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0:08
I'm Steve Brunton and I'm really excited to introduce you to one of my favorite topics in all of math, which is Differential Equations. So this is a short course I'm putting together this is actually part of a larger sequence in my engineering math course that I've been offering for about the last 10 years. This is one of my favorite courses to teach: math to engineers, and part one -- kind of the entry level for this, as I see it -- is really this topic
0:42
of differential equations. So I fell in love with differential equations, it opened up a whole world of possibilities, of things you could model, of things you can understand, of things you can build with math with equations. And so I feel like this is the next logical step after Calculus. So if you like Calculus and if you like the way that you can model the world with how things change in time, this is the next step in that progression: differential equations.
1:13
We're going to use Calculus to model how systems change in time. So this is one of my favorite examples of a chaotic dynamical system or differential equation called the Lorenz system, which was introduced about a half a century ago, a little bit longer than that, to model chaotic convection in the atmosphere. So we'll talk about this a lot later. This course is meant to be about a seven week series in a university class, so that's about 20 hours of lectures
1:47
that I'm going to be putting together here. And all that I assume that you know know is that you have taken a Calculus class... I don't even assume that you remember most of your Calculus class I'm assuming that you are smart and motivated that you have seen some of these calculus Concepts before but maybe you don't remember them and I'm going to build everything up from scratch into this pretty rich and complex world of differential equations.
2:16
okay good um so in this video I'm going to kind of lay out what are the topics we're going to cover in the class what are some of the applications that we can open up and enable with this kind of of analysis of differential equations I'm inherently going to be folding in linear algebra because the way I see it really these two topics should be taught completely together you can't understand differential equations without linear algebra and differential equations are probably
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the best motivating example to understand linear algebra better so for me linear algebra is a is a piece it's a part of differential equations you can't separate those so we'll talk about topics applications I'll tell you kind of where this fits in so I've already told you this is kind of the next logical piece after calculus but I'll tell you also what's next Once you learn these differential equations what new fields of math are going to be open up to you and I'll tell you how
3:10
to get some of the resources that we're going to use throughout this class so in the video video description below I'm going to put links to all of the code for the entire class everything's going to be available in Python and Matlab there will be PDF notes for all of the lectures so every single video is going to have a lecture notes with it there will be a course website I'm planning on writing you know a short book on this topic so maybe by the time you watch this
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for those of you time travelers from the future you know maybe there are actually be you know a typed up book that you can you can reference but all of that will be in the the video description
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good um and I'll tell you why I think this is important and interesting it's one of my favorite topics okay so overview of objects that are going to be covered in this uh this course and this kind of 20 hours of material that fits into this larger engineering math framework I'm going to start with a simple review of calculus this will be super brief super short but you know again I'm going to assume you've taken calculus but maybe you don't remember it
4:15
or maybe it wasn't taught you know in a way that was motivated by applications so I'm going to you know start really really simple remind you what the derivative power you know the derivative is power rule chain rule but I'm going to also really kind of early on focus on the Taylor series so you have some intuition for what this this means and how this works because we're going to use Taylor series quite a lot and how we understand you know Solutions of differential equations
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we're actually going to derive exponential functions and Sines and cosines as Solutions of differential equations using Taylor series so this isn't going to be very long but it's very important so if you need to brush up on calculus definitely you know don't skip these lectures
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and then I'm going to jump right into a system simple simple ordinary differential equations so really simple uh differential equations of a single variable X so X is some quantity that's changing in time that I care about maybe it's the population of bunnies and those population of bunnies is growing and growing and growing so the rate of change of X my population the number of bunnies in my population is going to equal it's going to be a function of my
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current population so if I have a hundred bunnies they're going to grow faster than if I only have 10 bunnies if I have a thousand buddies they'll grow even faster still so the rate of change of my population in time is a function it's a linear proportional to the number of bunnies I have in my population currently and so this is going to be like the simplest simplest entry ode ordinary differential equation and from this we're going to look at what the solutions are
5:54
