The follow-up to explain how this fits into the central limit theorem: ua-cam.com/video/d_qvLDhkg00/v-deo.html That video, in turn, benefits from a little prerequisite knowledge about convolutions, which I cover here: ua-cam.com/video/IaSGqQa5O-M/v-deo.html
5:08 just some feedback- I dont think you should display the integral expressed as a function of time and not explain that. Too often y’all are just too smart and drop big ideas like they are obvious. It is not obvious how your function can be expressed as x then swap out x for t. Maybe its just poor “math grammar” that science has adopted…idk - bc philosophically- if you need the integral function to be expressed as time (?acceleration) and the function is purely based on space, that is quite interesting
I can help you my friend I work as a voiceover ...I am arabian ...I can do voiceover. I will present it to you "free" as a thank you gift, in support of the channel
Grant, as a lowly college lecturer with insufficient funds to donate to your cause, I must nonetheless congratulate you on another masterpiece. Your visualizations are second to none and your teaching is beyond fantastic. Thank you for your contributions to mathematics.
I think pi gets really sad whenever e is not around. It's not just love. It's a rather complex relationship. No wonder, both of them seem to be a bit irrational. Especially e gets fouriously impotent when pi is not around, despite pi's negativity.
Grant, I’m a mathematician and math educator. I’ve of course seen the integration proof of the computation of the area under e^{-x^2}, but never in my life have I either seen or come up with such an elegant demonstration for why we MUST expect pi to show up in the Gaussian. Thank you, sir. Actually this gives me an idea for an in-class activity for my future analysis students…
1/sqrt(2pi) is also "incidentally" the normalization factor when you want to build an orthonormal basis of an infinite dimensional L2 space from the functions e^(ikx). This factor is what it takes to make the norm of each such element =1. And that's also how the area of the Normal gets calibrated to 1. So you could approach this from a number of angles... (Disclaimer, not a mathematician, but just sayin'.)
1:45 put such a wide smile on my face and reminded me why I love this channel so much. So often the explanation for why a thing is is entirely proof based. I love proofs, and coming up with proofs is a wonderful experience of problem solving, but on their own they cannot *satisfy* my disbelief. Stuff like this, using the concepts and *reasoning* /within/ the proofs to make a point, is exactly what I love about math. Thank you, grant!
I do think the goals of understanding and proof should be thought of as separate. Both are deeply important and sometimes they coincide. It's often incredibly fun and enriching to follow a good proof with the exercise of saying "right then, so where on earth did that come from?"
Well, it certainly makes learning this material easier. From experience, it was much more difficult to learn it only by textbooks, professors and interacting with other students. But grad school was one of my favorite ways to spend time.
Yeah. Using math to solve math does make it far less real feeling. You just have to take it at its face value. Trying to relate these numbers and concepts back to the real world is hard (which is basically the whole point of this channel) but is very satisfying to see.
@@3blue1brown While I'd generally agree (especially when cumbersome rigorous notation may hide the underlying principles of a proof), very satisfying proofs often seem to combine their concepts and the necessary notation elegantly and effortlessly. It is a bit like witnessing skilled craftsmanship -- using the exact right tool for a job leads to not only a beautiful result, but a very satisfying process to get there in the first place. Thank you very much for capturing that beauty!
Him saying "feel less out of the blue" at 22:10 after deriving the proof visually and in BLUE is like the 10th bonus point for this channel. I love it so much
Really .. the best source of the feeling of epiphany this channel is. People love to criticize the internet.. but 30 years ago, you would've had to pay a bunch of $ to aquire such regular epiphanies
Just watched the video again with your Korean AI voice. As a Korean, I'm genuinely blown away-it sounds incredibly natural! Imagining how this will broaden accessibility to your fantastic math content is truly exciting 👏
That we live in an age where educators have the opportunity to unpack the meaning and history behind some of the greatest mathematical discoveries for a substantially large audience is a privilege that we should all be infinitely grateful for.
I view mathematics as a series of signs you are following on a road map plugging in the integers needed to get to your destination and the visual explains why you need to understand these signs. This is a great way to teach...
I love everything about your videos. Your amazing animations, masterful scripts, pleasant and well recorded voice, tight editing. It all comes together to create some of the best educational content the world has ever seen. Thank you for sharing this for free and enriching the intellectual lives of so many people. ❤
This makes me think: a series on statistics would be excellent. I am sure there is a lot of visualisation behind all the sum of squares and F statistics, of interactions and everything, that are never taught. Even "serious" books barely talk about the intuition of the sum of squares beyond how they are derived from LRT.
