title

Differential Equations and Dynamical Systems: Overview

description

This video presents an overview lecture for a new series on Differential Equations & Dynamical Systems. Dynamical systems are differential equations that describe any system that changes in time. Applications include fluid dynamics, elasticity and vibrations, weather and climate systems, epidemiology, biomechanics, space mission design, and control theory.
I assume that students have taken some calculus (but might not remember it) and are interested in modeling the real world.
Playlist:
Course Website [will be updated soon]: http://faculty.washington.edu/sbrunton/me564/
@eigensteve on Twitter
eigensteve.com
databookuw.com
%%% CHAPTERS %%%
0:00 Introduction and Overview
3:54 Overview of Topics
10:41 Balancing Classic and Modern Techniques
13:11 What's After Differential Equations?
17:18 Cool Applications
22:38 Chaos
26:32 Sneak Peak of Next Topics

detail

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of things you can understand, of things you can build with math,', 'start': 47.92, 'duration': 7.942}, {'end': 56.602, 'text': 'with equations.', 'start': 55.862, 'duration': 0.74}, {'end': 61.623, 'text': 'And so kind of I feel like this is the next logical step after calculus.', 'start': 57.242, 'duration': 4.381}, {'end': 68.529, 'text': 'So, if you like calculus and if you like the way that you can model the world with how things change in time,', 'start': 62.063, 'duration': 6.466}, {'end': 71.831, 'text': 'this is kind of the next step in that progression.', 'start': 68.529, 'duration': 3.302}, {'end': 72.712, 'text': 'differential equations.', 'start': 71.831, 'duration': 0.881}, {'end': 76.676, 'text': "We're going to use calculus to model how systems change in time.", 'start': 72.752, 'duration': 3.924}, {'end': 86.186, 'text': 'OK, so this this is kind of one of my favorite examples of a chaotic dynamical system or differential equation, called the Lorentz system,', 'start': 77.276, 'duration': 8.91}, {'end': 93.835, 'text': 'which was introduced about a half a century ago, a little bit longer than that, to model chaotic convection in the atmosphere.', 'start': 86.186, 'duration': 7.649}, {'end': 95.977, 'text': "So we'll talk about this a lot later.", 'start': 93.995, 'duration': 1.982}, {'end': 97.859, 'text': 'This course.', 'start': 97.198, 'duration': 0.661}, {'end': 104.204, 'text': 'is meant to be about a seven week series in a university class.', 'start': 98.88, 'duration': 5.324}, {'end': 109.728, 'text': "So that's about 20 hours of lectures that I'm going to be putting together here.", 'start': 105.025, 'duration': 4.703}], 'summary': 'Introduction to differential equations, a 7-week course with 20 hours of lectures.', 'duration': 72.89, 'max_score': 36.838, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u836838.jpg'}, {'end': 189.29, 'src': 'embed', 'start': 129.18, 'weight': 4, 'content': [{'end': 135.284, 'text': "And I'm going to build everything up from scratch into this pretty rich and complex world of differential equations.", 'start': 129.18, 'duration': 6.104}, {'end': 143.75, 'text': "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.", 'start': 136.264, 'duration': 7.486}, {'end': 150.956, 'text': 'what are some of the applications that we can open up and enable with this kind of of analysis of differential equations?', 'start': 143.75, 'duration': 7.206}, {'end': 158.902, 'text': "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.", 'start': 150.956, 'duration': 7.946}, {'end': 161.685, 'text': "you can't understand differential equations without linear algebra,", 'start': 158.902, 'duration': 2.783}, {'end': 167.286, 'text': 'and differential equations are probably the best motivating example to understand linear algebra better.', 'start': 162.305, 'duration': 4.981}, {'end': 172.087, 'text': "So for me, linear algebra is a piece, it's a part of differential equations.", 'start': 167.426, 'duration': 4.661}, {'end': 173.127, 'text': "You can't separate those.", 'start': 172.127, 'duration': 1}, {'end': 175.948, 'text': "So we'll talk about topics, applications.", 'start': 173.667, 'duration': 2.281}, {'end': 178.808, 'text': "I'll tell you kind of where this fits in.", 'start': 177.228, 'duration': 1.58}, {'end': 184.529, 'text': "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.", 'start': 178.848, 'duration': 5.681}, {'end': 189.29, 'text': 'Once you learn these differential equations, what new fields of math are gonna be open up to you.', 'start': 184.669, 'duration': 4.621}], 'summary': 'Building a rich world of differential equations, incorporating linear algebra, and exploring applications and new fields of math.', 'duration': 60.11, 'max_score': 129.18, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8129180.jpg'}], 'start': 9.042, 'title': 'Fundamentals of differential equations', 'summary': 'Introduces a comprehensive understanding of differential equations through a seven-week series with approximately 20 hours of lectures, emphasizing applications such as the lorentz system and chaotic convection in the atmosphere. it also covers the transition from calculus to differential equations, emphasizing the importance of understanding linear algebra alongside, and highlights the applications and new fields of math opened up by learning differential equations.', 'chapters': [{'end': 109.728, 'start': 9.042, 'title': 'Introduction to differential equations', 'summary': 'Introduces a short course on differential equations, a fundamental topic in math and engineering, aiming to provide a comprehensive understanding of how systems change 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the learning process.']