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Whenever my university taught me the Fourier (and the Taylor) series, it genuinely felt like I was witnessing something incredible and fundamental about math. Generalization is king, and this series is the king of generalization.
Gaussian quadrature is like that, they belong tô the realm of brilhante simple ideias, I undering how manny of this ideias are still waiting for us tô imagine❤❤
Z transform has entered the chat. So did fractional Fourier transforms. And, ug, wavelets. So has the spherical FT…no one calls it that, really: spherical harmonics, but it’s an FT on a sphere.
What impressed me most is the use of FFT algorithm, popularized by Cooley and Tukey in 1965, was first invented by Gauss 1.5 centuries prior to that(which he didn't publish because he thought it was useless) and he even predated fourier on representations of functions as infinite harmonic series. He had a lot of "This theorem was discovered by [insert name], but it turned out to have been proven by Gauss 10 years prior" moments, hence the phrase "you're smart but you're no Gauss". He really just needs a better PR team, akin to those of Newton's
I think a lot of these stories surrounding Gauss are apocryphal and rooted more in wishful thinking rather than facts. Not denying that Gauss was a great mathematician tho.
@@Neater_profile Gauss and Euler had mathematical abilities so far beyond your comprehension that tales of them are interpreted by wishful thinking even if they had no computers to check the results. Euler stumped his teachers by adding numbers with a system instead of being a linear, step by step sheeple like the rest.
Wonderful, I'm french and the auto generated subtitles keep my focus. Fourier is a true genius, one of the first geniuses that Normale Sup and X created
Dear Will, thank you. You have answered a question, I have been pondering for the past 42 years. As I watched your video, I was teleported back, "through space and time" to the summer of 1982. I was studying Fourier analysis and I had an epiphany, the first time my "wave function collapsed". I simply realized,"If you give me any function, any function f(x), I can express it in terms of a simple combination of sines and cosines." - Pure mathematics at its best, QED. Or as Sidney Coleman said it, "The career of a young Theoretical Physicist consists of treating the harmonic oscillator in ever increasing levels of abstraction."
I very recently realised that since one can also expand f in terms of a power series, thereby “projecting” the function from one basis to another. Absolutely beautiful stuff.
So you understand why saying, “my wave function expanded” is the same statement as “my wave function collapse”, but in the canonically complementary domain?
Dear Dr Will Wood. Can you explain the relationship between equation at 4:27 and Newton's cooling law? At first glance it seems to make sense, but in Newton Law of Cooling there is no spacial variable? Also the unit of 2 equations is not the same. For Newton's law of cooling, the unit of dQ/dt is Watt, but for the second equation, the unit is W/m. Can you help explain this?
Very nice video, I like that you were more holistic in your exposition and this was a succinct and well motivated video. As an idea, a similar video on Galois would go down well, you could do him justice.
Hey, man! Amazing video! Loved the background story!!! I would like to know if you can do the same for the Laplace Transform. I did a lot of digging through the years and I actually figured that it just came to be what it is from trial and error. However, I am aware that there is a way to derive it from Fourier Transform. Anyway, would be awesome to see you covering these topics as well!
Nice video! One nit, around 6:00, dT is a pretty bad choice of notation as you do not mean an increment in temperature but an increment in the _derivative_ of temperature.
I was very confused around 6 min. I had to watch it several times and I didn't get it until the end of that sub-segmant when he declared it was the first order Taylor expansion. I still don't see clearly how he got there
@@timothyvanrhein5230it’s just weird bc it’s a Taylor expansion on the derivative. Every time I see the heat equation, I think about this point with the 2nd derivative. But why does it appear in the Schrödinger eq? Momentum is the linear derivative wrt to position, and then energy is the square…idk.
Thank you Dr. Will! You're providing a precious resource by providing an insight into the intellectual maneuvers and methods of the minds which shaped our world, Awesome Video :D
Great video! It would have been really nice to see the actual approximation as a 3D function (the values over the x-y plane), not only the section at x=0.