Grant has said in the comments section of the other videos in this series that he anticipates making a stats series soon! These videos will be a part of it.
Even though I've seen this proof like a hundred times, it still brings a smile each time especially when animated as beautifully as here! And 3b1b did not disappoint, this really is a new perspective on it and I can't wait for the next video.
Great explanation, as usual! Minor historical correction: at 21:40 you say that Maxwell "independently stumbled upon the same derivation" as Herschel. The current scholarly consensus is that Maxwell read Herschel's paper and adapted his proof to the kinetic theory of gases, so it was not independently discovered. For details (as well as details that may be useful for your next promised video) see B. Gyenis (2017) "Maxwell and the normal distribution" in Studies in History and Philosophy of Modern Physics vol 57, doi: 10.1016/j.shpsb.2017.01.001 .
You know I actually learned the proof about spheres you mentioned by hand proving it for my harmonic analysis class last semester! I was hoping to be able to learn more about abstract harmonic analysis and/or wavelets when I set out in that course, but it actually wound up being a very classical class, and featured many diversions on historical analytic number theory, special functions, and computational techniques. It was not at all what I was expecting at the outset, but it was fascinating! Also, I'll mention that I was one of the people who got to meet you at JMM, and got a picture! I've been following your content for years, and you're still such an inspiration to the way that I communicate math to non-mathematicians, even if I know most of the content that you post these days! I still find it very interesting and always a good review though :)
YES YES YES! Thank you so much! When I took statistics, I always had this sense that I was missing something, because I never had the same intuition for it that I did for other areas of math. Something about this video just made everything click. Keep up the good work!
This video is breathtakingly marvelous! You elegantly answered several lingering questions that I thought I held separately and brought it all together in a visually stunning package. It was like a gift from heaven. You remind me of why I love mathematics so much. I'll be rehearsing this lesson in my mind for years to come. You've earned every penny of Patreon support I've given to date on this one video alone, and I have enjoyed so many others. Many, many thanks for sharing your extraordinary gifts with us.
you are part of the reason why the world will be smarter and achieve more. the impact you and your channel will have on people. I wish I had this growing up
I have done engineering from a very lowly college but still, my engineering math teacher was so succinct in teaching exactly how you have taught with animation at that time I didn't care enough but now watching your video just made me realise there are good teachers in every corner but we just pass them and don't appreciate their hard work. Your channel always helps in learning new things and re-learning what is hidden inside our minds. Thank you so much for your contribution.
These gaussian integrals are all over the place in quantum field theory! My QFT homework also taugt me a fun relationship between the area of the unit sphere in D dimensions and the Gamma function. 2 pi^(D/2) / \Gamma (D/2) . Would love to hear your thoughts on renormalization sometime that stuff is WILD. The Zeta function even shows up sometimes. I've started to make my own visualizations for physics and I find them so helpful. Great video!
The problem at the end was a delight to solve honestly. Part 2 was a really fun extension of the initial trick in the video. Thanks for sharing it. The video itself was also equally great.
I think this is the best video you've ever made. I absolutely love the distinction you make between proofs that are beautiful (slick, elegant, clean) and proofs that provide intuition (clearly use the assumptions that they start with). I can't wait for the finale!
I absolutely love you and your videos. I'm a math and computer science major. Between the statistics of building neural networks and the math of fourier analysis that I've been studying in detail in school, these last three videos have given me such a synthesis and crystallization. Everything is connected, and math is the language that describes those patterns!
Great video. I love the storytelling in this video. To me, one of the fun parts of working in the field of math, is the social aspect of sharing knowledge and the explaining and reasoning which gives me joy, and the pictures of the people talking in a café adds to that feeling. I have a freind, who everytime we meet, demands me to explain some math to him, so that he can get his mind blown. This video made me think of that.