}], 'duration': 180.248, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u89042.jpg', 'highlights': ['The course is a part of a larger sequence in the engineering math course, with about 20 hours of lectures planned for a seven-week series in a university class.', "The chapter emphasizes the significance of differential equations in modeling systems' changes in time, building on the concepts of calculus and offering a wide range of possibilities for understanding and constructing mathematical models.", 'The introduction of the Lorentz system, a chaotic dynamical system, demonstrates the practical application of differential equations in modeling chaotic convection in the atmosphere, providing a real-world context for the course content.', 'The speaker expresses enthusiasm and passion for teaching differential equations, emphasizing its importance in understanding and modeling real-world phenomena for engineers, making it a fundamental course in engineering math.', 'The importance of understanding linear algebra alongside differential equations, with the assertion that these two topics should be taught together.', 'Emphasizing the significance of differential equations as the next logical piece after calculus, and identifying the new fields of math that will be opened up by learning differential equations.', 'The intention to build up from scratch into the world of differential equations and the incorporation of linear algebra throughout the learning process.']}, {'end': 575.994, 'segs': [{'end': 221.898, 'src': 'embed', 'start': 189.87, 'weight': 0, 'content': [{'end': 194.114, 'text': "And I'll tell you how to get some of the resources that we're going to use throughout this class.", 'start': 189.87, 'duration': 4.244}, {'end': 200.841, 'text': "So in the video description below, I'm going to put links to all of the code for the entire class.", 'start': 194.535, 'duration': 6.306}, {'end': 203.924, 'text': 'Everything is going to be available in Python and MATLAB.', 'start': 201.481, 'duration': 2.443}, {'end': 206.706, 'text': 'There will be PDF notes for all of the lectures.', 'start': 204.384, 'duration': 2.322}, {'end': 210.87, 'text': 'So every single video is going to have lecture notes with it.', 'start': 206.747, 'duration': 4.123}, {'end': 212.812, 'text': 'There will be a course website.', 'start': 211.511, 'duration': 1.301}, {'end': 221.898, 'text': "I'm planning on writing a short book on this topic, so maybe by the time you watch this for those of you time travelers from the future,", 'start': 213.352, 'duration': 8.546}], 'summary': 'Access python and matlab code, pdf lecture notes, and a course website.', 'duration': 32.028, 'max_score': 189.87, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8189870.jpg'}, {'end': 296.31, 'src': 'embed', 'start': 254.002, 'weight': 5, 'content': [{'end': 259.024, 'text': "but maybe you don't remember it or maybe it wasn't taught in a way that was motivated by applications.", 'start': 254.002, 'duration': 5.022}, {'end': 268.027, 'text': "So I'm gonna start really really simple remind you what the derivative power, the derivative is power rule, chain rule.", 'start': 259.264, 'duration': 8.763}, {'end': 272.501, 'text': "but I'm gonna also, really kind of early on, focus on the Taylor series.", 'start': 268.027, 'duration': 4.474}, {'end': 275.702, 'text': 'so you have some intuition for what this means and how this works.', 'start': 272.501, 'duration': 3.201}, {'end': 281.724, 'text': "Because we're gonna use Taylor series quite a lot in how we understand solutions of differential equations.", 'start': 276.602, 'duration': 5.122}, {'end': 289.707, 'text': "We're actually going to derive exponential functions and sines and cosines as solutions of differential equations using Taylor series.", 'start': 281.744, 'duration': 7.963}, {'end': 292.729, 'text': "So this isn't gonna be very long, but it's very important.", 'start': 290.628, 'duration': 2.101}, {'end': 296.31, 'text': "So if you need to brush up on calculus, definitely don't skip these lectures.", 'start': 292.949, 'duration': 3.361}], 'summary': 'Focus on teaching derivative rules and taylor series for understanding differential equations and functions.', 'duration': 42.308, 'max_score': 254.002, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8254002.jpg'}, {'end': 474.485, 'src': 'embed', 'start': 432.316, 'weight': 1, 'content': [{'end': 436.48, 'text': 'Lots of examples, lots of cool things you can do with systems of ordinary differential equations.', 'start': 432.316, 'duration': 4.164}, {'end': 442.425, 'text': 'You can model pendula, masses on springs, lots of cool systems you can model.', 'start': 436.88, 'duration': 5.545}, {'end': 446.582, 'text': "and i'm, this is where the linear algebra starts to come in.", 'start': 443.9, 'duration': 2.682}, {'end': 450.786, 'text': 'so when we have systems of ordinary differential equations, you know a is a matrix.', 'start': 446.582, 'duration': 4.204}, {'end': 457.531, 'text': "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,", 'start': 450.786, 'duration': 6.745}, {'end': 463.736, 'text': "and specifically, i'm going to derive the eigenvalues and eigenvectors of this a matrix,", 'start': 457.531, 'duration': 6.205}, {'end': 474.485, 'text': 'specifically as a useful set of coordinates or a useful set of quantities that are used to solve the system of ODEs.', 'start': 464.557, 'duration': 9.928}], 'summary': 'Linear algebra is used to derive eigenvalues and eigenvectors for solving systems of odes.', 'duration': 42.169, 'max_score': 432.316, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8432316.jpg'}, {'end': 559.258, 'src': 'embed', 'start': 533.237, 'weight': 4, 'content': [{'end': 540.964, 'text': "So when I talk about the populations of bunnies and wolves, bunnies are prey and wolves are predators, they don't really behave linearly.", 'start': 533.237, 'duration': 7.727}, {'end': 544.108, 'text': 'There are nonlinear interactions that are important to model.', 'start': 541.004, 'duration': 3.104}, {'end': 546.611, 'text': "And so we'll talk about those nonlinear systems.", 'start': 544.729, 'duration': 1.882}, {'end': 554.521, 'text': "That chaotic example, that movie I showed in my first slide, that's an example of the solution of a nonlinear ordinary differential equation.", 'start': 547.332, 'duration': 7.189}, {'end': 559.258, 'text': 'And a lot of my favorite systems really are in this realm here.', 'start': 555.074, 'duration': 4.184}], 'summary': 'Nonlinear interactions between bunny and wolf populations are important to model, with chaotic examples being solutions of nonlinear ordinary differential equations.', 'duration': 26.021, 'max_score': 533.237, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8533237.jpg'}], 'start': 189.87, 'title': 'Engineering math and introduction to odes', 'summary': 'Covers resources for engineering math class and an overview of topics. it also introduces odes, systems of odes, and the importance of eigenvalues