how to analyze them how to interpret them and then we're going to generalize that into much more complicated Rich differential equations so here dot just means derivative and time that's shorthand for D DT of X where X is the state of my system or some variable that I'm keeping track of as it evolves in time like a bunny population or like you know the radioactive decay of plutonium or uranium or something like that so we'll start off real simple and build everything from scratch
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and then we're going to get to more complicated examples of systems of ordinary differential equations so now instead of keeping track of one variable like the population of bunnies now maybe I'm going to keep track of lots of variables I care about maybe I am keeping track of the population of bunnies and wolves and you know a few other species so I have a you know multi-dimensional State X that has a few components you know X1 X2 X3
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and so now this differential equation is going to be a matrix system of differential equations where a is not a number it's a matrix that tells me how my my system changes in time as a linear function of that system okay so systems of Odes Builds on these simple ones but very very important lots of examples lots of cool things you can do with systems of ordinary differential equations you can model pendula masses on Springs lots of cool systems you can model
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and I'm this is where the linear algebra starts to come in so when we have systems of ordinary differential equations you know a is a matrix so we're going to be talking about linear algebra and that's essentially the study of you know matrices and operators and their properties and specifically I'm going to derive the eigenvalues and eigenvectors of this a matrix specifically as a useful set of coordinates or a useful set of of quantities that are used to
7:53
solve the system of Odes so we're going to show what it means to make the system easy to solve and we're going to find that the way that we make that system easy to solve is through an eigenvector coordinate transformation so we're going to derive eigenvalues and eigenvectors and motivate them as how we solve systems of Odes so if you took a linear algebra class in the past and you were confused by eigenvalues and eigenvectors or you thought they were unmotivated
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that couldn't be farther from the truth they're very easy to understand they're extremely well motivated they are the way that we solve systems of Odes so I'll derive all of that from scratch
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and then we're going to go to more complicated differential equations so instead of linear differential equations where everything where the rate of change of X is kind of linearly proportional to the state X we're going to talk about non-linear differential equations nonlinear systems most of the real world is actually non-linear so when I talk about the populations of bunnies and wolves you know bunnies or prey and wolves or Predators They Don't Really behave
9:00
linearly there are non-linear interactions that are important to model and so we'll talk about those non-linear systems that chaotic example that movie I showed in my first slide that's an example of a the solution of a non-linear ordinary differential equation and a lot of my favorite systems really are in this uh this realm here so the motion of the planets and the solar system is a rich chaotic non-linear system your metabolism in your body is a non-linear dynamical system
9:29
you know presumably your brain is a very high dimensional complex non-linear system so we'll talk about those as well and throughout all of this I'm going to be pinning this to numerics and computations so we're not only going to be talking about how do you do all of this with pencil and paper or on a whiteboard we're also going to you know fire up Jupiter notebooks in python or Matlab and we're going to work through how do you actually simulate these differential equations how
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can we plot our Solutions and confirm that they um you know match our intuition from eigenvalues and eigenvectors okay so everything I'm going to do is going to be pinned to computations and I'm going to do everything in Python everything in Matlab all of the codes are going to be online freely available and open okay so you can kind of pick or choose your language of choice good uh so
10:20
that that's the high level overview this is again approximately seven weeks about 20 hours a little bit less than 20 hours of content I'm planning and this fits into the larger framework of um you know engineering mathematics this is the math you need to know if you want to do you know real science and engineering um kind of at the next level Beyond calculus okay um and again I I always like to point out you know a lot of what I'm going to be telling you about is pretty classical
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pretty analytic it can almost feel a little dry or desiccated you know some of it's hundreds of years old a few hundreds of years old you know most of the differential equations that we deal with at least to begin with come from things like Newton's f equals m a and they're built on the calculus you know that was around since Newton and leibniz and so some of this you know can feel pretty classical
11:09
and that's okay that's fine we're gonna you know we need to build somewhere but we're going to use these techniques these pencil and paper techniques and I really find it important I want you I want to convey to you how we use the intuition we get from paper and pencil calculations for advanced topics like now in the modern era with computers we're not limited to paper and pencil we can simulate very very complex systems very high dimensional very non-linear things you
11:40
