title

But what is a Fourier series? From heat flow to drawing with circles | DE4

description

Fourier series, from the heat equation epicycles.
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/de4thanks
12 minutes of pure Fourier series animations: https://youtu.be/-qgreAUpPwM
Some viewers made apps that create circle animations for your own drawing. Check them out!
https://www.reddit.com/r/3Blue1Brown/comments/cvpdn7/make_your_own_fourier_circle_drawings/
https://isaacvr.github.io/coding/fourier_transform/
Thanks to Stuart@Biocinematics for the one-line sketch of Fourier via twitter. As it happens, he also has an educational YouTube channel:
https://www.youtube.com/channel/UCKOiJd9YCbv7LeL2LFOGiLQ
Small correction: at 9:33, all the exponents should have a pi^2 in them.
If you're looking for more Fourier Series content online, including code to play with to create this kind of animation yourself, check out these posts:
Mathologer
https://youtu.be/qS4H6PEcCCA
The Coding Train
https://youtu.be/Mm2eYfj0SgA
Jezmoon
http://www.jezzamon.com/fourier/index.html
For those of you into pure math looking to really dig into the analysis behind this topic, you might want to take a look at Stein Shakarchi's book "Fourier Analysis: An Introduction"
Timestamps:
0:00 - Drawing with circles
2:10 - The heat equation
6:25 - Interpreting infinite function sums
9:52 - Trig in the complex plane
14:11 - Summing complex exponentials
22:11 - Example: The step function
23:54 - Conclusion
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These animations are largely made using a custom open-source python library, manim. See the FAQ comments here:
https://www.3blue1brown.com/faq#manim
https://github.com/3b1b/manim
https://github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
https://github.com/3b1b/videos/
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.
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe
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detail

{'title': 'But what is a Fourier series? From heat flow to drawing with circles | DE4', 'heatmap': [{'end': 1089.218, 'start': 1067.458, 'weight': 1}], 'summary': 'Explores the complexity of a complex fourier series, the application of the heat equation, generalizing the fourier transform, expressing functions as a sum of terms, and the decomposition of drawings into rotating vectors, providing a comprehensive understanding of fourier series and its diverse applications.', 'chapters': [{'end': 152.101, 'segs': [{'end': 57.839, 'src': 'embed', 'start': 5.464, 'weight': 0, 'content': [{'end': 10.626, 'text': "Here we look at the math behind an animation like this one, what's known as a complex Fourier series.", 'start': 5.464, 'duration': 5.162}, {'end': 17.35, 'text': 'Each little vector is rotating at some constant integer frequency, and when you add them together tip to tail,', 'start': 11.267, 'duration': 6.083}, {'end': 20.011, 'text': 'the final tip draws out some shape over time.', 'start': 17.35, 'duration': 2.661}, {'end': 28.295, 'text': "By tweaking the initial size and angle of each vector, we can make it draw pretty much anything that we want, and here you'll see how.", 'start': 21.252, 'duration': 7.043}, {'end': 35.865, 'text': 'Before diving into it all, I want you to take a moment to just linger on how striking this is.', 'start': 31.397, 'duration': 4.468}, {'end': 40.855, 'text': 'This particular animation has 300 rotating arrows in total.', 'start': 37.268, 'duration': 3.587}, {'end': 43.981, 'text': 'Go full screen for this if you can, the intricacy is worth it.', 'start': 41.396, 'duration': 2.585}, {'end': 57.839, 'text': 'Think about this, the action of each individual arrow is perhaps the simplest thing you could imagine, rotation at a steady rate.', 'start': 50.834, 'duration': 7.005}], 'summary': 'Complex fourier series animation with 300 rotating arrows creating intricate shapes over time.', 'duration': 52.375, 'max_score': 5.464, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k5464.jpg'}, {'end': 106.386, 'src': 'embed', 'start': 77.891, 'weight': 1, 'content': [{'end': 83.252, 'text': "it's bizarre how the swarm acts with a kind of coordination to trace out some very specific shape.", 'start': 77.891, 'duration': 5.361}, {'end': 87.593, 'text': 'And unlike much of the emergent complexity you find elsewhere in nature,', 'start': 83.972, 'duration': 3.621}, {'end': 91.034, 'text': 'this is something that we have the math to describe and to control completely.', 'start': 87.593, 'duration': 3.441}, {'end': 94.895, 'text': 'Just by tuning the starting conditions, nothing more.', 'start': 91.674, 'duration': 3.221}, {'end': 101.476, 'text': 'we can make this swarm conspire in all of the right ways to draw anything that you want, provided that you have enough little arrows.', 'start': 94.895, 'duration': 6.581}, {'end': 106.386, 'text': "What's even crazier is that the ultimate formula for all of this is incredibly short.", 'start': 102.156, 'duration': 4.23}], 'summary': 'Swarm can be controlled to form specific shapes with math, just by tuning starting