Wow. just wow. I am using FFT since like 25 years ago and I never realized what a breakthrough was at the time. We are lucky he was not killed during the french revolution
1.1 Thank you for the informative video. In the video, I demonstrate the buckling of a spring sheet material that exhibits either a flat bell shape or a sine wave curve, bounded at the ends. Stress or compression is applied from the vertical axis. I question whether there is a correlation to statistics in this model. Additionally, I perform the same action in a V-shaped pattern. While some may compare this to plucking guitar strings, I contend that structures cannot be created with vibrating guitar strings or harmonic oscillators. In the model, "U" shaped waves emerge as the load increases, just before the wave-like function transitions to a higher energy level. By overlapping all the wave frequencies using Fourier Transforms, it seems to create a "U" shape or square waveform. If this model holds true, observing the sawtooth load versus deflection graph could provide significant insight into the events occurring during quantum jumps. These results can be replicated using a sheet of Mylar, the transparent plastic commonly found in school folders.Thanks for your interesting video. In the video below I buckle up a spring sheet of material. The shape looks like a flat bell shape or sine wave curve. It is bounded on the ends. I stress or compress it from the vertical axis. Is there any analogy in statistics that this models? I also do it in a V-shaped pattern. People say I am just plucked guitar strings. I said you can not make structures with vibrating guitar strings or harmonic oscillators. na01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fyoutu.be%2FwrBsqiE0vG4%3Fsi%3DwaT8lY2iX-wJdjO3&data=05%7C02%7C%7C95c4ee6902ca4276b64a08dcb9501371%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C638589000775490143%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0%7C%7C%7C&sdata=WYUkBNUzDfY1atRef6fYzzAyo0AG12XC0GEMxlLF4f0%3D&reserved=0 In the model, “U” shape waves are produced as the loading increases and just before the wave-like function shifts to the next higher energy level. Over-lapping all the waves frequencies together using Fournier Transforms, I understand makes a “U” shape or square wave form. If this model has merit, seeing the sawtooth load verse deflection graph produced could give some real insight in what happened during the quantum jumps. You can reproduce my results using a sheet of Mylar* ( the clear plastic found in school folders.
That which is like to itself in differentiation and exponentiation must be directly related to the exponential function, and Gamma(z) is equal to it for certain values, and seems to oscillate between cosine and sine at multiples of 1/2. In fact, it seems to act like a generalization of exp(z), and Gamma does after all show up in the partial sum of exp(z) itself which would also seem to imply a way to possibly generalize factorial given a means to compute the nth digit of e in some base? So far, my guess is there's probably a sum of four independent terms involving the exponential which I hypothesize from the likeness and alternative representation of the simple sum of complexes z + w as z+w=\left(\sqrt{z}+i\sqrt{w} ight)\left(\sqrt{z}-i\sqrt{w} ight)=\frac{1}{2}\left(e^{-i\arccos\sqrt{z}}+e^{i\arccos\sqrt{z}}+i\left(e^{-i\arccos\sqrt{w}}+e^{i\arccos\sqrt{w}} ight) ight)\cdot\frac{1}{2}\left(e^{-i\arccos\sqrt{z}}+e^{i\arccos\sqrt{z}}-i\left(e^{-i\arccos\sqrt{w}}+e^{i\arccos\sqrt{w}} ight) ight). Consider also product_(n=0)^(k) (x + i n) and the particular products with which this product converges as k goes to infinity. All of this leads me to believe that perhaps there's some simple sort of representation by generalizing the imaginary unit if not the complexes in a particular way such that something simple along the lines of f(z)^n = Gamma(f(z) + n)/Gamma(f(z)). With that, and with being able to represent any Gamma(z) for z in the rectangular region [0, 1 + i] (or really any such region [n+ik, n+1 + ij] for integers n, k and j), both representing Gamma sufficiently with which to create some sort of symbolic arithmetic (provided certain comparative operations can be performed symbolically), as well as computing arbitrarily good approximations of Gamma(z), would be trivialized-and that's just what I'm looking for. Am still sad I didn't get addicted to complex arithmetic sooner 😔
You're right. Not a typo, just me being a bit loose with variable naming. Should've just given it a generic name like "a" or something in hindsight maybe!
