of where you might have seen these in your everyday life, where it. 100 00:09:37,950 --> 00:09:40,080 OK, so what about the wave equation? 101 00:09:40,080 --> 00:09:47,790 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, 102 00:09:47,790 --> 00:09:57,750 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. 103 00:09:57,750 --> 00:10:00,820 And then the question is just what are the big f g? 104 00:10:00,820 --> 00:10:09,330 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. 105 00:10:09,330 --> 00:10:16,020 C, You know that when you take a function of X minus A, that's like just translating the whole graph to the right. 106 00:10:16,020 --> 00:10:22,110 And if you do a function of X plus a constant, that's like moving the graph to the left. 107 00:10:22,110 --> 00:10:27,030 So what we're doing is we're moving the graph Big F to the right with an amount that depends 108 00:10:27,030 --> 00:10:35,050 on time with this constant C and we move in the graph Big D to the left again at some speed C. 109 00:10:35,050 --> 00:10:42,640 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 110 00:10:42,640 --> 00:10:51,150 understand what this transmission or this this sorry troubling waves do when you're playing your music. 111 00:10:51,150 --> 00:10:59,470 I'm sure that it's a familiar experience to parents and to children alike that you often get complaints about your music. 112 00:10:59,470 --> 00:11:03,230 And the question is, why is that what's going on there? 113 00:11:03,230 --> 00:11:09,490 So what we have is we have our music in our bedroom, which we think we are playing at a reasonable volume, OK? 114 00:11:09,490 --> 00:11:16,190 And it's hitting the wall and we hear it all being reflected by our parents hear stuff being transmitted. 115 00:11:16,190 --> 00:11:20,710 And the question is what bits are being transmitted and what bits are being reflected. 116 00:11:20,710 --> 00:11:24,220 And again, for a calculation that you can do after the third day course, 117 00:11:24,220 --> 00:11:32,290 you find that the ratio of what's transmitted to what's reflected is inversely proportional to the frequency omega. 118 00:11:32,290 --> 00:11:40,420 OK. So what does that mean? Well, it means that if you have very low frequencies, that transmission to reflection is very large. 119 00:11:40,420 --> 00:11:45,760 And if you have very high frequency, the transmission to reflection is very small. 120 00:11:45,760 --> 00:11:52,810 OK, so as we maybe what you see in everyday life that high frequencies tend to bounce off the wall, low frequencies are transmitted. 121 00:11:52,810 --> 00:12:00,460 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. 122 00:12:00,460 --> 00:12:04,930 OK, so there are lots of other solution types of solution of the wave equation. 123 00:12:04,930 --> 00:12:10,900 For example, if you think about a drum, the solutions are irrelevant when you hit a drummer. 124 00:12:10,900 --> 00:12:15,430 What are called normal modes and normal modes depend on the shape of the boundary, 125 00:12:15,430 --> 00:12:19,120 the shape of your drum, but they also depend on the way that you hit them. 126 00:12:19,120 --> 00:12:26,350 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. 127 00:12:26,350 --> 00:12:30,430 If you hit it in slightly other ways, you might get combinations of different moods. 128 00:12:30,430 --> 00:12:35,710 And these are the modes of a circular drum, so you get this sort of this hat shape. 129 00:12:35,710 --> 00:12:40,810 And again, each shape has a different frequency associated with it. 130 00:12:40,810 --> 00:12:48,360 And again, the techniques are used to calculate these are covered in this optional third vehicles called waves and compressible flight. 131 00:12:48,360 --> 00:12:56,490 What's really important is that the shape of the object is important in determining the frequencies of which it likes to oscillate, 132 00:12:56,490 --> 00:13:04,200 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. 133 00:13:04,200 --> 00:13:06,210 So you can see that in the violin, 134 00:13:06,210 --> 00:13:13,710 where you can put powder on the surface of violin as it vibrates and you get certain nodes of the oscillation patterns. 135 00:13:13,710 --> 00:13:20,700 OK. And just as a simple demonstration of this, you could imagine taking a simple mug, 136 00:13:20,700 --> 00:13:27,270 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. 137 00:13:27,270 --> 00:13:43,770 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? 