conditions.', 'duration': 28.495, 'max_score': 77.891, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k77891.jpg'}, {'end': 164.732, 'src': 'embed', 'start': 138.189, 'weight': 4, 'content': [{'end': 143.013, 'text': "I would like to teach you about Fourier series in a way that doesn't depend on you coming from those chapters,", 'start': 138.189, 'duration': 4.824}, {'end': 148.118, 'text': 'but if you have at least a high-level idea for the problem from physics which originally motivated this piece of math,', 'start': 143.013, 'duration': 5.105}, {'end': 152.101, 'text': 'it gives some indication for just how unexpectedly far-reaching Fourier series are.', 'start': 148.118, 'duration': 3.983}, {'end': 160.428, 'text': 'All you need to know is that we had a certain equation which tells us how the temperature distribution on a rod would evolve over time.', 'start': 152.982, 'duration': 7.446}, {'end': 164.732, 'text': 'And incidentally, it also describes many other phenomena unrelated to heat.', 'start': 161.369, 'duration': 3.363}], 'summary': 'Fourier series extends beyond heat equations, with broad applications.', 'duration': 26.543, 'max_score': 138.189, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k138189.jpg'}], 'start': 5.464, 'title': 'Complex fourier series math', 'summary': 'Explores the complexity of a complex fourier series, describing the creation of intricate shapes through coordinated chaos using 300 rotating arrows, ultimately controlled and described by a remarkably concise formula.', 'chapters': [{'end': 152.101, 'start': 5.464, 'title': 'Complex fourier series math', 'summary': 'Explores the complexity of a complex fourier series, where 300 rotating arrows create intricate shapes through coordinated chaos, describable and controllable by math, with the ultimate formula being incredibly short.', 'duration': 146.637, 'highlights': ['The animation features 300 rotating arrows, each contributing to the intricate shape drawn over time.', 'The mind-boggling complexity of the coordinated chaos is described and controlled completely by tuning the starting conditions.', "The ultimate formula for the swarm's behavior is incredibly short, allowing it to draw anything provided there are enough little arrows.", 'Fourier series is a more general rotating vector phenomenon, with the special case being the breakdown of functions of real numbers into a sum of sine waves.', 'Fourier series are unexpectedly far-reaching and were developed as a solution to the heat equation problem in physics.']}], 'duration': 146.637, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k5464.jpg', 'highlights': ['The animation features 300 rotating arrows, contributing to the intricate shape.', 'The mind-boggling complexity of coordinated chaos is controlled by tuning starting conditions.', "The ultimate formula for the swarm's behavior is incredibly short, allowing it to draw anything.", 'Fourier series is a more general rotating vector phenomenon, with the special case being the breakdown of functions into a sum of sine waves.', 'Fourier series were developed as a solution to the heat equation problem in physics.']}, {'end': 598.393, 'segs': [{'end': 207.334, 'src': 'embed', 'start': 152.982, 'weight': 0, 'content': [{'end': 160.428, 'text': 'All you need to know is that we had a certain equation which tells us how the temperature distribution on a rod would evolve over time.', 'start': 152.982, 'duration': 7.446}, {'end': 164.732, 'text': 'And incidentally, it also describes many other phenomena unrelated to heat.', 'start': 161.369, 'duration': 3.363}, {'end': 170.896, 'text': "And while it's hard to directly use this equation to figure out what will happen to an arbitrary heat distribution,", 'start': 165.352, 'duration': 5.544}, {'end': 175.34, 'text': "there's a simple solution if the initial function just happens to look like a cosine wave,", 'start': 170.896, 'duration': 4.444}, {'end': 178.062, 'text': "with the frequency tuned so that it's flat at each endpoint.", 'start': 175.34, 'duration': 2.722}, {'end': 181.604, 'text': 'Specifically, as you graph what happens over time.', 'start': 178.482, 'duration': 3.122}, {'end': 187.869, 'text': 'these waves simply get scaled down exponentially, with higher frequency waves having a faster exponential decay.', 'start': 181.604, 'duration': 6.265}, {'end': 198.462, 'text': "The heat equation happens to be what's known in the business as a linear equation, meaning if you know two solutions and you add them up,", 'start': 190.771, 'duration': 7.691}, {'end': 200.204, 'text': 'that sum is a new solution.', 'start': 198.462, 'duration': 1.742}, {'end': 207.334, 'text': 'You can even scale them each by some constant, which gives you some dials to turn to construct a custom function solving the equation.', 'start': 200.925, 'duration': 6.409}], 'summary': 'Heat equation evolves cosine wave distribution exponentially over time.', 'duration': 54.352, 'max_score': 152.982, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k152982.jpg'}, {'end': 251.731, 'src': 'embed', 'start': 230.214, 