This was bugging me too. Not only the units but the maths does not work as well. @DrWillWood please correct it. Not to be an asshole but it just threw me a bit off.
its amazing that fourier dreamed this all up 200 years ago while napoleon was conquering europe.....there seems to be a tendency toward great intellectual discoveries when a nation is in the highest geopolitical ascendancy in its history
The integral you present is not, I'm pretty sure, equal to 0 for m != n. That is only true if m + n is even. For it to be true for all m!=n the boundaries have to be from 0 to π. In the next slide you have already crossed out the terms with m+n=2k+1. Observe the terms: it goes 1+5=6, 3+5 and so on... Because if this the mistake is hard to catch but when trying to prove the integral you present its pretty easy to understand. Just thought I'd point it out because u put so much work into the video! Hope it helps!!
Cant see any bound between Fourier's lifestory and his maths solution. I dont mean that autor was wrong when added history to this video, but it need better connection of scenario parts.
Involved in the Reign of Terror.......imprisoned and survived prison? So, he wasn't "involved" in the Reign of Terror but he WAS imprisoned during the Reign of Terror. Why was this?
Only problem i've seen with the video is it's assertion that you can derive fourier's law from newton's law of cooling. You can not, in the video he slipped in dT/dx instead of just dT, which is newton's original formulation, such a move is unjustified though
Fouriers Law for Cooling can & does include a dx term in the denominator on the right side, try a Google Search to see equations for Fouriers Law for Cooling.
Sorry, but I'm sure the explanations were clearer when I studied Fourier 45 years ago in Electrical Engineering math at University, that or I'm just getting old.
![The Most Useful Curve in Mathematics [Logarithms]](/img/n.gif)
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I'm french and study in Fourier Institut at Grenoble, France. Cool to see the story of the brilliant man who gave his name to my institut !
Whenever my university taught me the Fourier (and the Taylor) series, it genuinely felt like I was witnessing something incredible and fundamental about math. Generalization is king, and this series is the king of generalization.
Gaussian quadrature is like that, they belong tô the realm of brilhante simple ideias, I undering how manny of this ideias are still waiting for us tô imagine❤❤
Your last sentence sounds like one of Gauss
@@kwiky5643 man I love the channeling of New ideas , being mentally possesed by pure creativity
Z transform has entered the chat.
So did fractional Fourier transforms.
And, ug, wavelets.
So has the spherical FT…no one calls it that, really: spherical harmonics, but it’s an FT on a sphere.
I never realised how old Fourier actually is! Great video!!!
What impressed me most is the use of FFT algorithm, popularized by Cooley and Tukey in 1965, was first invented by Gauss 1.5 centuries prior to that(which he didn't publish because he thought it was useless) and he even predated fourier on representations of functions as infinite harmonic series.
He had a lot of "This theorem was discovered by [insert name], but it turned out to have been proven by Gauss 10 years prior" moments, hence the phrase "you're smart but you're no Gauss". He really just needs a better PR team, akin to those of Newton's
That's really cool if it's real
I think a lot of these stories surrounding Gauss are apocryphal and rooted more in wishful thinking rather than facts. Not denying that Gauss was a great mathematician tho.
@@Neater_profile Gauss and Euler had mathematical abilities so far beyond your comprehension that tales of them are interpreted by wishful thinking even if they had no computers to check the results.
Euler stumped his teachers by adding numbers with a system instead of being a linear, step by step sheeple like the rest.
@@colorx6030 partially true.