I love how this channel is a gateway for amateur mathematics enjoyers (like me) to gain an intuition on a topic and be able to contextualize and explain these (admittedly rather complex) concepts. I, for example, am a high school student who loves learning about math but my teachers refuse to talk about anything other than basic high school math and give us hundreds of pages of excersises to mechanically compute. On the other hand, I cannot really understand the dense mathematical textbooks for university-level students as firstly, I do not even know the motivation or reasoning behind the proofs and derivations therein, but I also do not understand what the notation means in practice. However, your videos provide the bridge that lets an uninitiated, amateur mathematician understand and learn about these things. For example, after watching your calculus series like 3 times I finally understand the stuff in calculus textbooks and I could follow the proofs and the rigorous definitions there. Therefore, I think it’s no underestimation that you have changed the course of the most mathematics enthusiasts’ lives, including me, of any person ever. Thank you for your hard work!
It's teacher appreciation week, and I must name you one of the best teachers I have ever had, Mr Grant! Your calculus and linear algebra series opened doors for me, thank you so so much!!
Visualization is without doubt, the tool for learning mathematics ...enhanced understanding ,Intuition Development ,Improved Retention and engagement and motivation...outstanding Job !!!
I keep finding in math that learning the history of how an idea was discovered serves to illuminate the idea in general, and makes the "modern" version of that idea so much more satisfying as well as easier to work with. Thank you as always.
If anyone is wondering how b^x can be written as e^(cx), it can be done because b^x = e^(ln(b^x)) = e^(x*ln(b)) = e^(x*c) = e^(cx), where c = ln(b) *Edit:* If c is negative, it implies that b lies between 0 and 1.
@@nothinginteresting1662 actually that was pretty easy but I am grateful If you are okay with solving my doubt which is. 14:44 how the hell f(x,y)=g(x)h(y)
@@im-Anarchy g(x) and h(y) are distributions of x and y respectively. Then it is shown that g and h are equivalent because there is no difference after switching the axes. The reason for having g(x)h(y) is that x and y are independent of each other. If they were not, it would have reduced to a single variable function in either x or y.
@@nothinginteresting1662 that's ok now let me ask a personal question are you a student or working profesinal , entrepreneur. are you successful and happy, because I want some career guidence
The Herschel-Maxwell derivation is also a very nice justification for why Gaussian convolutions are separable (i.e. you can apply a gaussian blur to an image by blurring it only in the horizontal direction and then blurring that vertically).
Yeah that was fun. I remember me skipping class a lot in Uni and just cramming double integrals without going to class just trying to solve the exercises. One of them was triple integral for sphere volume and i had no good parametrisation with nice bounds in mind. And then i remembered a book about a tank crew positioning the turret by rotating the canon with 2 angles for azimuth and zenith. That was when i naively discovered polar transforms and even funnier I just naively represented x y and z as terms of R alfa and beta then pretended I did a variable change and just calculated dx dy and dz plugged in and somehow got the right result. Many years had passed until i found out the jacobian from algebra was actually not put there to torture us😂
i wish I could be a mathematician like you. The ability to intuitively explain these stuff is very uncommon and it really stems out of your admiration and curiosity towards the subject. Thank You!!
This has actually made me understand statistics better than my university when discussing Radiation measurements. It wasn't alone as I went out of my way to look up further information beyond the supplied reading material, but it definitely helped and motivated for me to do such, thanks for that.
I’m glad you put “beyond integral tricks” in the title, otherwise I would have scrolled past lol. I didn’t know about Herschel’s derivation. This is a great video, thanks for making it!
Ohhh i just realised that that's why the maxwell boltzmann distribution is what it is. I like this a lot more than the proof we saw in uni, which iirc used quantum mechanics (which maxwell didn't know about). Very cool thanks
Again, I'm blown away with how Grant is all about how someone could rediscover maths and is suspicious of tricks in doing maths. Without doubt, the best maths channel on UA-cam. He just gets what learning maths is all about: how would someone rediscover for themselves why something is the way it is- to feel the maths in their bones, not just remember tricks. Just brilliant.
I LOVE THIS CHANNEL!! I have always wondered what pi was doing there. Thank you so much Grant for throwing back the covers. You truly have reawakened my wonder for Math
5:32 One way I’ve solved the integral of e^(-x^2) is to use the Taylor expansion of e^x (this was during my university days). As mentioned, the anti derivative is non-elementary
I was this week reading a paper on how to create neural networks that outputted a measurement and a covariance. And that pi was really something weird in the loss function that I could not understand. Thanks for making math more bearable!
This is weirdly relevant for me, cause I'm doing a course on Statistical Mechanics and we're dealing with this exact problem of deriving the gaussian distribution. Thanks a lot 3b1b!