'weight': 6, 'content': [{'end': 240.442, 'text': "One important thing I'd like you to notice is that when you combine these waves because the higher frequency ones decay faster the sum that you construct will tend to smooth out over time,", 'start': 230.214, 'duration': 10.228}, {'end': 245.406, 'text': 'as all the high frequency terms quickly go to zero, leaving only the low frequency terms dominating.', 'start': 240.442, 'duration': 4.964}, {'end': 251.731, 'text': 'So, in a funny way, all of the complexity in the evolution of this heat distribution which the heat equation implies,', 'start': 246.047, 'duration': 5.684}], 'summary': 'Combining waves results in smoothing over time, with low frequency terms dominating.', 'duration': 21.517, 'max_score': 230.214, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k230214.jpg'}, {'end': 423.042, 'src': 'embed', 'start': 393.625, 'weight': 1, 'content': [{'end': 401.408, 'text': "And it's true, any finite sum of sine waves will never be perfectly flat, except for a constant function, nor will it be discontinuous.", 'start': 393.625, 'duration': 7.783}, {'end': 405.409, 'text': 'But Fourier thought more broadly, considering infinite sums.', 'start': 402.088, 'duration': 3.321}, {'end': 414.773, 'text': 'In the case of our step function, it turns out to be equal to this infinite sum, where the coefficients are 1, negative 1, third plus 1,', 'start': 406.21, 'duration': 8.563}, {'end': 423.042, 'text': 'fifth minus 1, seventh and so on, for all the odd frequencies, and all of it is rescaled by 4, divided by pi.', 'start': 414.773, 'duration': 8.269}], 'summary': 'Fourier considered infinite sums for step function, equal to sum with odd coefficients rescaled by 4/pi.', 'duration': 29.417, 'max_score': 393.625, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k393625.jpg'}, {'end': 586.269, 'src': 'embed', 'start': 559.069, 'weight': 5, 'content': [{'end': 565.451, 'text': "These are exactly the kind of questions which real analysis is built to answer, but it falls a bit deeper in the weeds than I'd like to go here,", 'start': 559.069, 'duration': 6.382}, {'end': 568.052, 'text': "so I'll relegate that all to links in the video's description.", 'start': 565.451, 'duration': 2.601}, {'end': 577.14, 'text': 'The upshot is that when you take the heat equation solutions associated with these cosine waves and you add them all up all infinitely many of them,', 'start': 568.692, 'duration': 8.448}, {'end': 582.005, 'text': 'you do get an exact solution describing how the step function will evolve over time.', 'start': 577.14, 'duration': 4.865}, {'end': 586.269, 'text': 'And if you had done this in 1822, you would have become immortal for doing so.', 'start': 582.626, 'duration': 3.643}], 'summary': 'Real analysis answers questions about heat equation solutions, leading to an exact solution for step function evolution over time with cosine waves.', 'duration': 27.2, 'max_score': 559.069, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k559069.jpg'}], 'start': 152.982, 'title': 'Heat equation, cosine waves, and fourier series', 'summary': "Discusses the heat equation's application in describing temperature distribution on a rod and the behaviour of cosine wave initial function, as well as the use of fourier series to represent complex initial conditions and exact solutions of the heat equation.", 'chapters': [{'end': 207.334, 'start': 152.982, 'title': 'Heat equation and cosine waves', 'summary': "Discusses the heat equation's ability to describe temperature distribution on a rod, including its applicability to various phenomena, and the simple solution for a cosine wave initial function, where higher frequency waves exhibit faster exponential decay.", 'duration': 54.352, 'highlights': ['The heat equation describes temperature distribution on a rod and other unrelated phenomena.', 'The simple solution for the heat equation occurs when the initial function is a cosine wave with a frequency tuned to be flat at each endpoint.', 'Higher frequency waves exhibit faster exponential decay as time progresses.', 'The heat equation is a linear equation, allowing new solutions to be formed by adding or scaling existing solutions.']}, {'end': 598.393, 'start': 209.212, 'title': 'Fourier series and heat equation', 'summary': 'Explores how fourier series can be used to represent complex initial conditions as a sum of sine waves, enabling the exact solution of the heat equation and its associated cosine waves to describe the evolution of a step function over time.', 'duration': 389.181, 'highlights': ['Fourier series can be used to represent complex initial conditions as a sum of sine waves.', 'The heat equation solutions associated with cosine waves can be added to get an exact solution describing the evolution of the step function.', 'Infinite sums of sine waves can represent a discontinuous flat function like a step function.']