138 00:13:43,770 --> 00:13:49,770 OK. So what's happening there is that by tapping the mug in different places, you're exciting, 139 00:13:49,770 --> 00:13:54,510 different modes of oscillation, and each of those modes of oscillation has a different frequency. 140 00:13:54,510 --> 00:13:58,770 What's special about the mug? Well, it's certainly symmetric. 141 00:13:58,770 --> 00:14:02,700 So if I just tap in a different place, it shouldn't make any difference. But of course, there's a handle. 142 00:14:02,700 --> 00:14:12,510 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. 143 00:14:12,510 --> 00:14:14,580 So I've talked about these different modes, 144 00:14:14,580 --> 00:14:22,800 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. 145 00:14:22,800 --> 00:14:29,810 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. 146 00:14:29,810 --> 00:14:34,110 Actually, the second I have made of a drum with this strange boundary shake. 147 00:14:34,110 --> 00:14:39,890 OK. So I've talked about why the frequency of the oscillation depends on the shape, 148 00:14:39,890 --> 00:14:42,170 and then you might if you're a mathematician, you might start to think, Well, 149 00:14:42,170 --> 00:14:49,340 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, 150 00:14:49,340 --> 00:14:53,570 maybe I can also tell what shape the drum is going to have. 151 00:14:53,570 --> 00:15:01,230 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 152 00:15:01,230 --> 00:15:09,360 that that actually you can have drums with very different shapes or apparently different shapes that have exactly the same frequencies. 153 00:15:09,360 --> 00:15:13,790 Okay, so there's a slightly odd shaped drum, but to actually prove that the spectra, 154 00:15:13,790 --> 00:15:19,080 the frequencies of these two drums is identical is a little bit involved. 155 00:15:19,080 --> 00:15:23,190 OK, so what about other examples of the wave equation in that you may have come across? 156 00:15:23,190 --> 00:15:33,790 Well, you may remember a few years ago there was a lot of excitement about. Sorry about the discovery of gravitational waves. 157 00:15:33,790 --> 00:15:43,830 So this is a simulation of two black holes colliding. OK, and what you're seeing is the sort of background of space time. 158 00:15:43,830 --> 00:15:49,590 As they get closer and closer, the time left until the collision is shown in the top left. 159 00:15:49,590 --> 00:15:52,920 And when we get to zero, it will rapidly slow down. 160 00:15:52,920 --> 00:15:58,740 But what you might be able to see or just start to see is that actually the colour is changing in space, 161 00:15:58,740 --> 00:16:05,810 and that's really because gravitational waves are being emitted and that's what was detected at the end of 2015. 162 00:16:05,810 --> 00:16:13,890 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 163 00:16:13,890 --> 00:16:21,710 to the way in which ripples move on the surface of the pond after you drop a stain into it. 164 00:16:21,710 --> 00:16:24,140 OK, so these these are the gravitational waves, 165 00:16:24,140 --> 00:16:28,550 another example that you may have actually seen already today is what's called the phantom traffic jam, 166 00:16:28,550 --> 00:16:34,310 and that's the situation in which you're travelling on the motorway or on any road and everything's going fine. 167 00:16:34,310 --> 00:16:39,440 And then all of a sudden the traffic stops and you think, OK, maybe there's been an accident or someone's broken down. 168 00:16:39,440 --> 00:16:47,300 And then after a few minutes, equally, suddenly everyone starts moving again and you never see any sign of an accident or a breakdown. 169 00:16:47,300 --> 00:16:53,480 So the question is what's going on there? And the answer is that people are imperfect drivers and as a demonstration of this. 170 00:16:53,480 --> 00:17:00,050 The BBC One show a few years ago did sort of experiment where they asked a series of drivers 171 00:17:00,050 --> 00:17:04,760 to just drive around this nice circle and told them to all stick to 10 miles an hour, 172 00:17:04,760 --> 00:17:11,600 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. 173 00:17:11,600 --> 00:17:14,780 And, of course, because they're trying to avoid banging into each other as well, 174 00:17:14,780 --> 00:17:24,500 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. 175 00:17:24,500 --> 00:17:29,450 OK, now of course, the mathematics behind this is slightly different to the mathematics of the wave equation. 176 00:17:29,450 --> 00:17:39,950 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. 177 00:17:39,950 --> 00:17:42,800 What I want to do now is to move on to the diffusion equation. 