title

But what is a partial differential equation? | DE2

description

The heat equation, as an introductory PDE.
Strogatz's new book: https://amzn.to/3bcnyw0
Special thanks to these supporters: http://3b1b.co/de2thanks
An equally valuable form of support is to simply share some of the videos.
Timestamps:
0:00 - Introduction
3:29 - Partial derivatives
6:52 - Building the heat equation
13:18 - ODEs vs PDEs
14:29 - The laplacian
16:04 - Book recommendation
Typo corrections:
- 1:33 - it should be “Black-Scholes”
- 16:21 - it should read "scratch an itch".
If anyone asks, I purposefully leave at least one typo in each video, like a Navajo rug with a deliberate imperfection as an artistic statement about the nature of life ;)
And to continue my unabashed Strogatz fanboyism, I should also mention that his textbook on nonlinear dynamics and chaos was also a meaningful motivator to do this series, as you'll hopefully see with the topics we build to.
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Animations made using manim, a scrappy open source python library. https://github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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detail

{'title': 'But what is a partial differential equation? | DE2', 'heatmap': [{'end': 437.647, 'start': 413.416, 'weight': 0.951}, {'end': 707.816, 'start': 676.535, 'weight': 0.711}, {'end': 1023.322, 'start': 1003.849, 'weight': 0.84}], 'summary': 'Explores the heat equation, fourier series, and partial derivatives for analyzing temperature distribution in a one-dimensional rod over time, emphasizing practical applications and the concept of partial differential equations in modeling temperature changes.', 'chapters': [{'end': 200.076, 'segs': [{'end': 44.098, 'src': 'embed', 'start': 4.17, 'weight': 1, 'content': [{'end': 11.655, 'text': 'After seeing how we think about ordinary differential equations in Chapter 1, we turn now to an example of a partial differential equation,', 'start': 4.17, 'duration': 7.485}, {'end': 12.516, 'text': 'the heat equation.', 'start': 11.655, 'duration': 0.861}, {'end': 14.277, 'text': 'To set things up.', 'start': 13.376, 'duration': 0.901}, {'end': 20.401, 'text': 'imagine you have some object like a piece of metal and you know how the heat is distributed across it at any one moment.', 'start': 14.277, 'duration': 6.124}, {'end': 23.863, 'text': "that is what's the temperature of every individual point along this plate.", 'start': 20.401, 'duration': 3.462}, {'end': 30.788, 'text': 'The question is how will this distribution change over time as the heat flows from warmer spots to cooler ones?', 'start': 24.644, 'duration': 6.144}, {'end': 38.314, 'text': 'The image on the left shows the temperature of an example plate using color, with the graph of that temperature being shown on the right.', 'start': 31.67, 'duration': 6.644}, {'end': 44.098, 'text': "To take a concrete one-dimensional example, let's say, you have two different rods at different temperatures,", 'start': 39.155, 'duration': 4.943}], 'summary': 'Introduction to the heat equation and its application to the distribution of temperature over time in a one-dimensional example.', 'duration': 39.928, 'max_score': 4.17, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz84170.jpg'}, {'end': 118.972, 'src': 'embed', 'start': 80.183, 'weight': 2, 'content': [{'end': 83.425, 'text': "And don't worry, we'll learn how to read the equations you're seeing now in just a minute.", 'start': 80.183, 'duration': 3.242}, {'end': 89.674, 'text': 'Variations of the heat equation show up in many other parts of math and physics, like Brownian motion,', 'start': 84.406, 'duration': 5.268}, {'end': 93.1, 'text': 'the Black-Scholes equations from finance and all sorts of diffusion.', 'start': 89.674, 'duration': 3.426}, {'end': 97.266, 'text': 'So there are many dividends to be had from a deep understanding of this one setup.', 'start': 93.34, 'duration': 3.926}, {'end': 101.067, 'text': 'In the last video, we looked at ways of building understanding,', 'start': 98.246, 'duration': 2.821}, {'end': 106.048, 'text': 'while acknowledging the truth that most differential equations are simply too difficult to actually solve.', 'start': 101.067, 'duration': 4.981}, {'end': 114.591, 'text': 'And indeed, PDEs tend to be even harder than ODEs, largely because they involve modeling infinitely many values changing in concert.', 'start': 106.729, 'duration': 7.862}, {'end': 118.972, 'text': 'But our main character for today is an equation that we actually can solve.', 'start': 115.171, 'duration': 3.801}], 'summary': 'Learning to read heat equations, used in math and physics, with applications in finance and diffusion. some pdes can be solved.', 'duration': 