}], 'duration': 445.411, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k152982.jpg', 'highlights': ['The heat equation is a linear equation, allowing new solutions to be formed by adding or scaling existing solutions.', 'Infinite sums of sine waves can represent a discontinuous flat function like a step function.', 'The simple solution for the heat equation occurs when the initial function is a cosine wave with a frequency tuned to be flat at each endpoint.', 'Fourier series can be used to represent complex initial conditions as a sum of sine waves.', 'The heat equation describes temperature distribution on a rod and other unrelated phenomena.', 'The heat equation solutions associated with cosine waves can be added to get an exact solution describing the evolution of the step function.', 'Higher frequency waves exhibit faster exponential decay as time progresses.']}, {'end': 988.97, 'segs': [{'end': 653.391, 'src': 'embed', 'start': 620.226, 'weight': 0, 'content': [{'end': 624.569, 'text': 'More importantly, it sets a good foundation for the ideas that will come up later on in the series,', 'start': 620.226, 'duration': 4.343}, {'end': 627.772, 'text': 'like the Laplace transform and the importance of exponential functions.', 'start': 624.569, 'duration': 3.203}, {'end': 635.976, 'text': "We'll still think of functions whose input is some real number on a finite interval, say from 0 up to 1 for simplicity.", 'start': 629.571, 'duration': 6.405}, {'end': 640.44, 'text': 'But whereas something like a temperature function will have outputs on the real number line,', 'start': 636.617, 'duration': 3.823}, {'end': 644.504, 'text': 'this broader view will let the outputs wander anywhere in the two-dimensional complex plane.', 'start': 640.44, 'duration': 4.064}, {'end': 653.391, 'text': 'You might think of such a function as a drawing, with a pencil tip tracing out different points in the complex plane as the input ranges from 0 to 1.', 'start': 645.124, 'duration': 8.267}], 'summary': 'Introduction to the broader view of functions and their outputs in the complex plane.', 'duration': 33.165, 'max_score': 620.226, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k620226.jpg'}, {'end': 748.978, 'src': 'embed', 'start': 716.081, 'weight': 1, 'content': [{'end': 722.79, 'text': 'But in the broader context of functions with complex number outputs, this oscillation on the horizontal line is what a sine wave looks like.', 'start': 716.081, 'duration': 6.709}, {'end': 732.932, 'text': 'Similarly, the pair of rotating vectors with frequencies 2 and negative 2 will add another sine wave component and so on,', 'start': 725.109, 'duration': 7.823}, {'end': 738.694, 'text': 'with the sine waves we were looking for earlier now corresponding to pairs of vectors rotating in opposite directions.', 'start': 732.932, 'duration': 5.762}, {'end': 748.978, 'text': 'So the context that Fourier originally studied breaking down real valued functions into sine waves is a special case of the more general idea of 2D drawings and rotating vectors.', 'start': 739.614, 'duration': 9.364}], 'summary': 'Fourier originally studied breaking down real valued functions into sine waves as a special case of the more general idea of 2d drawings and rotating vectors.', 'duration': 32.897, 'max_score': 716.081, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k716081.jpg'}, {'end': 813.527, 'src': 'embed', 'start': 784.572, 'weight': 4, 'content': [{'end': 791.295, 'text': 'As the input t ticks forward with time, this value walks around the unit circle at a rate of one unit per second.', 'start': 784.572, 'duration': 6.723}, {'end': 800.079, 'text': "In the next video you'll see a quick intuition for why exponentiating imaginary numbers walks around circles like this from the perspective of differential equations.", 'start': 792.275, 'duration': 7.804}, {'end': 807.062, 'text': 'And beyond that, as the series progresses, I hope to give you some sense for why complex exponentials like this are actually very important.', 'start': 800.619, 'duration': 6.443}, {'end': 813.527, 'text': 'In theory, you could describe all of the Fourier series stuff purely in terms of vectors and never breathe a word of i,', 'start': 807.742, 'duration': 5.785}], 'summary': 'Imaginary numbers walk around the unit circle at a rate of one unit per second, demonstrating their importance in fourier series.', 'duration': 28.955, 'max_score': 784.572, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k784572.jpg'}, {'end': 946.863, 'src': 'embed', 'start': 904.535, 'weight': 2, 'content': [{'end': 911.077, 'text': 'Similarly, the one going two rotations per second is e to the 2 times 2 pi i times t,', 'start': 904.535, 'duration': 6.542}, {'end': 916.578, 'text': 'where that 2 times 2 pi in the exponent describes how much distance is covered in one second.', 'start': 911.077, 'duration': 5.501}, {'end': 929.396, 'text': 'And we go on like this over all integers, both positive and negative, with a general formula of e to the n times 2 pi times i t.', 'start': 920.748, 'duration': 8.648}, {'end': 935.381, 'text': 'Notice, this makes it more consistent to write that constant vector as e to the 0 times 2 pi times i t,', 'start': 929.396, 'duration': 5.985}, {'end': 939.564, 'text': 