178 00:17:42,800 --> 00:17:48,890 It's often also called the heat equation because it's used to describe the mathematics of how heat flows. 179 00:17:48,890 --> 00:17:53,000 And I want to go through the derivation of the heat equation a little bit just because 180 00:17:53,000 --> 00:17:57,290 it gives you an idea of a different physical principles and Newton's second law. 181 00:17:57,290 --> 00:18:03,200 So what we do is we think about a metal bar, OK, that's got there's no heat loss through the sites. 182 00:18:03,200 --> 00:18:09,020 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. 183 00:18:09,020 --> 00:18:09,740 OK. 184 00:18:09,740 --> 00:18:17,690 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, 185 00:18:17,690 --> 00:18:24,620 this little cylinder with the dust curves on either side vary between time T and time T plus Delta G. 186 00:18:24,620 --> 00:18:29,000 Well, if the temperature has changed and that means there must be a change in the internal energy. 187 00:18:29,000 --> 00:18:34,580 The change in internal energy is a change in temperature times the area times the length times, 188 00:18:34,580 --> 00:18:42,810 the density times what's called the specific heat capacity. How can that energy have appeared a contest have appeared from nowhere. 189 00:18:42,810 --> 00:18:48,670 It must have either flown in from the left or not flown out from the right. 190 00:18:48,670 --> 00:18:53,430 So really it's a difference between how much heat flows in from the left and how much heat flows 191 00:18:53,430 --> 00:18:59,100 in from the right that will determine how much the temperature changes in an instant of time. 192 00:18:59,100 --> 00:19:04,110 So I've just introduced this Q of X, and I haven't told you anything about what it is, 193 00:19:04,110 --> 00:19:08,760 but Fourier told us that actually heat flows from hot to cold and in mathematics, 194 00:19:08,760 --> 00:19:16,260 we write that as Q the heat flux, the flow of heat is minus d t by the X case. 195 00:19:16,260 --> 00:19:22,050 Heat flows from high T again down to low T. 196 00:19:22,050 --> 00:19:29,040 So if I substitute this into the right hand side here and then 10 X do that, Delta X tends to zero. 197 00:19:29,040 --> 00:19:34,990 And Delta T 10 to zero, then what I find is exactly this diffusion equation. 198 00:19:34,990 --> 00:19:40,420 Well, what can I say about how that or the way in which the solution to this behave? 199 00:19:40,420 --> 00:19:47,200 Well, the way in which you will have seen the consequences of the heat equation in everyday life is when cooking, 200 00:19:47,200 --> 00:19:51,520 and it's typically important in our house because I really like this brownie recipe. 201 00:19:51,520 --> 00:19:57,160 OK. And so what you see when you look at the recipe for brownies is, as you expect, it might tell you the ingredients. 202 00:19:57,160 --> 00:20:00,880 It tells you what you need to make this brownie. But then somewhat surprisingly, 203 00:20:00,880 --> 00:20:10,240 I think the first instruction is that you should lightly grease a 20 centimetre shallow square cake tin and line the base. 204 00:20:10,240 --> 00:20:12,580 So that wasn't what I was expecting. I was expecting to do some mixing. 205 00:20:12,580 --> 00:20:17,610 So the question then is, well, why does it tell us what size tin we should use? 206 00:20:17,610 --> 00:20:22,680 And the real particular problem in our house is that we don't have a 20 centimetre cake tin. 207 00:20:22,680 --> 00:20:30,210 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. 208 00:20:30,210 --> 00:20:35,580 Obviously, one solution would just be to make two sort of scale the ingredients, but that would mean having less brownie. 209 00:20:35,580 --> 00:20:42,830 So that's not really a solution. We want to think about how we should change the cooking of the brownie to accommodate this. 210 00:20:42,830 --> 00:20:52,310 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, 211 00:20:52,310 --> 00:20:58,430 the top and bottom are getting heated that he has to diffuse through to the centre 212 00:20:58,430 --> 00:21:01,870 and say that the centre gets sufficiently hot to cook the eggs and whatever else. 213 00:21:01,870 --> 00:21:07,310 OK. So basically, what you're waiting for is you're waiting for the temperature to diffuse through the thickness. 214 00:21:07,310 --> 00:21:09,290 That means we're going to use a diffusion equation, 215 00:21:09,290 --> 00:21:17,050 and the diffusion equation tells us that the temperature changes with time according to the derivative with respect to the thickness of a y. 216 00:21:17,050 --> 00:21:19,410 Well, second derivative with respect to the thickness. 