38.789, 'max_score': 80.183, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz880183.jpg'}, {'end': 200.076, 'src': 'embed', 'start': 176.226, 'weight': 0, 'content': [{'end': 182.431, 'text': "What you're seeing now uses only 100 circles, and I think you'd agree that the deviations from the real shape are negligible.", 'start': 176.226, 'duration': 6.205}, {'end': 188.224, 'text': "What's mind-blowing is that, just by tweaking the initial size and angle of each vector,", 'start': 183.339, 'duration': 4.885}, {'end': 191.267, 'text': 'that gives you enough control to approximate any curve that you want.', 'start': 188.224, 'duration': 3.043}, {'end': 198.034, 'text': 'At first, this might seem like an idle curiosity.', 'start': 195.492, 'duration': 2.542}, {'end': 200.076, 'text': 'A neat art project, but little more.', 'start': 198.535, 'duration': 1.541}], 'summary': 'Using 100 circles, control vector angle and size to approximate any curve.', 'duration': 23.85, 'max_score': 176.226, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8176226.jpg'}], 'start': 4.17, 'title': 'Heat equation and fourier series', 'summary': 'Introduces the heat equation illustrating temperature distribution changes over time and covers the concept of fourier series with practical applications in approximating arbitrary shapes.', 'chapters': [{'end': 63.611, 'start': 4.17, 'title': 'Heat equation and temperature distribution', 'summary': 'Introduces the heat equation as an example of a partial differential equation, illustrating how the temperature distribution changes over time as heat flows from warmer to cooler spots using the example of two rods at different temperatures.', 'duration': 59.441, 'highlights': ['The heat equation is introduced as an example of a partial differential equation, showcasing the change in temperature distribution over time as heat moves from warmer areas to cooler ones.', 'Illustration of temperature distribution using an example of two rods at different temperatures and how the temperature equalizes over time when they are brought into contact.', 'Explanation of how the temperature changes over time as the heat flows from the hot rod to the cooler one, showcasing the dynamics of temperature distribution.']}, {'end': 200.076, 'start': 63.611, 'title': 'Understanding heat equation and fourier series', 'summary': 'Covers the concept of heat equation and its applications in various fields, the challenges of solving differential equations, and an introduction to fourier series, which has practical applications in approximating arbitrary shapes using rotating vectors.', 'duration': 136.465, 'highlights': ['The animation demonstrates how multiple rotating vectors, each rotating at a constant integer frequency, can trace out an arbitrary shape, with 100 circles showing negligible deviations from the real shape, highlighting the practical application of Fourier series in approximating curves.', 'The chapter discusses the prevalence of variations of the heat equation in math and physics, including Brownian motion, the Black-Scholes equations from finance, and diffusion, showcasing its wide-ranging applications.', 'It emphasizes the difficulty in solving most differential equations and highlights the complexity of partial differential equations (PDEs) due to modeling infinitely many values changing in concert.']}], 'duration': 195.906, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz84170.jpg', 'highlights': ['The animation demonstrates how multiple rotating vectors can trace out an arbitrary shape, with 100 circles showing negligible deviations from the real shape.', 'The heat equation is introduced as an example of a partial differential equation, showcasing the change in temperature distribution over time as heat moves from warmer areas to cooler ones.', 'The chapter discusses the prevalence of variations of the heat equation in math and physics, including Brownian motion, the Black-Scholes equations from finance, and diffusion, showcasing its wide-ranging applications.', 'Explanation of how the temperature changes over time as the heat flows from the hot rod to the cooler one, showcasing the dynamics of temperature distribution.', 'Illustration of temperature distribution using an example of two rods at different temperatures and how the temperature equalizes over time when they are brought into contact.', 'It emphasizes the difficulty in solving most differential equations and highlights the complexity of partial differential equations (PDEs) due to modeling infinitely many values changing in concert.']