and eigenvectors.', 'chapters': [{'end': 345.479, 'start': 189.87, 'title': 'Engineering math: calculus and differential equations', 'summary': 'Covers resources available for the class, including code links, pdf notes, and a planned short book, and provides an overview of the topics to be covered, such as a brief review of calculus and the focus on simple ordinary differential equations.', 'duration': 155.609, 'highlights': ['The chapter provides links to all the code for the entire class, available in Python and MATLAB, along with PDF notes for all lectures, and a planned short book on the topic.', 'An overview of the topics to be covered includes a brief review of calculus, focusing on the derivative power rule, chain rule, and Taylor series, which will be used to understand solutions of differential equations.', 'The chapter also emphasizes the importance of not skipping the lectures on calculus as they are crucial for understanding the subsequent material on differential equations.']}, {'end': 575.994, 'start': 346.099, 'title': 'Introduction to odes and systems', 'summary': 'Covers the simplest ordinary differential equations, progressing to systems of odes, including examples of linear and nonlinear systems, and the importance of eigenvalues and eigenvectors in solving odes.', 'duration': 229.895, 'highlights': ['The chapter starts with the simplest ordinary differential equation and progresses to more complex examples, including systems of ODEs and the importance of eigenvalues and eigenvectors. The chapter begins with the simplest ordinary differential equation and progresses to more complex examples, including systems of ODEs and the importance of eigenvalues and eigenvectors in solving ODEs.', 'The discussion extends to systems of ordinary differential equations, involving multiple variables and matrix systems of differential equations. The discussion extends to systems of ordinary differential equations, involving multiple variables and matrix systems of differential equations.', 'The chapter delves into the study of linear algebra, specifically deriving eigenvalues and eigenvectors as a means to solve systems of ODEs. The chapter delves into the study of linear algebra, specifically deriving eigenvalues and eigenvectors as a means to solve systems of ODEs.', 'The chapter also covers nonlinear differential equations and systems, highlighting their prevalence in real-world scenarios such as predator-prey interactions and chaotic systems. The chapter also covers nonlinear differential equations and systems, highlighting their prevalence in real-world scenarios such as predator-prey interactions and chaotic systems.']}], 'duration': 386.124, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8189870.jpg', 'highlights': ['The chapter provides links to all the code for the entire class, available in Python and MATLAB, along with PDF notes for all lectures, and a planned short book on the topic.', 'The chapter starts with the simplest ordinary differential equation and progresses to more complex examples, including systems of ODEs and the importance of eigenvalues and eigenvectors.', 'The discussion extends to systems of ordinary differential equations, involving multiple variables and matrix systems of differential equations.', 'The chapter delves into the study of linear algebra, specifically deriving eigenvalues and eigenvectors as a means to solve systems of ODEs.', 'The chapter also covers nonlinear differential equations and systems, highlighting their prevalence in real-world scenarios such as predator-prey interactions and chaotic systems.', 'An overview of the topics to be covered includes a brief review of calculus, focusing on the derivative power rule, chain rule, and Taylor series, which will be used to understand solutions of differential equations.', 'The chapter also emphasizes the importance of not skipping the lectures on calculus as they are crucial for understanding the subsequent material on differential equations.']}, {'end': 1030.886, 'segs': [{'end': 641.387, 'src': 'embed', 'start': 577.646, 'weight': 0, 'content': [{'end': 582.029, 'text': "And throughout all of this, I'm going to be pinning this to numerics and computations.", 'start': 577.646, 'duration': 4.383}, {'end': 587.153, 'text': "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.", 'start': 582.149, 'duration': 5.004}, {'end': 596.48, 'text': "we're also going to fire up Jupyter notebooks in Python or MATLAB and we're going to work through how do you actually simulate these differential equations?", 'start': 587.153, 'duration': 9.327}, {'end': 603.685, 'text': 'How can we plot our solutions and confirm that they match our intuition from eigenvalues and eigenvectors?', 'start': 596.52, 'duration': 7.165}, {'end': 610.736, 'text': "Okay so, everything I'm gonna do is gonna be pinned to computations and I'm gonna do everything in Python, everything in MATLAB.", 'start': 603.925, 'duration': 6.811}, {'end': 615.263, 'text': 'all of the codes are gonna be online, freely available and open, okay?', 'start': 610.736, 'duration': 4.527}, {'end': 617.988, 'text': 'So you can kind of pick or choose your language of choice.', 'start': 615.323, 'duration': 2.665}, {'end': 621.827, 'text': "Good, so that's the high level overview.", 'start': 619.124, 'duration': 2.703}, {'end': 628.554, 'text': "This is, again, approximately seven weeks, about 20 hours, a little bit less than 20 hours of content I'm planning.", 'start': 621.847, 'duration': 6.707}, {'end': 634.48, 'text': 'And this fits into the larger framework of engineering mathematics.', 'start': 629.114, 'duration': 5.366}, {'end': 641.387, 'text': 'This is the math you need to know if you wanna do real science and engineering kind of at the next level beyond calculus.', 'start': 634.52, 'duration': 6.867}], 'summary': 'Numerics and computations in python and matlab, 7 weeks, 20 hours, essential for advanced science and engineering.', 'duration': 63.741, 'max_score': 577.646, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8577646.jpg'}, {'end': 754.952, 'src': 'embed', 'start': 725.982, 'weight': 3, 'content': [{'end': 731.488, 'text': 'we build a tremendous amount of intuition for how the world works and how differential equations and dynamical systems,', 'start': 725.982, 'duration': 5.506}, {'end': 737.074, 'text': 'how things change in time