could never solve by hand with paper and pencil things that you know even Isaac Newton couldn't solve analytically the flow you know the fluid flow past a cylinder we have to simulate this or run experiments but the intuition we build for the classic systems that you can solve with paper with eigenvalues and eigenvectors and linear systems and things like that these ordinary differential
12:03
equations I'm going to introduce you to we build a tremendous amount of intuition for how the world works and how differential equations and dynamical systems how things change in time from those analytic classic Solutions and we're going to carry that intuition with us when we do advanced modern computational methods or even more recently data-driven analysis and machine learning methods
12:28
so it's very important I'm not just telling you about things that were discovered 300 years ago but I want you to know how we can take our you know analytics Solutions things we can actually solve in closed form and how we can translate that to you know really hard Modern Problems things like you know Advanced simulations and advanced machine learning models where you know no one can write down the solution to these we need computers for these but we can still gain
12:57
intuition from the classics so that's a really important perspective here of how that fits in good okay um so again this is the overview of topics I've already walked you through these
13:11
um and then the next thing I want to tell you in this video is how this fits in to you know once you you learn this material you invest this 20 hours what does that open up to you what are the next things and I have most of these uh next steps actually filmed up so once you learn this you can go kind of learn the next things okay so what's next
13:34
um you know at least when I teach this as a two-quarter sequence for my students I have two weeks on complex analysis so Solutions of differential equations are naturally going to involve um you know complex numbers and complex functions so we'll spend you know not in this sequence but in a future sequence we'll spend two weeks of complex analysis uh we'll introduce Vector calculus and partial differential equations so generalizing these ordinary differential
14:03
equations which are just you know a single derivative uh in terms of the independent variable now we're going to look at multiple derivatives uh of different independent variables so I can take you know derivatives of a function in time and in space and relate those that's what's known as partial differential equation a partial differential equation and that's inherently built on the language of vector calculus so this for me is actually a unit Vector calculus and partial
14:29
differential equations go together in the same way that ordinary differential equations and linear algebra go together so I'm actually teaching this as a unit again that's after this kind of intro to differential equations you can then learn complex analysis Vector calculus and pdes uh we're also in this two-quarter sequence going to introduce the Fourier and Laplace transform very closely related to differential equations especially partial differential equations and
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I'm not just going to talk about kind of again classic analytic you know hundreds of years old Fourier and Laplace transforms I'm also going to talk about how you do this in a computer with Discrete Fourier transforms and things like that how do you use those for you know not just solving differential equations how do you use those for things like audio processing or image compression so this is kind of one of my favorite topics is really
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this is transitioning from you know differential equations and solving physics problems to maybe solving data science problems how do I compress images and audio how do I do things like that and that segue is very naturally into kind of and this is only a teaser in my two-quarter uh engineering math sequence but I have a you know whole kind of boot camps and courses on this alone
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once we have learned about Fourier transforms and Laplace transforms and we had that kind of understanding of what linear algebra means it's really natural to introduce modern data-driven techniques um you know data decompositions and machine learning things like the singular value decomposition which generalize the Fourier transform to more complex and Rich systems so this is usually one of my students one of my my favorite topics and one of the favorite topics
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of my students when I teach it this is kind of all culminating towards these Advanced topics and I'll also point out the differential equations you're going to learn in this sequence in this in this course are going to be very very applicable to control systems so if you want to learn about control systems and you want to kind of dive into the control boot camp this is the starting point
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is really learning about differential equations and linear algebra so that you feel comfortable so these are all of the topics that are kind of opened up after you take this course so I'm super excited like for me this opened up the whole world of engineering applied math and physics and I hope uh you know my dream this would be the absolute best outcome is if even a small fraction of you
16:57
feel the same way that this opens up possibilities for you as well okay good so I hope you can tell that like I love this I'm passionate about it this is what I I want to do is to teach you this math okay so that's what's next um and I think for the rest of this video I'm just going to give you some cool examples and applications of things uh you know cool applications that you can start studying and understanding once you have a handle on differential equations
17:27
you know one of my passions and and Fields of Applied interest is fluid mechanics I find fluids really interesting because like literally we live and work inside of a fluid and so do all of our machines your heart is pumping fluid through your veins um you know every almost every trillion dollar industry has a working fluid at the heart of it if you think about energy Transportation defense Health you know all of these have fluids wind turbines and you