}, {'end': 498.234, 'segs': [{'end': 252.198, 'src': 'embed', 'start': 200.797, 'weight': 1, 'content': [{'end': 206.223, 'text': 'In fact, the math that makes this possible is the same as the math describing the physics of heat flow.', 'start': 200.797, 'duration': 5.426}, {'end': 211.572, 'text': "But we're getting ahead of ourselves.", 'start': 210.151, 'duration': 1.421}, {'end': 214.494, 'text': 'Step one is simply to build up the heat equation.', 'start': 211.952, 'duration': 2.542}, {'end': 219.818, 'text': "And for that, let's start by being clear about what the function we're analyzing is exactly.", 'start': 214.894, 'duration': 4.924}, {'end': 224.961, 'text': "We have a rod in one dimension, and we're thinking of it as sitting on an x-axis.", 'start': 220.598, 'duration': 4.363}, {'end': 231.486, 'text': 'So each point of that rod is labeled with a unique number, x.', 'start': 225.362, 'duration': 6.124}, {'end': 236.95, 'text': 'The temperature is some function of that position, T shown here as a graph above it.', 'start': 231.486, 'duration': 5.464}, {'end': 244.736, 'text': 'But really, since the value changes over time, we should think of this function as having one more input, t, for time.', 'start': 238.834, 'duration': 5.902}, {'end': 252.198, 'text': 'You could, if you wanted, think of this input space as really being two-dimensional, representing space and time together,', 'start': 245.696, 'duration': 6.502}], 'summary': 'Explaining the math behind heat flow and building the heat equation for a one-dimensional rod with temperature as a function of position and time.', 'duration': 51.401, 'max_score': 200.797, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8200797.jpg'}, {'end': 313.425, 'src': 'embed', 'start': 276.356, 'weight': 3, 'content': [{'end': 278.698, 'text': "Be mindful, when you're studying equations like these,", 'start': 276.356, 'duration': 2.342}, {'end': 286.045, 'text': "of whether time is being represented with its own axis or if it's being represented with literal changes over time, say in an animation.", 'start': 278.698, 'duration': 7.347}, {'end': 294.492, 'text': 'Last chapter we looked at some systems where just a handful of numbers changed over time, like the angle and angular velocity of a pendulum,', 'start': 287.066, 'duration': 7.426}, {'end': 297.114, 'text': 'describing that change in the language of derivatives.', 'start': 294.492, 'duration': 2.622}, {'end': 303.118, 'text': 'But when we have an entire function changing with time, the mathematical tools become slightly more intricate.', 'start': 297.654, 'duration': 5.464}, {'end': 311.144, 'text': "Because we're thinking of this temperature function with multiple dimensions to its input space, in this case one for space and one for time.", 'start': 303.578, 'duration': 7.566}, {'end': 313.425, 'text': 'there are multiple different rates of change at play.', 'start': 311.144, 'duration': 2.281}], 'summary': 'Studying equations representing time, space, and change in math and physics.', 'duration': 37.069, 'max_score': 276.356, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8276356.jpg'}, {'end': 401.782, 'src': 'embed', 'start': 370.904, 'weight': 7, 'content': [{'end': 373.928, 'text': 'One is a small change to temperature after a small change in time.', 'start': 370.904, 'duration': 3.024}, {'end': 379.214, 'text': 'The other is a small change to temperature after a small step in space.', 'start': 375.329, 'duration': 3.885}, {'end': 385.008, 'text': 'To reiterate a point I made in the calculus series,', 'start': 382.506, 'duration': 2.502}, {'end': 393.835, 'text': "I do think it's healthy to initially read derivatives like this as a literal ratio between a small change to the function's output and the small change to the input that caused it.", 'start': 385.008, 'duration': 8.827}, {'end': 401.782, 'text': 'Just keep in mind that what this notation is meant to encode is the limit of that ratio for smaller and smaller nudges to the input,', 'start': 394.296, 'duration': 7.486}], 'summary': 'Derivatives represent small changes in temperature over time and space, emphasizing the concept as a literal ratio between function output and input changes.', 'duration': 30.878, 'max_score': 370.