from those analytic classic solutions.', 'start': 731.488, 'duration': 5.586}, {'end': 745.563, 'text': "and we're going to carry that intuition with us when we do advanced modern computational methods or even, more recently,", 'start': 737.074, 'duration': 8.489}, {'end': 748.386, 'text': 'data-driven analysis and machine learning methods.', 'start': 745.563, 'duration': 2.823}, {'end': 754.952, 'text': "So it's very important to me that I'm not just telling you about things that were discovered 300 years ago,", 'start': 748.946, 'duration': 6.006}], 'summary': 'Intuition from classic solutions aids modern computational and data-driven analysis.', 'duration': 28.97, 'max_score': 725.982, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8725982.jpg'}, {'end': 908.996, 'src': 'embed', 'start': 878.049, 'weight': 4, 'content': [{'end': 881.171, 'text': "Again, that's after this kind of intro to differential equations.", 'start': 878.049, 'duration': 3.122}, {'end': 884.792, 'text': 'You can then learn complex analysis, vector calculus, and PDEs.', 'start': 881.191, 'duration': 3.601}, {'end': 895.063, 'text': "We're also, in this two-quarter sequence, going to introduce the Fourier and Laplace transform, very closely related to differential equations,", 'start': 886.374, 'duration': 8.689}, {'end': 896.805, 'text': 'especially partial differential equations.', 'start': 895.063, 'duration': 1.742}, {'end': 900.93, 'text': "And I'm not just going to talk about kind of, again, classic analytic equations.", 'start': 897.806, 'duration': 3.124}, {'end': 903.832, 'text': 'you know, hundreds of years old Fourier and Laplace transforms.', 'start': 901.01, 'duration': 2.822}, {'end': 908.996, 'text': "I'm also gonna talk about how you do this in a computer with discrete Fourier transforms and things like that.", 'start': 904.292, 'duration': 4.704}], 'summary': 'Course covers differential equations, complex analysis, vector calculus, pdes, fourier and laplace transforms, including computer applications.', 'duration': 30.947, 'max_score': 878.049, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8878049.jpg'}], 'start': 577.646, 'title': 'Numerical simulations of differential equations and engineering mathematics overview', 'summary': 'Will focus on simulating differential equations using python or matlab and covers a seven-week course with approximately 20 hours of content, emphasizing classic mathematical techniques for advanced computational methods and machine learning models.', 'chapters': [{'end': 617.988, 'start': 577.646, 'title': 'Numerical simulations of differential equations', 'summary': 'Will focus on simulating differential equations using python or matlab, with a strong emphasis on computations and providing freely available codes for practical application.', 'duration': 40.342, 'highlights': ['The focus will be on simulating differential equations using Python or MATLAB, with an emphasis on computations and practical applications.', 'The codes for simulations will be freely available online for practical use.', 'The chapter will provide an opportunity to confirm the solutions using eigenvalues and eigenvectors and visualize the results through plotting.']}, {'end': 1030.886, 'start': 619.124, 'title': 'Engineering mathematics overview', 'summary': 'Covers a seven-week course with approximately 20 hours of content, emphasizing the importance of classic mathematical techniques in building intuition for advanced modern computational methods and machine learning models, ultimately opening up opportunities in complex analysis, vector calculus, partial differential equations, fourier and laplace transforms, data-driven techniques, and control systems.', 'duration': 411.762, 'highlights': ['The course covers a seven-week curriculum with approximately 20 hours of content. The course duration is approximately seven weeks with about 20 hours of content.', 'Emphasis on the importance of classic mathematical techniques in building intuition for advanced modern computational methods and machine learning models. The course emphasizes the importance of classic mathematical techniques in building intuition for advanced modern computational methods and machine learning models.', 'The course opens up opportunities in complex analysis, vector calculus, partial differential equations, Fourier and Laplace transforms, data-driven techniques, and control systems. The course opens up opportunities in complex analysis, vector calculus, partial differential equations, Fourier and Laplace transforms, data-driven techniques, and control systems.']}], 'duration': 453.24, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u8577646.jpg', 'highlights': ['The course covers a seven-week curriculum with approximately 20 hours of content.', 'The focus will be on simulating differential equations using Python or MATLAB, with an emphasis on computations and practical applications.', 'The codes for simulations will be freely available online for practical use.', 'The course emphasizes the importance of classic mathematical techniques in building intuition for advanced modern computational methods and machine learning models.', 'The course opens up opportunities in complex analysis, vector calculus, partial differential equations, Fourier and Laplace transforms, data-driven techniques, and control systems.', 'The chapter will provide an opportunity to confirm the solutions using eigenvalues and eigenvectors and visualize the results through plotting.']