@3Blue1Brown please post a video about the different areas of math and how someone with a basic understanding of calculus can proceed to self-learn various advanced concepts in mathematics like real analysis, etc
From the moment I first learned about the fascinating concept of the Gaussian Integral, I had hoped that 3Blue1Brown would produce a video on the subject. And true to form, the video they created was truly exceptional. Grant's expertise and presentation style were truly captivating, and I am grateful for the opportunity to have learned from such a talented educator. Thanks Grant.
I really appreciate that you elaborate some trivial things, as, for example in 8:40. You spell out exactly, what taking an antiderivative value at point \inf means. Your manner of spelling things out in concise yet meaningful way is extremely helpful Keep up your incredible work, Grant!
I love this, I made a project about volumes of spheres in higher dimensions in the past, and watching this video added a lot more clarity and understanding! Ty for the great content!
Thanks are just a little word to describe how helpful your videos are. I learnt mathematics when I started following you. Now, I am blessed to be able to understand this video, which is more than 25 mins. While in the past, I was not even able to understand why we sum or take a root of a variable. Thank you, Grant.
23:25 "The Central Limit Theorem which is all about adding together many different independent variables." One interesting artifact of the CLT is the requirement of adding the variables. If, for instance, you multiply the variables together, you get a lognormal distribution instead. You can see this for yourself by calculating the expected probabilities of two dice whose result is multiplied. Therefore, the CLT is most applicable for data that varies by only a single order of magnitude. If you add two dice, the largest result is only 6X the smallest result which is inside a single order of magnitude, but if you multiply the dice, the largest result is 36X the smallest result. So one way to determining whether your data really follows a normal distribution is to calculate the "order of magnitude" present in the data. Because the normal distribution decays so quickly it can't model widely varying data very well and you will likely need a fat-tailed distribution. This has bitten many "experts" when the probability of extreme events is much larger than what is implied by a normal distribution.
A vaguely interesting little thing I realized a while back is you can write that as e^(-x)^(x) ... So there's a sense in which there's an exponential going in one direction, and then another going in the opposite direction. In this way, it feels to me to reflect 2 opposing exponential forces... Another way to look at it that is a little more circle-y is e^(i*pi)^(i*pi)... It works for matrices too (applying the linear transformation to rotate by 90 degrees twice yields something gaussian), and of course the infinite sums, but I guess that's all just kind of just recapitulating definitions. I spent way more time than I should have trying to figure out if there's anything deeper there and I couldn't find anything though
@@ct---cp8li because if they are independent we can say that function f(x,y) is basically some function in x and some function in y clubbed together...really i too dont know exactly why but its kinda feel like it should be true will be glad hear if got any other satisfying answer...
@@ct---cp8li I couldn't figure out how f2 turned into g * h . Then I remembered this is probability, and that g and h are the probability of two independent events, so of course it turns into a product.
Hi Grant! I'm an Industrial Engineering junior who's interested in pure math and statistics and altough I'm comfortable with their applications, I lack most of the technical knowledge and cenceptual understanding to fully absorb the essence of the notions' origins and proofs in these fields. I was surfing on the web about natural logarithms of complex numbers, which somehow lead me back to the central limit theorem and I pondered upon the very questions you examine in this video-essay. It's so refreshing to find such an expert source with a very fluid and graphic teaching method, explaining all the miniscule details that I was specifically curious of and many more like them about such sneak and elusive concepts. You're a real gem! Thank you for broadening the minds and enthusiasms of millions like me!
I couldn't figure out how f2 turned into g * h at 14:58. Then I remembered this is probability, and that g and h are the probability of two independent events, so of course it turns into a product.
24:56 that's a few seconds of results that I never knew I needed - screen printed for later. Yes, I knew all this from err... 50 years ago but I never saw how it fitted together, so satisfying - thanks.
1:30 I remember when I saw for the first time that you could derive the formula for higher dimensional spheres in that way. It was in Peskin&Schroeder's book on QFT. They have this two-liner in the middle of a computation where they show that result, and I remember thinking why I'd never thought of that before.
It also pops up in introduction to statistical mechanics in the derivation of Boltzmann entropy of an ideal gas. In the microcanonical ensemble we coun't all the states that have the same energy or put otherwise are in the same shell of a multidimensional sphere. I like that this video exemplified that the Gaussian stems from radially symmetric and uncorrelated, which is just the ideal gas.