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8370904.jpg'}, {'end': 478.201, 'src': 'heatmap', 'start': 413.416, 'weight': 0, 'content': [{'end': 416.577, 'text': 'The heat equation is written in terms of these partial derivatives.', 'start': 413.416, 'duration': 3.161}, {'end': 423.14, 'text': 'It tells us that the way this function changes with respect to time depends on how it changes with respect to space.', 'start': 417.018, 'duration': 6.122}, {'end': 429.383, 'text': "More specifically, it's proportional to the second partial derivative with respect to x.", 'start': 423.601, 'duration': 5.782}, {'end': 430.063, 'text': 'At a high level.', 'start': 429.383, 'duration': 0.68}, {'end': 437.647, 'text': 'the intuition is that at points where the temperature distribution curves, it tends to change more quickly in the direction of that curvature.', 'start': 430.063, 'duration': 7.584}, {'end': 443.311, 'text': 'Since a rule like this is written using partial derivatives, we call it a partial differential equation.', 'start': 438.567, 'duration': 4.744}, {'end': 451.299, 'text': 'This has the funny result that to an outsider, the name sounds like a tamer version of ordinary differential equations, when, quite to the contrary,', 'start': 443.972, 'duration': 7.327}, {'end': 456.643, 'text': 'partial differential equations tend to tell a much richer story than ODEs and are much harder to solve in general.', 'start': 451.299, 'duration': 5.344}, {'end': 463.408, 'text': 'The general heat equation applies to bodies in any number of dimensions, which would mean more inputs to our temperature function.', 'start': 457.364, 'duration': 6.044}, {'end': 467.57, 'text': "But it'll be easiest for us to stay focused on the one-dimensional case of a rod.", 'start': 463.868, 'duration': 3.702}, {'end': 473.674, 'text': 'As it is, graphing this in a way which gives time its own axis already pushes our visuals into the third dimension.', 'start': 468.171, 'duration': 5.503}, {'end': 478.201, 'text': 'So I threw out this equation, but where does this come from??', 'start': 475.359, 'duration': 2.842}], 'summary': 'The heat equation describes changes in temperature with respect to time and space, expressed using partial derivatives, and is more complex than ordinary differential equations.', 'duration': 61.183, 'max_score': 413.416, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8413416.jpg'}], 'start': 200.797, 'title': 'Heat equation and partial derivatives', 'summary': 'Covers the process of building the heat equation to analyze temperature distribution of a one-dimensional rod over time, as well as the representation of functions with multiple dimensions and the calculation of partial derivatives, emphasizing different rates of change with respect to space and time. it also explains the concept of partial derivatives in the context of the heat equation and its application to the change in temperature with respect to time and space.', 'chapters': [{'end': 276.036, 'start': 200.797, 'title': 'Heat equation and temperature analysis', 'summary': 'Explains the process of building the heat equation to analyze the temperature of a one-dimensional rod over time, using the function of position and time to graph the temperature distribution.', 'duration': 75.239, 'highlights': ['The temperature of a one-dimensional rod is analyzed using a function of position and time, which can be graphed as a surface representing space and time together, or simply as a graph of temperature changing with time.', 'The process involves building up the heat equation to describe the physics of heat flow, using math similar to that used in describing heat flow physics.']}, {'end': 370.284, 'start': 276.356, 'title': 'Multidimensional rates of change', 'summary': 'Explores the intricacies of representing functions with multiple dimensions and the calculation of partial derivatives, emphasizing the different rates of change with respect to space and time.', 'duration': 93.928, 'highlights': ['The chapter explains the complexities of representing functions with multiple dimensions, particularly when an entire function changes with time, involving multiple rates of change.', 'It discusses the concept of partial derivatives, emphasizing the different rates of change with respect to space and time, and the notation changes involved.', 'The chapter also highlights the calculation of rates of change with respect to x and time, illustrating the intricacies involved in understanding how a temperature function changes over time.']}, {'end': 498.234, 'start': 370.904, 'title': 'Understanding heat equation and partial derivatives', 'summary': 'Explains the concept of partial derivatives in the context of the heat equation, highlighting its application to the change in temperature with respect to time and space, and the complexity and richness of partial differential equations compared to ordinary differential equations.', 'duration': 127.33, 'highlights': ['The heat equation explains how the function changes with respect to time and space, with specific emphasis on the second partial derivative with respect to x, demonstrating the relationship between temperature distribution and curvature.', 'Partial differential equations tend to tell a much richer story than ordinary differential equations and are much harder to solve in general, presenting a more complex and challenging aspect of mathematical modeling.', 'The general heat equation applies to bodies in any number of dimensions, but the chapter focuses on the one-dimensional case of a rod, demonstrating the multidimensional application of the equation.', "The concept of partial derivatives is initially explained as a literal ratio between a small change in the function's output and the small change in the input, providing a foundational understanding of the derivative concept."]