Thanks! I was looking for this from a long time!!
Thank you! Appreciate the support 🙂
Wonderful, I'm french and the auto generated subtitles keep my focus. Fourier is a true genius, one of the first geniuses that Normale Sup and X created
way too underrated, you explained it well
Amazing quote for Fourier in the beginning ! Thank you!
What a gem of a video, I really enjoyed the animations and explanation. Very well made!!
just finished studying everything I think I need from the heat equation to FFT and this is a nice dessert to wrap things all up...
This is the best video about the Fourier Series I've ever watched! Finally a video that links heat to the series!!!!
Dear Will, thank you.
You have answered a question, I have been pondering for the past 42 years.
As I watched your video, I was teleported back, "through space and time" to the summer of 1982.
I was studying Fourier analysis and I had an epiphany, the first time my "wave function collapsed".
I simply realized,"If you give me any function, any function f(x), I can express it in terms of a simple combination of sines and cosines." - Pure mathematics at its best, QED.
Or as Sidney Coleman said it, "The career of a young Theoretical Physicist consists of treating the harmonic oscillator in ever increasing levels of abstraction."
Exactly.
I very recently realised that since one can also expand f in terms of a power series, thereby “projecting” the function from one basis to another. Absolutely beautiful stuff.
So you understand why saying, “my wave function expanded” is the same statement as “my wave function collapse”, but in the canonically complementary domain?
awesome video you have represented the beauty of doing Physics and for the first time I saw the derivation of heat equation
Dear Dr Will Wood. Can you explain the relationship between equation at 4:27 and Newton's cooling law? At first glance it seems to make sense, but in Newton Law of Cooling there is no spacial variable? Also the unit of 2 equations is not the same. For Newton's law of cooling, the unit of dQ/dt is Watt, but for the second equation, the unit is W/m. Can you help explain this?
Thank you so much for your videos!.
Very nice video, I like that you were more holistic in your exposition and this was a succinct and well motivated video.
As an idea, a similar video on Galois would go down well, you could do him justice.
your creativity and passion shine through every project!
Hey, man! Amazing video! Loved the background story!!!
I would like to know if you can do the same for the Laplace Transform. I did a lot of digging through the years and I actually figured that it just came to be what it is from trial and error. However, I am aware that there is a way to derive it from Fourier Transform.
Anyway, would be awesome to see you covering these topics as well!
Nice video!
One nit, around 6:00, dT is a pretty bad choice of notation as you do not mean an increment in temperature but an increment in the _derivative_ of temperature.
I was very confused around 6 min. I had to watch it several times and I didn't get it until the end of that sub-segmant when he declared it was the first order Taylor expansion. I still don't see clearly how he got there
@@timothyvanrhein5230 Yeah same. He kinda skimmed over the whole maths explanation and it's not easy for someone who's never seen this kind of stuff
@@timothyvanrhein5230it’s just weird bc it’s a Taylor expansion on the derivative. Every time I see the heat equation, I think about this point with the 2nd derivative.
But why does it appear in the Schrödinger eq? Momentum is the linear derivative wrt to position, and then energy is the square…idk.
Thank you Dr. Will! You're providing a precious resource by providing an insight into the intellectual maneuvers and methods of the minds which shaped our world, Awesome Video :D
Great video!
It would have been really nice to see the actual approximation as a 3D function (the values over the x-y plane), not only the section at x=0.
One of the best explanations!
Standing on one's head at job interviews forms a lasting impression.
Well done vid on a person people should know a lot more about. 😀👍
thank you DR.
Fascinating!
Love your music, keep it up!
Hi Dr. Wood!
Great teaching!
I don't understand at 6:04 why it's the second derivative. Isn't that used to determine the inflection points? Did I miss something in maths class?
Wow. just wow. I am using FFT since like 25 years ago and I never realized what a breakthrough was at the time.
We are lucky he was not killed during the french revolution
great work!