217 00:21:19,410 --> 00:21:25,970 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, 218 00:21:25,970 --> 00:21:31,010 what matters is the differentiating is a little bit like dividing by time and 219 00:21:31,010 --> 00:21:35,390 differentiating in respect to why it's a little bit like dividing by the thickness h. 220 00:21:35,390 --> 00:21:40,340 So I differentiate respect to time once I divide by time, once I differentiate with respect to y twice. 221 00:21:40,340 --> 00:21:43,340 I divide by the thickness twice and the important things. 222 00:21:43,340 --> 00:21:50,780 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. 223 00:21:50,780 --> 00:21:55,490 OK, so that's the sort of conclusion from this very simple mathematical model. 224 00:21:55,490 --> 00:21:59,780 And of course, the recipe is already hinted fixes a particular volume of mix. 225 00:21:59,780 --> 00:22:05,390 So if I change my area of my tin, then I'm going to change the thickness of my brownie mix. 226 00:22:05,390 --> 00:22:11,150 And that's going to mean that I'm going to change the cooking time in proportion to the thickness squared. 227 00:22:11,150 --> 00:22:16,520 So in particular, if I use my 15 centimetre cake tin, it's going to mean that I need to cook three times longer. 228 00:22:16,520 --> 00:22:20,750 OK, just because of the way the area scales and so on. Now, of course, 229 00:22:20,750 --> 00:22:24,920 there are lots more serious applications of the diffusion equation and one that's particularly important 230 00:22:24,920 --> 00:22:32,480 to the people doing research here is what's called by interaction between multiple chemical species. 231 00:22:32,480 --> 00:22:38,660 So what they do is they write down a different diffusion equation for the diffusion of a chemical one. 232 00:22:38,660 --> 00:22:42,980 And then for a chemical to and there's an interaction between these two chemicals. 233 00:22:42,980 --> 00:22:49,790 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. 234 00:22:49,790 --> 00:22:55,790 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, 235 00:22:55,790 --> 00:23:02,970 if you have a square, OK? And what you see is that very quickly, because you've got these two chemicals interacting, 236 00:23:02,970 --> 00:23:08,910 you get stripes of one chemical, followed by a stripe of another. If you solve it in a rectangular domain, what is it? 237 00:23:08,910 --> 00:23:13,660 Whereas if you solve it on a square domain, you tend to get spots. 238 00:23:13,660 --> 00:23:17,260 So as I say, this is something that's an area of active research. 239 00:23:17,260 --> 00:23:26,800 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. 240 00:23:26,800 --> 00:23:33,130 So if you look at big cats in particular, a lot of them tend to have spotty coats and stripy tails. 241 00:23:33,130 --> 00:23:40,490 And it's thought that this is really a consequence of the change in aspect ratio as you go along the tail. 242 00:23:40,490 --> 00:23:45,680 OK, so this is covered in a various third and fourth year courses in mathematical biology. 243 00:23:45,680 --> 00:23:51,440 The final equation is what's called the +s equation, which is used to govern the potential around electric fields, 244 00:23:51,440 --> 00:23:58,930 how fluids flow past aircraft wing and then also wings, and then also things like the evacuation of paint. 245 00:23:58,930 --> 00:24:05,370 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? 246 00:24:05,370 --> 00:24:11,170 So basically you you can see the sort of wet front propagating and so on. 247 00:24:11,170 --> 00:24:17,170 So the important characteristic of the plastics equation is that it really doesn't like curved regions. 248 00:24:17,170 --> 00:24:23,530 So you know that you might have heard in science or physics that when you have a very sharp conductor, 249 00:24:23,530 --> 00:24:29,560 charges accumulate in that region of very high curvature, and that's really a consequence of the plastics equation. 250 00:24:29,560 --> 00:24:34,750 The consequence of that is that tool. Shop buildings tend to get struck by lightning. 251 00:24:34,750 --> 00:24:37,960 But in your everyday life, there are some other examples as well. 252 00:24:37,960 --> 00:24:47,080 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, 253 00:24:47,080 --> 00:24:54,820 an evaporation rate and that pulls liquid to the edge and leaves behind a very dark ring of coffee right at the edge. 254 00:24:54,820 --> 00:24:58,810 Another example of thinking about cooking again is that when you cook potato wedges, 255 00:24:58,810 --> 00:25:07,480 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. 