}, {'end': 1272.647, 'segs': [{'end': 1075.78, 'src': 'embed', 'start': 1030.886, 'weight': 0, 'content': [{'end': 1039.071, 'text': "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.", 'start': 1030.886, 'duration': 8.185}, {'end': 1046.217, 'text': 'uh, you know, cool applications that you can start studying and understanding once you have a handle on differential equations.', 'start': 1039.071, 'duration': 7.146}, {'end': 1052.582, 'text': 'So, for me, one of my passions and fields of applied interest is fluid mechanics.', 'start': 1046.936, 'duration': 5.646}, {'end': 1060.649, 'text': 'I find fluids really interesting because literally we live and work inside of a fluid and so do all of our machines.', 'start': 1053.123, 'duration': 7.526}, {'end': 1063.632, 'text': 'Your heart is pumping fluid through your veins.', 'start': 1061.19, 'duration': 2.442}, {'end': 1069.156, 'text': 'almost every trillion dollar industry has a working fluid at the heart of it.', 'start': 1065.834, 'duration': 3.322}, {'end': 1075.78, 'text': 'If you think about energy, transportation, defense, health, all of these have fluids.', 'start': 1069.336, 'duration': 6.444}], 'summary': 'Fluid mechanics is vital in trillion-dollar industries like energy, transportation, and health.', 'duration': 44.894, 'max_score': 1030.886, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81030886.jpg'}, {'end': 1132.536, 'src': 'embed', 'start': 1106.146, 'weight': 2, 'content': [{'end': 1110.888, 'text': 'Some other cool examples of differential equations that come up all the time involve mixing.', 'start': 1106.146, 'duration': 4.742}, {'end': 1115.909, 'text': 'So mixing is a really big problem industrially and scientifically.', 'start': 1111.448, 'duration': 4.461}, {'end': 1116.75, 'text': 'How do you?', 'start': 1115.969, 'duration': 0.781}, {'end': 1119.05, 'text': "let's say, you're trying to make not acid.", 'start': 1116.75, 'duration': 2.3}, {'end': 1123.612, 'text': "let's say you're trying to make aspirin, not acid.", 'start': 1119.05, 'duration': 4.562}, {'end': 1126.373, 'text': "maybe you wanna make acid, but you're making aspirin.", 'start': 1123.612, 'duration': 2.761}, {'end': 1132.536, 'text': "in a big chemical vat and so you're adding all these ingredients and you wanna mix them as efficiently as possible.", 'start': 1127.593, 'duration': 4.943}], 'summary': 'Differential equations are crucial in optimizing mixing processes for industrial and scientific applications.', 'duration': 26.39, 'max_score': 1106.146, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81106146.jpg'}, {'end': 1189.342, 'src': 'embed', 'start': 1166.761, 'weight': 3, 'content': [{'end': 1175.347, 'text': 'so this is a jellyfish, and this is one of my cool, one of my favorite examples, where Researchers actually measured the vector field,', 'start': 1166.761, 'duration': 8.586}, {'end': 1182.553, 'text': 'kind of the fluid flow field behind this jellyfish, using, you know, measurement techniques, experimental measurement techniques,', 'start': 1175.347, 'duration': 7.206}, {'end': 1189.342, 'text': 'and then thinking about that fluid vector field or velocity field as a differential equation,', 'start': 1182.553, 'duration': 6.789}], 'summary': 'Researchers measured the fluid flow field behind a jellyfish using experimental techniques.', 'duration': 22.581, 'max_score': 1166.761, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81166761.jpg'}], 'start': 1030.886, 'title': 'Applications and significance of differential equations in fluid mechanics', 'summary': 'Explores practical applications of differential equations in fluid mechanics in various industries and highlights the significance of differential equations in understanding fluid dynamics with examples such as efficient chemical mixing and fluid flow fields behind jellyfish.', 'chapters': [{'end': 1075.78, 'start': 1030.886, 'title': 'Applications of differential equations in fluid mechanics', 'summary': 'Explores the practical applications of differential equations in fluid mechanics, highlighting the relevance of fluids in various industries such as energy, transportation, defense, and health.', 'duration': 44.894, 'highlights': ['Fluid mechanics is crucial in trillion dollar industries like energy, transportation, defense, and health, as they all rely on the behavior of fluids.', 'Understanding differential equations can lead to studying cool applications such as fluid mechanics, which is relevant to our daily lives and various industries.', 'The relevance of fluids is evident as they are integral components in our bodies and machines, making it a field of applied interest with vast real-world implications.']}, {'end': 1272.647, 'start': 1075.86, 'title': 'Fluid dynamics and differential equations', 'summary': 'Explores the significance of differential equations in understanding and solving fluid dynamics, highlighting examples such as the efficient mixing of chemicals and the study of fluid flow fields behind jellyfish through differential equations and test particle integration.', 'duration': 196.787, 'highlights': ['Differential equations govern the operation of wind turbines, automobiles, trains, planes, and mixing systems, and are essential in understanding fluid dynamics.', 'Efficient mixing of chemicals in industrial and scientific processes is governed by a set of differential equations, enabling the optimization of stirring techniques for maximal efficiency.', 'The study of fluid flow fields behind a jellyfish involves using differential equations and test particle integration to understand how particles are advected in the fluid, leading to scientific discoveries in jellyfish feeding behavior and the design of a mechanical jellyfish using tissue from a rodent heart.']}], 'duration': 241.761, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81030886.jpg', 'highlights': ['Fluid mechanics is crucial in trillion dollar industries like energy, transportation, defense, and health.', 'Understanding differential equations can lead to studying cool applications such as fluid mechanics.', 'Efficient mixing of chemicals in industrial and scientific processes is governed by a set of differential equations.', 'The study of fluid flow fields behind a jellyfish involves using differential equations and test particle integration.']