My compliments to the artist and the inspiration to use their beautiful watercolors to merge abstract mathematics and humanity. Beautiful artwork and wonderfully complimentary to the elegant graphics and insights.
I don't know if you have read the description, seems like it was mostly Ai generated, which isn't a bad thing but seems things are changing bit too fast
This matches Chapter 7.2 of Probability Theory: The Logic of Science by E T Jaynes. The chapter provides several motivations of Gaussians. So if anyone wants to anticipate the next video, I recommend reading it.
@3blue1brown As you’ve already talked about normal distribution, can you also talk about estimation theory (ex. maximum likelihood estimation, bayesian estimation) and hypothesis testing such as likelihood ratio test or wald test?
I watch those videos mostly because or the beauty of your explanations, I understand a bit of them but the holes in basic mathematics concepts I have after growing in a country with poor basic education will never let me fully appreciate the full extent of your message
Eugene Wigner's philosophical approach was very profound. The "unreasonable effectiveness of mathematics..." is an amazing paper to say the least ! Whenever I discuss math and the universe, the conversation always tends towards this result
Love what you do! It's great to have a mathematician ask the "Huh?" and why are these connected questions . In my studies the sense of wonder and sublime beauty of the connectedness (as well as original joy of the how of discovery by the original mathematicians) was wrung out of the process. You ask the "meta" questions!
Grant, you should do a video about the new proof that 2 high schoolers came up with for the Pythagorean Theorem! It’s in the news, easy to find and a great discover for to young mathematicians.
I'm so incredibly happy you're doing the probability videos. Been waiting for this series since first discovering your channel 5 year ago! The linear algebra one boosted my gpa by probably a full point lol
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
The follow-up to explain how this fits into the central limit theorem: ua-cam.com/video/d_qvLDhkg00/v-deo.html
That video, in turn, benefits from a little prerequisite knowledge about convolutions, which I cover here: ua-cam.com/video/IaSGqQa5O-M/v-deo.html
Still, why is the 1/2 in the exponent?
5:08 just some feedback- I dont think you should display the integral expressed as a function of time and not explain that. Too often y’all are just too smart and drop big ideas like they are obvious. It is not obvious how your function can be expressed as x then swap out x for t. Maybe its just poor “math grammar” that science has adopted…idk - bc philosophically- if you need the integral function to be expressed as time (?acceleration) and the function is purely based on space, that is quite interesting
I can help you my friend
I work as a voiceover ...I am arabian ...I can do voiceover. I will present it to you "free" as a thank you gift, in support of the channel
Grant, as a lowly college lecturer with insufficient funds to donate to your cause, I must nonetheless congratulate you on another masterpiece. Your visualizations are second to none and your teaching is beyond fantastic. Thank you for your contributions to mathematics.
raise this man to the top of the comments.
@@oszkarvarnagy7896 fax
Same
god damn ma man be talking like nepolean
@@jaw0449I’ll
I think pi gets really sad whenever e is not around. It's not just love. It's a rather complex relationship. No wonder, both of them seem to be a bit irrational. Especially e gets fouriously impotent when pi is not around, despite pi's negativity.
I feel like impotent is not a random word but another reference I am missing
@@david13579naranja potentiation is just another name for "raising a number to the power of another"
e^(i*pi)+1=0 is probably the most wholesome thing in the mathematical world
Their relationship truly transcends the ones of real life.
Bravo!
"Who ordered another dimension" 😂 classic mathematician path to solving a problem
Only PhD mathematicians have enough math money to order the 11 dimensions needed for string theory
@@InTrancedState We only get 3 and these elites get 11
Unfair
“Sir, this is a Wendy’s”
I'll have 4 dimensions with extra dip.
String theorists af
Grant, I’m a mathematician and math educator. I’ve of course seen the integration proof of the computation of the area under e^{-x^2}, but never in my life have I either seen or come up with such an elegant demonstration for why we MUST expect pi to show up in the Gaussian. Thank you, sir. Actually this gives me an idea for an in-class activity for my future analysis students…
1/sqrt(2pi) is also "incidentally" the normalization factor when you want to build an orthonormal basis of an infinite dimensional L2 space from the functions e^(ikx). This factor is what it takes to make the norm of each such element =1. And that's also how the area of the Normal gets calibrated to 1. So you could approach this from a number of angles... (Disclaimer, not a mathematician, but just sayin'.)