'which feels like an awfully complicated way to write the number 1, but at least it fits the pattern.', 'start': 935.381, 'duration': 4.183}, {'end': 946.863, 'text': 'The control that we have, the set of knobs and dials we get to turn, is the initial size and direction of each of these numbers.', 'start': 940.478, 'duration': 6.385}], 'summary': 'The motion is described by e^(n*2*pi*i*t), with n representing rotations per second.', 'duration': 42.328, 'max_score': 904.535, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k904535.jpg'}], 'start': 598.393, 'title': 'Generalizing fourier transform', 'summary': 'Discusses generalizing the fourier transform to functions with complex number outputs in the 2d plane, providing a foundation for laplace transform and exponential functions, and demonstrating how decomposing functions into rotating vectors can represent sine waves in a broader context.', 'chapters': [{'end': 818.652, 'start': 598.393, 'title': 'Generalizing fourier transform', 'summary': 'Discusses generalizing the fourier transform to functions with complex number outputs in the 2d plane, providing a foundation for laplace transform and exponential functions, and demonstrating how decomposing functions into rotating vectors can represent sine waves in a broader context.', 'duration': 220.259, 'highlights': ['The broader view of functions with complex number outputs sets a foundation for Laplace transform and exponential functions, allowing outputs to wander in the two-dimensional complex plane.', 'The decomposition of functions into rotating vectors represents sine waves in a broader context, with pairs of vectors rotating in opposite directions corresponding to the sine waves.', 'The complex exponential e to the i times t walks around the unit circle at a rate of one unit per second, demonstrating its importance in Fourier series and providing intuition for its significance.']}, {'end': 988.97, 'start': 818.652, 'title': 'Complex numbers and rotating vectors', 'summary': 'Discusses the significance of e to the i times t in describing rotating vectors, providing formulas for rotating vectors, and the control over their initial size and direction through complex constants.', 'duration': 170.318, 'highlights': ['The chapter discusses the significance of e to the i times t in describing rotating vectors, providing formulas for rotating vectors, and the control over their initial size and direction through complex constants.', 'The formula e to the 2 pi i times t represents a vector rotating one cycle every second, covering a distance of 2 pi along the circle as t goes from 0 to 1.', 'The control over the initial size and direction of each vector is achieved by multiplying it by a complex constant, allowing adjustments in length and angle.']}], 'duration': 390.577, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k598393.jpg', 'highlights': ['The broader view of functions with complex number outputs sets a foundation for Laplace transform and exponential functions, allowing outputs to wander in the two-dimensional complex plane.', 'The decomposition of functions into rotating vectors represents sine waves in a broader context, with pairs of vectors rotating in opposite directions corresponding to the sine waves.', 'The formula e to the 2 pi i times t represents a vector rotating one cycle every second, covering a distance of 2 pi along the circle as t goes from 0 to 1.', 'The chapter discusses the significance of e to the i times t in describing rotating vectors, providing formulas for rotating vectors, and the control over their initial size and direction through complex constants.', 'The complex exponential e to the i times t walks around the unit circle at a rate of one unit per second, demonstrating its importance in Fourier series and providing intuition for its significance.', 'The control over the initial size and direction of each vector is achieved by multiplying it by a complex constant, allowing adjustments in length and angle.']}, {'end': 1220.792, 'segs': [{'end': 1089.218, 'src': 'heatmap', 'start': 988.97, 'weight': 1, 'content': [{'end': 991.931, 'text': 'which determines its initial angle and its total magnitude.', 'start': 988.97, 'duration': 2.961}, {'end': 1002.495, 'text': 'Our goal is to express any arbitrary function f say this one that draws an eighth note as t goes from 0 to 1, as a sum of terms like this.', 'start': 992.751, 'duration': 9.744}, {'end': 1009.197, 'text': 'So we need some way of picking out these constants one by one given the data of the function itself.', 'start': 1003.195, 'duration': 6.002}, {'end': 1014.488, 'text': 'The easiest of these to find is the constant term.', 'start': 1012.185, 'duration': 2.303}, {'end': 1018.474, 'text': 'This term represents a sort of center of mass for the full drawing.', 'start': 1015.129, 'duration': 3.345}, {'end': 1024.723, 'text': 'If you were to sample a bunch of evenly spaced values for the input t, as it ranges from 0 to 1,', 'start': 1019.115, 'duration': 5.608}, {'end': 1030.609, 'text': 'the average of all the outputs of the function