Excellent! Thank you very much.
Epic video as usual; never fails to disappoint. You upload too little and too late 😔
1.1
Thank you for the informative video. In the video, I demonstrate the buckling of a spring sheet material that exhibits either a flat bell shape or a sine wave curve, bounded at the ends. Stress or compression is applied from the vertical axis. I question whether there is a correlation to statistics in this model. Additionally, I perform the same action in a V-shaped pattern. While some may compare this to plucking guitar strings, I contend that structures cannot be created with vibrating guitar strings or harmonic oscillators. In the model, "U" shaped waves emerge as the load increases, just before the wave-like function transitions to a higher energy level. By overlapping all the wave frequencies using Fourier Transforms, it seems to create a "U" shape or square waveform. If this model holds true, observing the sawtooth load versus deflection graph could provide significant insight into the events occurring during quantum jumps. These results can be replicated using a sheet of Mylar, the transparent plastic commonly found in school folders.Thanks for your interesting video.
In the video below I buckle up a spring sheet of material. The shape looks like a flat bell shape or sine wave curve. It is bounded on the ends. I stress or compress it from the vertical axis.
Is there any analogy in statistics that this models?
I also do it in a V-shaped pattern.
People say I am just plucked guitar strings. I said you can not make structures with vibrating guitar strings or harmonic oscillators.
na01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fyoutu.be%2FwrBsqiE0vG4%3Fsi%3DwaT8lY2iX-wJdjO3&data=05%7C02%7C%7C95c4ee6902ca4276b64a08dcb9501371%7C84df9e7fe9f640afb435aaaaaaaaaaaa%7C1%7C0%7C638589000775490143%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0%7C%7C%7C&sdata=WYUkBNUzDfY1atRef6fYzzAyo0AG12XC0GEMxlLF4f0%3D&reserved=0
In the model, “U” shape waves are produced as the loading increases and just before the wave-like function shifts to the next higher energy level.
Over-lapping all the waves frequencies together using Fournier Transforms, I understand makes a “U” shape or square wave form.
If this model has merit, seeing the sawtooth load verse deflection graph produced could give some real insight in what happened during the quantum jumps.
You can reproduce my results using a sheet of Mylar* ( the clear plastic found in school folders.
Boulders lined the side of the road foretelling what could come next.
That which is like to itself in differentiation and exponentiation must be directly related to the exponential function, and Gamma(z) is equal to it for certain values, and seems to oscillate between cosine and sine at multiples of 1/2. In fact, it seems to act like a generalization of exp(z), and Gamma does after all show up in the partial sum of exp(z) itself which would also seem to imply a way to possibly generalize factorial given a means to compute the nth digit of e in some base?
So far, my guess is there's probably a sum of four independent terms involving the exponential which I hypothesize from the likeness and alternative representation of the simple sum of complexes z + w as z+w=\left(\sqrt{z}+i\sqrt{w}
ight)\left(\sqrt{z}-i\sqrt{w}
ight)=\frac{1}{2}\left(e^{-i\arccos\sqrt{z}}+e^{i\arccos\sqrt{z}}+i\left(e^{-i\arccos\sqrt{w}}+e^{i\arccos\sqrt{w}}
ight)
ight)\cdot\frac{1}{2}\left(e^{-i\arccos\sqrt{z}}+e^{i\arccos\sqrt{z}}-i\left(e^{-i\arccos\sqrt{w}}+e^{i\arccos\sqrt{w}}
ight)
ight).
Consider also product_(n=0)^(k) (x + i n) and the particular products with which this product converges as k goes to infinity. All of this leads me to believe that perhaps there's some simple sort of representation by generalizing the imaginary unit if not the complexes in a particular way such that something simple along the lines of f(z)^n = Gamma(f(z) + n)/Gamma(f(z)). With that, and with being able to represent any Gamma(z) for z in the rectangular region [0, 1 + i] (or really any such region [n+ik, n+1 + ij] for integers n, k and j), both representing Gamma sufficiently with which to create some sort of symbolic arithmetic (provided certain comparative operations can be performed symbolically), as well as computing arbitrarily good approximations of Gamma(z), would be trivialized-and that's just what I'm looking for. Am still sad I didn't get addicted to complex arithmetic sooner 😔
Typo at 5:50.