}], 'duration': 297.437, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8200797.jpg', 'highlights': ['The heat equation explains how the function changes with respect to time and space, with specific emphasis on the second partial derivative with respect to x, demonstrating the relationship between temperature distribution and curvature.', 'The temperature of a one-dimensional rod is analyzed using a function of position and time, which can be graphed as a surface representing space and time together, or simply as a graph of temperature changing with time.', 'The process involves building up the heat equation to describe the physics of heat flow, using math similar to that used in describing heat flow physics.', 'The chapter explains the complexities of representing functions with multiple dimensions, particularly when an entire function changes with time, involving multiple rates of change.', 'Partial differential equations tend to tell a much richer story than ordinary differential equations and are much harder to solve in general, presenting a more complex and challenging aspect of mathematical modeling.', 'The chapter also highlights the calculation of rates of change with respect to x and time, illustrating the intricacies involved in understanding how a temperature function changes over time.', 'The general heat equation applies to bodies in any number of dimensions, but the chapter focuses on the one-dimensional case of a rod, demonstrating the multidimensional application of the equation.', "The concept of partial derivatives is initially explained as a literal ratio between a small change in the function's output and the small change in the input, providing a foundational understanding of the derivative concept.", 'It discusses the concept of partial derivatives, emphasizing the different rates of change with respect to space and time, and the notation changes involved.']}, {'end': 1036.733, 'segs': [{'end': 555.345, 'src': 'embed', 'start': 525.433, 'weight': 2, 'content': [{'end': 528.875, 'text': 'When this difference is greater than zero, T2 will tend to heat up.', 'start': 525.433, 'duration': 3.442}, {'end': 532.818, 'text': 'And the bigger the difference, the faster it heats up.', 'start': 530.636, 'duration': 2.182}, {'end': 541.125, 'text': "Likewise, if it's negative, T2 will tend to cool down, at a rate proportional to that difference.", 'start': 535.959, 'duration': 5.166}, {'end': 544.122, 'text': 'More formally,', 'start': 543.362, 'duration': 0.76}, {'end': 551.964, 'text': 'we write that the derivative of T2 with respect to time is proportional to the difference between this average value of its neighbors and its own value.', 'start': 544.122, 'duration': 7.842}, {'end': 555.345, 'text': 'Alpha here is simply a proportionality constant.', 'start': 552.724, 'duration': 2.621}], 'summary': 'T2 heats up when difference > 0, cools down when < 0.', 'duration': 29.912, 'max_score': 525.433, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8525433.jpg'}, {'end': 665.286, 'src': 'embed', 'start': 617.102, 'weight': 4, 'content': [{'end': 619.324, 'text': 'Here, take a moment to gut check that this makes sense.', 'start': 617.102, 'duration': 2.222}, {'end': 627.63, 'text': 'If those two differences are the same, then the average of T1 and T3 is the same as T2, so T2 will not tend to change.', 'start': 619.944, 'duration': 7.686}, {'end': 635.196, 'text': 'If delta T2 is bigger than delta T1, meaning the difference of the differences is positive.', 'start': 628.551, 'duration': 6.645}, {'end': 641.06, 'text': 'notice how the average of T1 and T3 is bigger than T2, so T2 tends to increase.', 'start': 635.196, 'duration': 5.864}, {'end': 650.537, 'text': 'And on the flip side, if the difference of the differences is negative, which means delta T2 is smaller than delta T1,', 'start': 642.492, 'duration': 8.045}, {'end': 653.899, 'text': 'it corresponds to an average of these neighbors being less than T2..', 'start': 650.537, 'duration': 3.362}, {'end': 661.483, 'text': 'We could be especially compact with our notation and write this whole term, the difference between the differences, as delta delta T1.', 'start': 654.659, 'duration': 6.824}, {'end': 665.286, 'text': 'This is known in the lingo as a second difference.', 'start': 662.984, 'duration': 2.302}], 'summary': 'Comparing differences