for those samples will be the constant term c0..', 'start': 1024.723, 'duration': 5.886}, {'end': 1039.214, 'text': 'Or, more accurately, as you consider finer and finer samples, the average of the outputs for these samples approaches c0 in the limit.', 'start': 1031.39, 'duration': 7.824}, {'end': 1050.908, 'text': "What I'm describing finer and finer sums of a function for samples of t from the input range is an integral, an integral of f from 0 to 1,.", 'start': 1039.954, 'duration': 10.954}, {'end': 1056.595, 'text': "Normally, since I'm framing this all in terms of averages, you would divide the integral by the length of the input range.", 'start': 1050.908, 'duration': 5.687}, {'end': 1058.377, 'text': 'But that length is 1.', 'start': 1057.035, 'duration': 1.342}, {'end': 1061.721, 'text': 'So in this case, taking an integral and taking an average are the same thing.', 'start': 1058.377, 'duration': 3.344}, {'end': 1065.986, 'text': "There's a very nice way to think about why this integral would pull out c0.", 'start': 1062.742, 'duration': 3.244}, {'end': 1071.143, 'text': 'Remember, we want to think of this function as a sum of rotating vectors.', 'start': 1067.458, 'duration': 3.685}, {'end': 1076.551, 'text': 'So consider this integral, this continuous average, as being applied to that whole sum.', 'start': 1071.704, 'duration': 4.847}, {'end': 1082.36, 'text': 'The average of a sum like this is the same as the sum over the averages of each part.', 'start': 1077.413, 'duration': 4.947}, {'end': 1089.218, 'text': 'You can read this move as a sort of subtle shift in perspective.', 'start': 1086.414, 'duration': 2.804}], 'summary': 'Express an arbitrary function as a sum of terms, finding the constant term c0 using integral and averages.', 'duration': 100.248, 'max_score': 988.97, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k988970.jpg'}, {'end': 1177.54, 'src': 'embed', 'start': 1146.207, 'weight': 0, 'content': [{'end': 1154.334, 'text': 'Specifically, if you multiply the whole function by e to the negative 2 times 2 pi i times t, think about what happens to each term.', 'start': 1146.207, 'duration': 8.127}, {'end': 1169.636, 'text': "Since multiplying exponentials results in adding what's in the exponent, the frequency term in each of our exponents gets shifted down by 2.", 'start': 1157.031, 'duration': 12.605}, {'end': 1177.54, 'text': 'So now, as we do our averages of each term, that c-1 vector spins around negative 3 times with an average of 0.', 'start': 1169.636, 'duration': 7.904}], 'summary': 'Multiplying the function by e^(-2*2*pi*i*t) shifts frequency down by 2, resulting in c-1 vector spinning around negative 3 times with an average of 0.', 'duration': 31.333, 'max_score': 1146.207, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k1146207.jpg'}], 'start': 988.97, 'title': 'Functions and integrals', 'summary': 'Delves into expressing functions as a sum of terms, finding the constant term representing the center of mass, and its connection to the integral, integral average technique applied to rotating vectors, and transforming functions to isolate specific terms such as c0 and c2.', 'chapters': [{'end': 1061.721, 'start': 988.97, 'title': 'Eighth note function analysis', 'summary': 'Discusses expressing a function as a sum of terms, finding the constant term representing the center of mass, and its connection to the integral of the function from 0 to 1.', 'duration': 72.751, 'highlights': ['Expressing a function as a sum of terms', 'Finding the constant term representing the center of mass', 'Connection between integral and average of function']}, {'end': 1129.432, 'start': 1062.742, 'title': 'Integral average and rotating vectors', 'summary': 'Discusses how the integral average technique applied to a sum of rotating vectors results in the extraction of c0, the constant term, due to the average of the rotating vectors being 0, except for the static c0 term.', 'duration': 66.69, 'highlights': ['The integral average technique applied to a sum of rotating vectors results in the extraction of c0, the constant term, due to the average of the rotating vectors being 0, except for the static c0 term.', "Considering the average of an individual vector as t ranges from 0 to 1 and then adding up all these averages results in the elimination of terms that aren't c0, while the constant term's average value is just whatever number it started on, which is c0.", 'The average of a sum of rotating vectors is the same as the sum over the averages of each part, leading to a clever way to eliminate terms other than c0 by leveraging the average property of the rotating vectors.']}, {'end': 1220.792, 'start': 1129.853, 'title': 'Transforming function to isolate a specific term', 'summary': 'Explains how to modify a function to isolate a specific term by multiplying it with a specific factor, resulting in the isolation of the desired term, as demonstrated using the example of c2.', 'duration': 90.939, 'highlights': ['Multiplying the function by e to the negative 2 times 2 pi i times t shifts the frequency term in each exponent down by 2, effectively isolating the c2 term.', "The modified function cleverly eliminates all terms other than c2 by averaging the vectors' rotations, providing a general formula for cn."]