You cannot add (dT/dx) and (dT). The units conflict.
You're right. Not a typo, just me being a bit loose with variable naming. Should've just given it a generic name like "a" or something in hindsight maybe!
This was bugging me too. Not only the units but the maths does not work as well. @DrWillWood please correct it. Not to be an asshole but it just threw me a bit off.
Im confused in that section too :v
Be the change that you want to see in the world.
Can you transfer heat through a photon? Or how about a frequency like gamma or infrared. Or is heat strictly bound to physical matter?
Can you tell about decomposition over Bernstein polynomials? Is it even possible?
its amazing that fourier dreamed this all up 200 years ago while napoleon was conquering europe.....there seems to be a tendency toward great intellectual discoveries when a nation is in the highest geopolitical ascendancy in its history
Bill ran from the giraffe toward the dolphin.
Brilliant!
One must be fond of people and trust them if one is not to make a mess of life.
Having to study this and Laplace transforms rn in school 😂
I'm a huge fan of Fourier's jelly for ten minutes.
This will only work as long as the PDE is linear, right?
What a legend
The integral you present is not, I'm pretty sure, equal to 0 for m != n. That is only true if m + n is even. For it to be true for all m!=n the boundaries have to be from 0 to π. In the next slide you have already crossed out the terms with m+n=2k+1. Observe the terms: it goes 1+5=6, 3+5 and so on... Because if this the mistake is hard to catch but when trying to prove the integral you present its pretty easy to understand. Just thought I'd point it out because u put so much work into the video! Hope it helps!!
what a genius
Cant see any bound between Fourier's lifestory and his maths solution. I dont mean that autor was wrong when added history to this video, but it need better connection of scenario parts.
It had been sixteen days since the zombies first attacked.
Today we gathered moss for my uncle's wedding.
Involved in the Reign of Terror.......imprisoned and survived prison?
So, he wasn't "involved" in the Reign of Terror but he WAS imprisoned during the Reign of Terror.
Why was this?
A intolerant and anticlerial quote is not the best way to start saying about math. However, the whole explanation was awesome.
Only problem i've seen with the video is it's assertion that you can derive fourier's law from newton's law of cooling. You can not, in the video he slipped in dT/dx instead of just dT, which is newton's original formulation, such a move is unjustified though
Fouriers Law for Cooling can & does include a dx term in the denominator on the right side, try a Google Search to see equations for Fouriers Law for Cooling.
Do a Google Search for Fouriers Law of Cooling; 7:44 there can be a dx term in the denominator on the right side of the equation.
Great content. The pronunciation is more like « Foorier ».
Crazy how times change. Today if you go to prison, you'll never get a job at a college or university.
That is a USA problem. Maybe visit the world. Learn something. Change your government.
@@Katchi_ Did you get triggered by something I wrote?
I just don't understand how anyone can come up with this
Never doubt human ingenuity.
Thinking about it. You know. That's how everything is done.
Amor fati and high self-efficacy
❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤
Who ate all the Pi's = 0
There's a message for you if you look up.
Sorry, but I'm sure the explanations were clearer when I studied Fourier 45 years ago in Electrical Engineering math at University, that or I'm just getting old.
The equation he wrote also isn’t homogenous
gil uh teen
Nope
why do you say "zee" and not "zed"? 🧐
It's standard in the US.
@@gaopinghu7332 yeah but he's clearly not speaking American English
Because it's one way to say it
@@yuseifudo6075 not in British English... Just funnily inconsistent, that's all