in t1, t2, and t3 to predict tendency for change.', 'duration': 48.184, 'max_score': 617.102, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8617102.jpg'}, {'end': 707.816, 'src': 'heatmap', 'start': 676.535, 'weight': 0.711, 'content': [{'end': 681.579, 'text': "And that factor doesn't really matter, because either way we're writing this equation in terms of some proportionality constant.", 'start': 676.535, 'duration': 5.044}, {'end': 688.004, 'text': 'The upshot is that the rate of change for the temperature of a point is proportional to the second difference around it.', 'start': 682.259, 'duration': 5.745}, {'end': 693.029, 'text': 'As we go from this finite context to the infinite continuous case.', 'start': 689.082, 'duration': 3.947}, {'end': 696.035, 'text': 'the analog of a second difference is the second derivative.', 'start': 693.029, 'duration': 3.006}, {'end': 703.453, 'text': 'Instead of looking at the difference between the temperature values at points some fixed distance apart,', 'start': 698.57, 'duration': 4.883}, {'end': 707.816, 'text': 'you instead consider what happens as you shrink the size of that step toward zero.', 'start': 703.453, 'duration': 4.363}], 'summary': 'Rate of temperature change proportional to second derivative in continuous case.', 'duration': 31.281, 'max_score': 676.535, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8676535.jpg'}, {'end': 787.337, 'src': 'embed', 'start': 760.619, 'weight': 1, 'content': [{'end': 768.486, 'text': 'Similarly, that slope decreases at points where the graph curves downwards, where the rate of change of this rate of change is negative.', 'start': 760.619, 'duration': 7.867}, {'end': 773.414, 'text': 'Tuck that away as a meaningful intuition for problems well beyond the heat equation.', 'start': 769.793, 'duration': 3.621}, {'end': 778.275, 'text': 'Second derivatives give a measure of how a value compares to the average of its neighbors.', 'start': 773.814, 'duration': 4.461}, {'end': 782.456, 'text': 'Hopefully that gives some satisfying added color to the equation.', 'start': 779.335, 'duration': 3.121}, {'end': 787.337, 'text': "It's already pretty intuitive when you read it as saying that curved points tend to flatten out,", 'start': 782.896, 'duration': 4.441}], 'summary': 'Second derivatives measure change in curved points, providing intuition for problems beyond the heat equation.', 'duration': 26.718, 'max_score': 760.619, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8760619.jpg'}, {'end': 931.09, 'src': 'embed', 'start': 903.677, 'weight': 0, 'content': [{'end': 908.921, 'text': 'This Laplacian can still be thought of as measuring how different is a point from the average of its neighbors.', 'start': 903.677, 'duration': 5.244}, {'end': 913.264, 'text': "But now, these neighbors aren't just left and right of it, they're all around.", 'start': 909.541, 'duration': 3.723}, {'end': 920.484, 'text': 'For the curious among you, I did a couple of videos during my time at Khan Academy on this operator if you want to go check them out.', 'start': 915.441, 'duration': 5.043}, {'end': 926.047, 'text': "For those of you with some multivariable calculus under your belt, it's nice to think about as the divergence of the gradient.", 'start': 920.944, 'duration': 5.103}, {'end': 928.148, 'text': "But you don't have to worry about that.", 'start': 926.708, 'duration': 1.44}, {'end': 931.09, 'text': "For our purposes, let's stay focused on the one-dimensional case.", 'start': 928.309, 'duration': 2.781}], 'summary': 'Laplacian measures point difference from neighbors, applicable in one-dimensional case.', 'duration': 27.413, 'max_score': 903.677, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8903677.jpg'}, {'end': 1024.623, 'src': 'heatmap', 'start': 994.284, 'weight': 3, 'content': [{'end': 997.826, 'text': 'When more people love math, the potential audience base for these videos gets bigger.', 'start': 994.284, 'duration': 3.542}, {'end': 1003.129, 'text': "And frankly, there are few better ways to get people loving the subject than to expose them to Strogatz's writing.", 'start': 998.346, 'duration': 4.783}, {'end': 1010.032, 'text': 'So if you have friends who you know who you think would enjoy the ideas of calculus, but maybe have been a bit intimidated by math in the past,', 'start': 1003.849, 'duration': 6.183}, {'end': 1015.355, 'text': 'this book does a really outstanding job, communicating the heart of the subject both substantively and accessibly.', 'start': 1010.032, 'duration': 5.323}, {'end': 1023.322, 'text': "Its main theme is the idea of constructing solutions to complex real-world problems from simple, idealized building blocks, which, as you'll see,", 'start': 1015.735, 'duration': 7.587}, {'end': 