}, {'end': 1761.063, 'segs': [{'end': 1342.612, 'src': 'embed', 'start': 1273.468, 'weight': 0, 'content': [{'end': 1276.029, 'text': 'And again, you can understand this is a dynamical system.', 'start': 1273.468, 'duration': 2.561}, {'end': 1281.233, 'text': 'It has a state that varies in time according to rules, F equals ma kinds of rules.', 'start': 1276.07, 'duration': 5.163}, {'end': 1289.398, 'text': 'And so you can write down a differential equation and understand how to design and optimize and predict the motion of this system.', 'start': 1281.333, 'duration': 8.065}, {'end': 1301.033, 'text': 'Other things you can do with differential equations is understand geophysical flows, ocean flows, climate systems.', 'start': 1291.325, 'duration': 9.708}, {'end': 1305.997, 'text': "Things like if there's an oil spill in the Gulf of Mexico, where is that oil going to go?", 'start': 1301.053, 'duration': 4.944}, {'end': 1315.144, 'text': 'Can you predict where things are going to go and have your models match again based on these differential equations and solution techniques to those differential equations?', 'start': 1306.057, 'duration': 9.087}, {'end': 1320.761, 'text': 'This is a cool simulation that I like by Philip Dutrois,', 'start': 1317.06, 'duration': 3.701}, {'end': 1330.044, 'text': "where he applied that same basic analysis to the entire global ocean model to figure out where are regions where mixing occurs and where mixing doesn't occur,", 'start': 1320.761, 'duration': 9.283}, {'end': 1330.725, 'text': 'and things like that.', 'start': 1330.044, 'duration': 0.681}, {'end': 1335.206, 'text': "Again, using basic techniques and differential equations that I'll tell you about in this class.", 'start': 1331.185, 'duration': 4.021}, {'end': 1342.612, 'text': 'And like I mentioned before, for me, fluids is really interesting, but the whole world is not just fluids.', 'start': 1336.989, 'duration': 5.623}], 'summary': 'Understanding dynamical systems using differential equations to predict motion and phenomena like geophysical flows and ocean currents.', 'duration': 69.144, 'max_score': 1273.468, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81273468.jpg'}, {'end': 1435.633, 'src': 'embed', 'start': 1407.73, 'weight': 3, 'content': [{'end': 1413.654, 'text': 'there are these kind of chaotic trajectories that you can leverage for energy efficient space travel,', 'start': 1407.73, 'duration': 5.924}, {'end': 1416.036, 'text': 'which is one of my favorite examples of a dynamical system.', 'start': 1413.654, 'duration': 2.382}, {'end': 1419.419, 'text': 'Again, a lot of our knowledge of dynamical systems.', 'start': 1416.076, 'duration': 3.343}, {'end': 1426.284, 'text': 'you know, if you go back to Newton and Leibniz and kind of the classics, a lot of this was motivated by understanding.', 'start': 1419.419, 'duration': 6.865}, {'end': 1427.865, 'text': 'you know planets in the solar system.', 'start': 1426.284, 'duration': 1.581}, {'end': 1431.688, 'text': 'And a lot of research still goes into this topic.', 'start': 1428.085, 'duration': 3.603}, {'end': 1435.633, 'text': "I'll talk about things like double pendula.", 'start': 1433.732, 'duration': 1.901}], 'summary': 'Chaotic trajectories enable energy-efficient space travel, a key example of dynamical systems.', 'duration': 27.903, 'max_score': 1407.73, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81407730.jpg'}, {'end': 1519.443, 'src': 'embed', 'start': 1494.92, 'weight': 4, 'content': [{'end': 1502.342, 'text': 'And so even like humans walking in bio locomotion, very much a dynamical system governed by differential equations.', 'start': 1494.92, 'duration': 7.422}, {'end': 1509.484, 'text': 'And we can understand how to control those systems and how to analyze those systems, again, using the techniques from this class.', 'start': 1502.982, 'duration': 6.502}, {'end': 1516.661, 'text': 'And once you can control these systems, you can do things like rehabilitation medicine.', 'start': 1512.138, 'duration': 4.523}, {'end': 1519.443, 'text': 'You can also do things like, you know, landing a rocket.', 'start': 1517.101, 'duration': 2.342}], 'summary': 'Study of dynamical systems for controlling and analyzing human locomotion and rocket landing.', 'duration': 24.523, 'max_score': 1494.92, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81494920.jpg'}, {'end': 1605.307, 'src': 'embed', 'start': 1579.221, 'weight': 6, 'content': [{'end': 1583.707, 'text': "You know, please let me know what's confusing, what made sense, what you liked, what you didn't like.", 'start': 1579.221, 'duration': 4.486}, {'end': 1586.751, 'text': 'You know, this is gonna be a living set of lectures.', 'start': 1583.787, 'duration': 2.964}, {'end': 1592.298, 'text': 'Again, all of the resources are gonna be made open, you know, PDFs of the notes and all of the codes.', 'start': 1587.512, 'duration': 4.786}, {'end': 1596.102, 'text': "I'm excited to walk you through all of this.", 'start': 1594.381, 'duration': 1.721}, {'end': 1599.404, 'text': "You know, I'm going to try to release all of these videos pretty soon.", 'start': 1596.122, 'duration': 3.282}, {'end': 1605.307, 'text': "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.", 'start': 1599.444, 'duration': 5.863}], 'summary': 'Open access to living lectures with pdf notes and codes. videos to be released soon.', 'duration': 26.086, 'max_score': 1579.221, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81579221.jpg'}, {'end': 1693.839, 'src': 'embed', 'start': 1661.271, 'weight': 7, 'content': [{'end': 1669.298, 'text': "I'm going to have a whole lecture on this, but I'm really excited to share the intuition about how eigenvalues and eigenvectors,", 'start': 1661.271, 'duration': 8.027}, {'end': 1673.622, 'text': 'I think are best understood when you think about solving differential equations.', 'start': 1669.298, 'duration': 4.324}, {'end': 1679.427, 'text': 'When you want to solve differential equations, eigenvalues and eigenvectors are like your most powerful tools.', 'start': 1674.382, 'duration': 5.045}, {'end': 1684.251, 'text': "That's how we think about and analyze and understand all of the dynamics of these differential equations.", 'start': 1679.427, 'duration': 4.824}, {'end': 1687.394, 'text': 'So this is going to be a killer lecture.', 'start': 1684.291, 'duration': 3.103}, {'end': 1688.074, 'text': 'Another one, you know.', 'start': 1687.394, 'duration': 0.68}, {'end': 1691.817, 'text': "I'm going to show you how you take a fluid flow like this.", 'start': 1688.074, 'duration': 3.743}, {'end': 1693.839, 'text': 'You know how you write down the differential equation.', 'start': 1691.817, 'duration': 2.022}], 'summary': 'Eigenvalues and eigenvectors are crucial for solving differential equations and understanding fluid dynamics.', 'duration': 32.568, 'max_score': 1661.271, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81661271.jpg'}, {'end': 1760.702, 'src': 'heatmap', 'start': 1752.495, 'weight': 1, 'content': [{'end': 1757.74, 'text': "And it's going to open up a huge, huge set of engineering applied math possibilities.", 'start': 1752.495, 'duration': 5.245}, {'end': 1759.121, 'text': "So I'm super excited.", 'start': 1758.22, 'duration': 0.901}, {'end': 1760.702, 