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know Automobiles and trains and planes and cars all operate in fluids and at the end of the day
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fluid dynamics are governed by a set of differential equations that you can write down essentially it generalizes Newton's second law of momentum conservation to fluids okay so this is an example of a differential equation that we can understand and solve after these techniques some other cool examples of differential equations that come up all the time involve mixing so mixing is a really big problem industrially and scientifically how do you let's say you're trying to make not
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acid let's say you're trying to make aspirin not acid maybe maybe you want to make acid but you're making aspirin you know and a big chemical vat and so you're adding all these ingredients and you want to mix them as efficiently as possible you know how do you put in stirring rods and things like that to mix this system as efficiently as possible again this is a dynamical system uh that that is governed by a set of differential equations you know that tell you how the system is
19:06
going to mix so that chaotic low rents a tractor I showed you in my first slide that's an example of a chaotic system that mixes uh efficiently in time uh and and once you you understand how to solve differential equations you can do things like Drop little test particles in these fluid systems in these Vector Fields so this is a jellyfish and this is one of my cool one of my favorite examples
19:31
where researchers actually measure the vector field kind of the fluid flow Field behind this jellyfish using uh you know measurement techniques experimental measurement techniques and then thinking about that fluid Vector field or velocity field as a differential equation you can think of where little food particles would add vect in that in that differential equation you could integrate those forward along this differential equation and figure out where they'll go
19:59
and so for example uh Sean shad and John DeBerry and Jerry Marsden in this really really cool paper showed that by integrating these particles you can find Regions where some particles get ejected into the propulsive wake of the jellyfish whereas other particles these green particles get kind of entrained into the Bell or the feeding region and so again this is they were using differential equations and the same exact techniques I'm going to teach you about in this series to
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integrate these test particles and to make this kind of scientific discovery of how jellyfish eat I can't help myself I'm also going to show this really cool kind of follow-on work uh by by John DeBerry and kit Parker where they actually design a mechanical jellyfish this is using I believe tissue from a mouse or a rat heart some rodent heart that they put in a fluid that has electrolytes and they're going to pulse this electrolyte solution
21:00
and kind of get this artificial jellyfish to uh to swim so this is when they start pulsing the the fluid you know they they put current through that fluid and this thing starts to pulse and again you can understand this is a dynamical system it has a state that varies in time according to rules f equals m a kinds of rules and so you can write down a differential equation and understand how
21:24
to design and optimize and predict the motion of this this system uh other things you can do with differential equations is understand you know geophysical flows ocean flows climate systems
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um you know things like if there's a an oil spill in the Gulf of Mexico where is that oil going to go can you predict uh you know where things are going to go and have your models match again based on these differential equations and solution techniques to those differential equations
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um this is a cool simulation uh that I like by Philip dutois where he applied that same basic analysis to the entire Global ocean model to figure out where our regions where mixing occurs and where mixing doesn't occur and things like that again using basic techniques and differential equations that I'll tell you about in this class
22:19
for me fluids is really interesting but the whole world is not just fluids there are you know climate processes and chemical processes you know combustion processes those are all systems that change in time according to rules and that's all that a differential equation is is a set of rules for how your system changes in time
22:39
uh good so in this we will also touch on chaos I've already eluded you know chaos for me is one of the most fascinating topics in all of Applied math because it comes up so often when we analyze systems maybe they seem to be orderly um for some parameters and when you change a parameter it becomes very chaotic and unpredictable so that's you know something we'll talk about is how can you simulate these systems how can you diagnose chaos how do you handle it numerically what are
23:07
some of the tools to analyze it um you know it turns out that the solar system the planets in the solar system have chaotic motion which I think is kind of interesting so they appear to be going around in ellipses around the sun which is true but if you are a a satellite or if you want to send a space probe to Jupiter there are these kind of chaotic trajectories that you can leverage for energy efficient space travel which is one of my favorite examples of
23:35
a dynamical system again a lot of our knowledge of dynamical systems you know if you go back to Newton and and leibniz and kind of the classics a lot of this was motivated by understanding you know planets in the solar system and a lot of research still uh goes into to this this topic