1024.623, 'text': 'is exactly what Fourier did.', 'start': 1023.322, 'duration': 1.301}], 'summary': "Strogatz's writing expands math audience, simplifies calculus concepts, and highlights fourier's approach.", 'duration': 30.339, 'max_score': 994.284, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8994284.jpg'}], 'start': 498.555, 'title': 'Heat diffusion model and partial differential equations', 'summary': "Covers the concept of heat diffusion model, explaining temperature changes and the derivative equation, and introduces partial differential equations, highlighting the role of second derivatives and their comparison to neighbors, with a reference to 'infinite powers' by steve strogatz.", 'chapters': [{'end': 681.579, 'start': 498.555, 'title': 'Heat diffusion model', 'summary': "Explains the concept of heat diffusion, where a point's temperature changes based on the average temperature of its neighboring points, expressed in a derivative equation and second differences.", 'duration': 183.024, 'highlights': ['The derivative of T2 with respect to time is proportional to the difference between the average value of its neighbors and its own value.', 'If delta T2 is bigger than delta T1, the average of T1 and T3 is bigger than T2, causing T2 to increase.', 'The second difference, represented as delta delta T1, quantifies how much T2 differs from the average of its neighbors.']}, {'end': 1036.733, 'start': 682.259, 'title': 'Understanding partial differential equations', 'summary': "Introduces the concept of partial differential equations and the intuition behind second derivatives, highlighting their role in measuring the rate of change and comparing a point to its neighbors, with a special mention of the book 'infinite powers' by steve strogatz.", 'duration': 354.474, 'highlights': ['The second derivative in the context of partial differential equations measures how a value compares to the average of its neighbors, providing a fundamental intuition for problems well beyond the heat equation.', "The concept of the Laplacian, a multivariable version of the second derivative, extends the idea of measuring a point's difference from its neighbors to multiple spatial dimensions.", "The mention of 'Infinite Powers' by Steve Strogatz as a valuable resource for those interested in gaining a deeper understanding and appreciation for calculus, emphasizing its potential to make more people love math and expand the audience base for educational videos."]}], 'duration': 538.178, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ly4S0oi3Yz8/pics/ly4S0oi3Yz8498555.jpg', 'highlights': ["The concept of the Laplacian extends the idea of measuring a point's difference from its neighbors to multiple spatial dimensions.", 'The second derivative in the context of partial differential equations measures how a value compares to the average of its neighbors, providing a fundamental intuition for problems well beyond the heat equation.', 'The derivative of T2 with respect to time is proportional to the difference between the average value of its neighbors and its own value.', "The mention of 'Infinite Powers' by Steve Strogatz as a valuable resource for those interested in gaining a deeper understanding and appreciation for calculus, emphasizing its potential to make more people love math and expand the audience base for educational videos.", 'If delta T2 is bigger than delta T1, the average of T1 and T3 is bigger than T2, causing T2 to increase.', 'The second difference, represented as delta delta T1, quantifies how much T2 differs from the average of its neighbors.']}], 'highlights': ['The heat equation is introduced as an example of a partial differential equation, showcasing the change in temperature distribution over time as heat moves from warmer areas to cooler ones.', 'The chapter discusses the prevalence of variations of the heat equation in math and physics, including Brownian motion, the Black-Scholes equations from finance, and diffusion, showcasing its wide-ranging applications.', 'The animation demonstrates how multiple rotating vectors can trace out an arbitrary shape, with 100 circles showing negligible deviations from the real shape.', "The concept of the Laplacian extends the idea of measuring a point's difference from its neighbors to multiple spatial dimensions.", 'The second derivative in the context of partial differential equations measures how a value compares to the average of its neighbors, providing a fundamental intuition for problems well beyond the heat equation.', 'The temperature of a one-dimensional rod is analyzed using a function of position and time, which can be graphed as a surface representing space and time together, or simply as a graph of temperature changing with time.']}