}], 'duration': 231.822, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k988970.jpg', 'highlights': ['Multiplying the function by e to the negative 2 times 2 pi i times t shifts the frequency term in each exponent down by 2, effectively isolating the c2 term.', 'Expressing a function as a sum of terms', 'Finding the constant term representing the center of mass', 'Connection between integral and average of function', 'The integral average technique applied to a sum of rotating vectors results in the extraction of c0, the constant term, due to the average of the rotating vectors being 0, except for the static c0 term.']}, {'end': 1465.645, 'segs': [{'end': 1281.415, 'src': 'embed', 'start': 1220.792, 'weight': 1, 'content': [{'end': 1225.655, 'text': 'and then performing an average which kills all of the moving vectors and leaves you only with the still part.', 'start': 1220.792, 'duration': 4.863}, {'end': 1227.097, 'text': "Isn't that crazy?", 'start': 1226.436, 'duration': 0.661}, {'end': 1230.479, 'text': "All of the complexity in these decompositions you're seeing,", 'start': 1227.537, 'duration': 2.942}, {'end': 1235.843, 'text': 'of drawings into sums of many rotating vectors is entirely captured in this little expression.', 'start': 1230.479, 'duration': 5.364}, {'end': 1240.757, 'text': "So when I'm rendering these animations, that's exactly what I'm having the computer do.", 'start': 1236.892, 'duration': 3.865}, {'end': 1248.446, 'text': 'It treats the path like a complex function and for a certain range of values in it computes this integral to find the coefficient.', 'start': 1241.297, 'duration': 7.149}, {'end': 1257.868, 'text': "c. For those of you curious about where the data for a path itself comes from, I'm going the easy route and just having the program read in an SVG,", 'start': 1248.446, 'duration': 9.422}, {'end': 1262.751, 'text': 'which is a file format that defines the image in terms of mathematical curves rather than with pixel values.', 'start': 1257.868, 'duration': 4.883}, {'end': 1268.614, 'text': 'So the mapping f from a time parameter to points in space basically comes predefined.', 'start': 1263.431, 'duration': 5.183}, {'end': 1278.233, 'text': "In what's shown right now, I'm using 101 rotating vectors, computing the values of n from negative 50 up to 50.", 'start': 1270.706, 'duration': 7.527}, {'end': 1281.415, 'text': 'In practice, each of these integrals is computed numerically,', 'start': 1278.233, 'duration': 3.182}], 'summary': 'Complex function computes integral for coefficient c, using 101 rotating vectors.', 'duration': 60.623, 'max_score': 1220.792, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k1220792.jpg'}, {'end': 1352.449, 'src': 'embed', 'start': 1298.83, 'weight': 0, 'content': [{'end': 1305.836, 'text': 'And after you compute these 101 constants, each one determines an initial angle and magnitude for the little vectors,', 'start': 1298.83, 'duration': 7.006}, {'end': 1309.48, 'text': 'and then you just set them all rotating, adding them tip to tail as they go,', 'start': 1305.836, 'duration': 3.644}, {'end': 1314.224, 'text': 'and the path drawn out by the final tip is some approximation of the original path you fed in.', 'start': 1309.48, 'duration': 4.744}, {'end': 1320.409, 'text': 'As the number of vectors used approaches infinity, the approximation path gets more and more accurate.', 'start': 1315.224, 'duration': 5.185}, {'end': 1336.181, 'text': 'To bring this all back down to earth.', 'start': 1334.52, 'duration': 1.661}, {'end': 1339.863, 'text': 'consider the example we were looking at earlier of a step function which, remember,', 'start': 1336.181, 'duration': 3.682}, {'end': 1345.505, 'text': 'was useful for modeling the heat dissipation between two rods at different temperatures after they come into contact.', 'start': 1339.863, 'duration': 5.642}, {'end': 1352.449, 'text': "Like any real number valued function, the step function is like a boring drawing that's confined to one dimension.", 'start': 1346.586, 'duration': 5.863}], 'summary': 'Computing 101 constants results in accurate path approximation using rotating vectors, approaching accuracy as the number of vectors used approaches infinity.', 'duration': 53.619, 'max_score': 1298.83, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k1298830.jpg'}], 'start': 1220.792, 'title': 'Drawing decomposition and fourier series', 'summary': 'Discusses decomposition of drawings into rotating vectors using integrals and numerical methods, and explains the role of step function in fourier series approximation, computation of coefficients, and its relation to cosine waves in the study of partial differential equations.', 'chapters': [{'end': 1320.409, 'start': 1220.792, 'title': 'Decomposition of drawings', 'summary': 'Discusses the decomposition of drawings into sums of rotating vectors, computed using integrals and numerical