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um I'll talk about things like you know double pendula this is one of my favorite tabletop examples of a chaotic dynamical system so this is actually me in a lecture in person about 10 years ago and this is my uh double pendulum setup that I built I'm just going to run this again
24:09
good maybe I'll even bring that into this lightboard studio I'm trying to figure out where I would mount it you know I'm going to give you a lecture on chaos and the double pendulum uh what it means and how to simulate it things like that if you have these differential equations and you understand how to use them you can do things like do control design and actually you know harness the Dynamics and overcome that chaos to do things like stabilize you know a trajectory
24:37
that you want and it's super interesting humans walking so I am an inverted double pendulum approximately as a walker if you you know shot novocaine or lidocaine into my legs I'd fall over because I'm unstable I'm approximately stable right now sometimes I'm more unstable than others and you know so even like humans walking and bio Locomotion very much a dynamical system governed by differential equations and we can understand how to control those
25:04
systems and how to analyze those systems again you know using the techniques from this class
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um and once you can control these systems you can do things like Rehabilitation medicine you can also do things like you know Landing a rocket this is an inverted pendulum control system that you're trying to you know you're trying to land this inverted pendulum with as little control as possible so again this is a pretty good example of a differential equation that we want to write down and analyze and eventually control the combustion in the rocket engine
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also a dynamical system you know the wind flowing over this thing the fluid flow environment also a differential equation so it really is kind of a rich uh almost everything you can think of as a differential equation at the end of the day and except things that are static and don't move okay you hope your bridge that you build is not a dynamical system but everything else really is
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okay good I'm trying to think if there's anything else that's really the very high level I've been looking forward to this class for a long time um you know I've been teaching this for 10 years this is one of the things I'm most passionate about and I'm super super excited to you know to give this to a broader audience um you know please let me know what's confusing what made sense what you liked what you didn't like you know this is going to be a living set of lectures again
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all of the resources are going to be made open you know PDFs of the notes and all of the codes
26:33
um I'm excited to walk you through all of this um you know I'm going to try to release all of these videos pretty soon so you're going to see this and then hopefully within a very short amount of time all of this is going to be following so I'll also have playlists again in the description so you can go to that playlist so you don't get lost about where you know what's the next video you can kind
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of follow this in in the right order uh good okay um just some sneak peeks of things I'm going to show you because I have a little bit of time left um you know I'm gonna again I've already mentioned this but this is one of my absolute favorite lectures to give because it made so much sense to me it was something I struggled with and I had bad explanation after bad explanation and you know it finally clicked when I started thinking about eigenvalues and eigenvectors as a coordinate
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transformation so eigenvectors the eigenvectors of of a matrix is a coordinate transformation it's a matrix that changes my coordinates so that my differential equation x dot equals ax becomes diagonal which means it's easy to solve so you know this is just a teaser I'm going to have a whole lecture on this but I'm really excited to share the intuition you know about how eigenvalues
27:48
and eigenvectors I think are best understood when you think about solving differential equations and when you want to solve differential equations eigenvalues and eigenvectors are like your most powerful tools that's how we think about and analyze and understand all of the Dynamics of these differential equations so this is going to be a killer lecture another one you know I'm going to show you how you take a fluid flow like this you know how you write down the differential
28:13
equation let's say I have a little particle here I drop a little dust particle X and it moves around in this fluid flow field U so U is literally a velocity field a vector field that varies in Space X and in time t and we're going to look at how that velocity field that Vector field of the fluid is a differential equation for how a passive particle would move if I drop a little dust particle or a smoke particle and in this class we're going to look at how
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to numerically integrate those test particles forward throughout that flow field and extract really interesting useful quantities like these cool movies and other things like it
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for now you know stay tuned this is going to be a lot we're going to build up real simple you know from basic calculus that you either have seen or hopefully remember and then we're going to hit the ground running it's going to go slow but it's going to build up quick and it's going to open up a huge huge set of engineering applied math possibilities so I'm super excited stay tuned and I'll see you next time thanks foreign
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