'text': "Stay tuned, and I'll see you next time.", 'start': 1759.221, 'duration': 1.481}], 'summary': 'Exciting engineering opportunities ahead.', 'duration': 8.207, 'max_score': 1752.495, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81752495.jpg'}], 'start': 1273.468, 'title': 'Applications of differential equations in various systems', 'summary': 'Explores the application of differential equations in understanding dynamical systems such as geophysical flows, ocean models, and chaotic systems, including predicting oil spill movement, simulating global ocean mixing, and showcasing unpredictable behavior in systems like the solar system and human locomotion. additionally, it provides a preview of upcoming lectures, focusing on eigenvalues, eigenvectors, and numerical integration of fluid flow, aiming to make these topics accessible and exciting for a broader audience.', 'chapters': [{'end': 1342.612, 'start': 1273.468, 'title': 'Differential equations in geophysical systems', 'summary': 'Discusses the application of differential equations in understanding dynamical systems, such as geophysical flows and ocean models, including predicting the movement of oil spills, and simulating global ocean mixing.', 'duration': 69.144, 'highlights': ['The application of differential equations in understanding and predicting the motion of dynamical systems, including geophysical flows and climate systems, offers valuable insights and optimization techniques.', 'The use of differential equations and solution techniques to predict the movement of oil spills in the Gulf of Mexico demonstrates the practical application of this mathematical framework in real-world scenarios.', 'The simulation by Philip Dutrois, applying differential equations to the global ocean model, provides insights into regions of mixing and non-mixing, showcasing the versatility of these techniques in studying geophysical systems.']}, {'end': 1558.745, 'start': 1342.712, 'title': 'Dynamics and chaos in systems', 'summary': 'Discusses the concept of differential equations as rules for system changes in time, including examples of chaotic and unpredictable behavior in various systems like the solar system, double pendula, and human locomotion.', 'duration': 216.033, 'highlights': ['The solar system and planets have chaotic motion, with chaotic trajectories that can be leveraged for energy efficient space travel. The planets in the solar system exhibit chaotic motion, with chaotic trajectories that can be utilized for energy efficient space travel, providing an interesting example of a dynamical system.', 'Human locomotion and bio locomotion are governed by differential equations, and understanding how to control these systems can have applications in rehabilitation medicine and rocket landing. Human locomotion and bio locomotion are dynamical systems governed by differential equations, and knowledge of controlling these systems can be applied in fields such as rehabilitation medicine and rocket landing.', 'Understanding differential equations enables control design to harness dynamics and overcome chaos, allowing stabilization of desired trajectories. Comprehension of differential equations facilitates control design to utilize dynamics and mitigate chaos, enabling stabilization of desired trajectories.']}, {'end': 1761.063, 'start': 1560.515, 'title': 'Exciting lecture teasers', 'summary': 'Gives a preview of upcoming lectures, including open resources and a focus on eigenvalues, eigenvectors, and numerical integration of fluid flow, aiming to make these topics accessible and exciting for a broader audience.', 'duration': 200.548, 'highlights': ['The chapter emphasizes the open availability of resources, including PDF notes and codes, for the upcoming lectures, aiming to make the material accessible to a broader audience.', 'The lecturer expresses enthusiasm for teaching, highlighting a particular focus on eigenvalues and eigenvectors as powerful tools for solving differential equations, which has been a passion for over 10 years.', 'A teaser is given about upcoming lectures, specifically focusing on understanding eigenvalues and eigenvectors in the context of solving differential equations, aiming to provide intuition and clear explanation for these concepts.', 'The lecturer previews the topic of numerically integrating fluid flow, emphasizing the practical applications and outcomes of the study, such as extracting useful quantities and creating visual representations of the flow field, indicating the potential for real-world engineering and applied math possibilities.']}], 'duration': 487.595, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/9fQkLQZe3u8/pics/9fQkLQZe3u81273468.jpg', 'highlights': ['The use of differential equations and solution techniques to predict the movement of oil spills in the Gulf of Mexico demonstrates the practical application of this mathematical framework in real-world scenarios.', 'The simulation by Philip Dutrois, applying differential equations to the global ocean model, provides insights into regions of mixing and non-mixing, showcasing the versatility of these techniques in studying geophysical systems.', 'The application of differential equations in understanding and predicting the motion of dynamical systems, including geophysical flows and climate systems, offers valuable insights and optimization techniques.', 'The solar system and planets have chaotic motion, with chaotic trajectories that can be leveraged for energy efficient space travel.', 'Human locomotion and bio locomotion are governed by differential equations, and understanding how to control these systems can have applications in rehabilitation medicine and rocket landing.', 'Understanding differential equations enables control design to harness dynamics and overcome chaos, allowing stabilization of desired trajectories.', 'The chapter emphasizes the open availability of resources, including PDF notes and codes, for the upcoming lectures, aiming to make the material accessible to a broader audience.', 'The lecturer expresses enthusiasm for teaching, highlighting a particular focus on eigenvalues and eigenvectors as powerful tools for solving differential equations, which has been a passion for over 10 years.', 'The lecturer previews the topic of numerically integrating fluid flow, emphasizing the practical applications and outcomes of the study, such as extracting useful quantities and creating visual representations of the flow field, indicating the potential for real-world engineering and applied math possibilities.', 'A teaser is given about upcoming lectures, specifically focusing on understanding eigenvalues and eigenvectors in the context of solving differential equations, aiming to provide intuition and clear explanation for these concepts.']