256 00:25:07,480 --> 00:25:11,260 And that's because these sharp edges are where the evaporation happens most. 257 00:25:11,260 --> 00:25:15,790 That means as the smallest water content and hence the highest temperature. 258 00:25:15,790 --> 00:25:25,380 And so those regions get burnt quickest. Again, that's just a consequence of the +s equation not liking sharp corners. 259 00:25:25,380 --> 00:25:30,690 OK, so of course, I've told you about some applications, but there are other aspects of applied maths, 260 00:25:30,690 --> 00:25:37,770 including how we deal with problems that we can't solve analytically. And as you might expect, that involves using a computer. 261 00:25:37,770 --> 00:25:44,730 But the important thing is to use the computer intelligently and that's again crudely this the field of numerical analysis. 262 00:25:44,730 --> 00:25:48,780 And there are a series of courses on numerical analysis in the undergraduate course. 263 00:25:48,780 --> 00:25:56,970 At Oxford, for example, the second year numerical analysis course really asks questions about how should we deal with matrices efficiently? 264 00:25:56,970 --> 00:26:01,560 How should we think about a function on a computer? We can't tell it the whole function. 265 00:26:01,560 --> 00:26:06,060 We have to tell it the functions, behaviour at certain points. So how do we do that in a clever way? 266 00:26:06,060 --> 00:26:09,090 And then in the third year, there are various options again, 267 00:26:09,090 --> 00:26:15,420 which are really about how to control the errors that you make by solving a problem on a computer. 268 00:26:15,420 --> 00:26:18,990 And again, there's two different courses one that, again, loosely speaking, 269 00:26:18,990 --> 00:26:27,020 is focussed on solving the diffusion equation and then the second one that's loosely speaking focussed on solving the +s equation. 270 00:26:27,020 --> 00:26:31,280 OK, so I've told you some examples of problems of plague mass in the real world, 271 00:26:31,280 --> 00:26:34,640 and I hope I've given you a sense of at least why I study applied maths, 272 00:26:34,640 --> 00:26:38,900 which is that I think it really gives you a new perspective on the world around you. 273 00:26:38,900 --> 00:26:43,340 So whoever that's thinking about why potato wedges are bent on the edges or 274 00:26:43,340 --> 00:26:49,240 why a boat travelling alone to sea has a particular wave pattern behind it? 275 00:26:49,240 --> 00:26:53,590 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 276 00:26:53,590 --> 00:26:56,920 to do maths and actually doing applied maths is kind of got the best of both worlds. 277 00:26:56,920 --> 00:27:05,470 You sort of get to learn about interesting scientific problems and to develop mathematical models of those problems and then to solve them. 278 00:27:05,470 --> 00:27:09,730 And of course, because it's mathematics, the key thing is that you're allowed to make abstractions. 279 00:27:09,730 --> 00:27:16,750 Just think about that model of the brownie. Of course, the real brownie is not an infinitely thin, infinitely long, thin slab, 280 00:27:16,750 --> 00:27:23,150 but it tells me the essential mathematics that's going on in the cooking of a brownie. 281 00:27:23,150 --> 00:27:26,180 I realise that there are a lot of parents in the audience as well as I just wouldn't to say 282 00:27:26,180 --> 00:27:30,770 a few words about why the skills learnt and applied maths are useful in the job market, 283 00:27:30,770 --> 00:27:39,020 essentially. And there are various reasons. For example, the way in which fluid flows are described mathematically is used a lot in finance. 284 00:27:39,020 --> 00:27:44,630 A key thing that we get students to do is really to learn how to code, how to deal with data. 285 00:27:44,630 --> 00:27:52,370 And again, that's a key skill. And crucially, a lot of our students go into a whole range of different careers, 286 00:27:52,370 --> 00:27:57,560 whether it's in finance, software engineering, cryptography, teaching and so on. 287 00:27:57,560 --> 00:28:03,140 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, 288 00:28:03,140 --> 00:28:10,610 there's a very nice short book called Applied Maths, a very short introduction by Alan Greeley, which is a very short book. 289 00:28:10,610 --> 00:28:16,070 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. 290 00:28:16,070 --> 00:28:20,240 But I also mention it because I am sort of my boss, and it's good to keep him happy. 291 00:28:20,240 --> 00:28:42,502 Thank you very much.