methods, to generate an approximation path that becomes more accurate as the number of vectors used approaches infinity.', 'duration': 99.617, 'highlights': ['The complexity in decomposing drawings into rotating vectors is captured in the expression used for computing the coefficients, which are determined by numerically computing integrals over a predefined time parameter and space points.', 'The program reads in an SVG file format to define the image in terms of mathematical curves rather than with pixel values, making the mapping from a time parameter to points in space predefined.', 'Using 101 rotating vectors, the values of n are computed from negative 50 up to 50, and each of these integrals is computed numerically by chopping up the unit interval into small pieces and adding up the values for each one of them.', 'After computing the constants from the integrals, each one determines an initial angle and magnitude for the vectors, which are then set rotating and added tip to tail to generate an approximation path of the original drawing.', 'The approximation path becomes more accurate as the number of vectors used approaches infinity.']}, {'end': 1465.645, 'start': 1334.52, 'title': 'Fourier series and heat dissipation', 'summary': "Explains the step function's role in fourier series approximation, the computation of coefficients, and its relation to cosine waves, providing insights into the study of partial differential equations and its significance in solving differential equations.", 'duration': 131.125, 'highlights': ["The step function's role in Fourier series approximation, where the vector sum stays close to 1 for the first half of the cycle and quickly jumps to negative 1 for the second half, provides practical insights into modeling heat dissipation (relevant for the study of partial differential equations).", 'The computation of coefficients in Fourier series approximation involves computing integrals, offering an opportunity for ambitious viewers to engage in exact calculus rather than relying on numerical methods.', 'The connection between breaking down a function as a combination of exponentials and using it to solve a differential equation demonstrates the significance of exponential functions in differential equations, highlighting their role in various forms and shapes.']}], 'duration': 244.853, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/r6sGWTCMz2k/pics/r6sGWTCMz2k1220792.jpg', 'highlights': ['The approximation path becomes more accurate as the number of vectors used approaches infinity.', 'Using 101 rotating vectors, the values of n are computed from negative 50 up to 50, and each of these integrals is computed numerically by chopping up the unit interval into small pieces and adding up the values for each one of them.', 'The program reads in an SVG file format to define the image in terms of mathematical curves rather than with pixel values, making the mapping from a time parameter to points in space predefined.', 'The complexity in decomposing drawings into rotating vectors is captured in the expression used for computing the coefficients, which are determined by numerically computing integrals over a predefined time parameter and space points.', "The step function's role in Fourier series approximation, where the vector sum stays close to 1 for the first half of the cycle and quickly jumps to negative 1 for the second half, provides practical insights into modeling heat dissipation (relevant for the study of partial differential equations)."]}], 'highlights': ['Fourier series were developed as a solution to the heat equation problem in physics.', 'The broader view of functions with complex number outputs sets a foundation for Laplace transform and exponential functions, allowing outputs to wander in the two-dimensional complex plane.', 'The decomposition of functions into rotating vectors represents sine waves in a broader context, with pairs of vectors rotating in opposite directions corresponding to the sine waves.', "The ultimate formula for the swarm's behavior is incredibly short, allowing it to draw anything.", 'The approximation path becomes more accurate as the number of vectors used approaches infinity.', 'The program reads in an SVG file format to define the image in terms of mathematical curves rather than with pixel values, making the mapping from a time parameter to points in space predefined.', 'Infinite sums of sine waves can represent a discontinuous flat function like a step function.', 'The complexity in decomposing drawings into rotating vectors is captured in the expression used for computing the coefficients, which are determined by numerically computing integrals over a predefined time parameter and space points.', 'The integral average technique applied to a sum of rotating vectors results in the extraction of c0, the constant term, due to the average of the rotating vectors being 0, except for the static c0 term.', 'The control over the initial size and direction of each vector is achieved by multiplying it by a complex constant, allowing adjustments in length and angle.']}