}], 'highlights': ['The course is a part of a larger sequence in the engineering math course, with about 20 hours of lectures planned for a seven-week series in a university class.', "The chapter emphasizes the significance of differential equations in modeling systems' changes in time, building on the concepts of calculus and offering a wide range of possibilities for understanding and constructing mathematical models.", 'The introduction of the Lorentz system, a chaotic dynamical system, demonstrates the practical application of differential equations in modeling chaotic convection in the atmosphere, providing a real-world context for the course content.', 'The speaker expresses enthusiasm and passion for teaching differential equations, emphasizing its importance in understanding and modeling real-world phenomena for engineers, making it a fundamental course in engineering math.', 'The importance of understanding linear algebra alongside differential equations, with the assertion that these two topics should be taught together.', 'Emphasizing the significance of differential equations as the next logical piece after calculus, and identifying the new fields of math that will be opened up by learning differential equations.', 'The intention to build up from scratch into the world of differential equations and the incorporation of linear algebra throughout the learning process.', 'The chapter provides links to all the code for the entire class, available in Python and MATLAB, along with PDF notes for all lectures, and a planned short book on the topic.', 'The chapter starts with the simplest ordinary differential equation and progresses to more complex examples, including systems of ODEs and the importance of eigenvalues and eigenvectors.', 'The discussion extends to systems of ordinary differential equations, involving multiple variables and matrix systems of differential equations.', 'The chapter delves into the study of linear algebra, specifically deriving eigenvalues and eigenvectors as a means to solve systems of ODEs.', 'The chapter also covers nonlinear differential equations and systems, highlighting their prevalence in real-world scenarios such as predator-prey interactions and chaotic systems.', 'An overview of the topics to be covered includes a brief review of calculus, focusing on the derivative power rule, chain rule, and Taylor series, which will be used to understand solutions of differential equations.', 'The chapter also emphasizes the importance of not skipping the lectures on calculus as they are crucial for understanding the subsequent material on differential equations.', 'The course covers a seven-week curriculum with approximately 20 hours of content.', 'The focus will be on simulating differential equations using Python or MATLAB, with an emphasis on computations and practical applications.', 'The codes for simulations will be freely available online for practical use.', 'The course emphasizes the importance of classic mathematical techniques in building intuition for advanced modern computational methods and machine learning models.', 'The course opens up opportunities in complex analysis, vector calculus, partial differential equations, Fourier and Laplace transforms, data-driven techniques, and control systems.', 'The chapter will provide an opportunity to confirm the solutions using eigenvalues and eigenvectors and visualize the results through plotting.', 'Fluid mechanics is crucial in trillion dollar industries like energy, transportation, defense, and health.', 'Understanding differential equations can lead to studying cool applications such as fluid mechanics.', 'Efficient mixing of chemicals in industrial and scientific processes is governed by a set of differential equations.', 'The study of fluid flow fields behind a jellyfish involves using differential equations and test particle integration.', 'The use of differential equations and solution techniques to predict the movement of oil spills in the Gulf of Mexico demonstrates the practical application of this mathematical framework in real-world scenarios.', 'The simulation by Philip Dutrois, applying differential equations to the global ocean model, provides insights into regions of mixing and non-mixing, showcasing the versatility of these techniques in studying geophysical systems.', 'The application of differential equations in understanding and predicting the motion of dynamical systems, including geophysical flows and climate systems, offers valuable insights and optimization techniques.', 'The solar system and planets have chaotic motion, with chaotic trajectories that can be leveraged for energy efficient space travel.', 'Human locomotion and bio locomotion are governed by differential equations, and understanding how to control these systems can have applications in rehabilitation medicine and rocket landing.', 'Understanding differential equations enables control design to harness dynamics and overcome chaos, allowing stabilization of desired trajectories.', 'The chapter emphasizes the open availability of resources, including PDF notes and codes, for the upcoming lectures, aiming to make the material accessible to a broader audience.', 'The lecturer expresses enthusiasm for teaching, highlighting a particular focus on eigenvalues and eigenvectors as powerful tools for solving differential equations, which has been a passion for over 10 years.', 'The lecturer previews the topic of numerically integrating fluid flow, emphasizing the practical applications and outcomes of the study, such as extracting useful quantities and creating visual representations of the flow field, indicating the potential for real-world engineering and applied math possibilities.', 'A teaser is given about upcoming lectures, specifically focusing on understanding eigenvalues and eigenvectors in the context of solving differential equations, aiming to provide intuition and clear explanation for these concepts.']}