1:45 put such a wide smile on my face and reminded me why I love this channel so much. So often the explanation for why a thing is is entirely proof based. I love proofs, and coming up with proofs is a wonderful experience of problem solving, but on their own they cannot *satisfy* my disbelief. Stuff like this, using the concepts and *reasoning* /within/ the proofs to make a point, is exactly what I love about math. Thank you, grant!
fun times
I do think the goals of understanding and proof should be thought of as separate. Both are deeply important and sometimes they coincide. It's often incredibly fun and enriching to follow a good proof with the exercise of saying "right then, so where on earth did that come from?"
Well, it certainly makes learning this material easier. From experience, it was much more difficult to learn it only by textbooks, professors and interacting with other students. But grad school was one of my favorite ways to spend time.
Yeah. Using math to solve math does make it far less real feeling. You just have to take it at its face value. Trying to relate these numbers and concepts back to the real world is hard (which is basically the whole point of this channel) but is very satisfying to see.
@@3blue1brown While I'd generally agree (especially when cumbersome rigorous notation may hide the underlying principles of a proof), very satisfying proofs often seem to combine their concepts and the necessary notation elegantly and effortlessly.
It is a bit like witnessing skilled craftsmanship -- using the exact right tool for a job leads to not only a beautiful result, but a very satisfying process to get there in the first place. Thank you very much for capturing that beauty!
Who else came here just to listen Grant speak Korean?
Me!!
That'd be me, even tho it not being my native language
Yep ;)
Me (but I also wanna know why pi shows up in the normal distribution)
🙋♂️
Him saying "feel less out of the blue" at 22:10 after deriving the proof visually and in BLUE is like the 10th bonus point for this channel. I love it so much
Three parts blue 🔵, one part brown 🟤
Really .. the best source of the feeling of epiphany this channel is. People love to criticize the internet.. but 30 years ago, you would've had to pay a bunch of $ to aquire such regular epiphanies
Does your bully call you blue butt too?
what are the other 9
Just watched the video again with your Korean AI voice. As a Korean, I'm genuinely blown away-it sounds incredibly natural! Imagining how this will broaden accessibility to your fantastic math content is truly exciting 👏
I should inform you that this is AI lol
@@lifinaleWOW I think he had no idea when he wrote that
@@lifinaleYes… they said his AI voice…
@@Karlswebb they edited the post…
You all edited your posts.
That we live in an age where educators have the opportunity to unpack the meaning and history behind some of the greatest mathematical discoveries for a substantially large audience is a privilege that we should all be infinitely grateful for.
I view mathematics as a series of signs you are following on a road map plugging in the integers needed to get to your destination and the visual explains why you need to understand these signs. This is a great way to teach...
Hear hear
The Korean version sounds very natural. The pipeline works incredibly well!
More than perhaps any video in 3b1b, this one shows how learning math history makes one a better mathematician. What a great lesson!
The same beauty in math and history: the best part is the story behind the facts.
How true!
I love everything about your videos. Your amazing animations, masterful scripts, pleasant and well recorded voice, tight editing. It all comes together to create some of the best educational content the world has ever seen. Thank you for sharing this for free and enriching the intellectual lives of so many people. ❤
Finally, the much awaited 3b1b statistics series is on a roll!
I doubt anyone can possibly make a better visualization for explaining this proof. The quality of your videos is truly on another level
You can always add bombs and blisters :')
This makes me think: a series on statistics would be excellent. I am sure there is a lot of visualisation behind all the sum of squares and F statistics, of interactions and everything, that are never taught. Even "serious" books barely talk about the intuition of the sum of squares beyond how they are derived from LRT.
yess please, who else need more stats videos? +1 here
Grant has said in the comments section of the other videos in this series that he anticipates making a stats series soon! These videos will be a part of it.
Even though I've seen this proof like a hundred times, it still brings a smile each time especially when animated as beautifully as here! And 3b1b did not disappoint, this really is a new perspective on it and I can't wait for the next video.
Great explanation, as usual! Minor historical correction: at 21:40 you say that Maxwell "independently stumbled upon the same derivation" as Herschel. The current scholarly consensus is that Maxwell read Herschel's paper and adapted his proof to the kinetic theory of gases, so it was not independently discovered. For details (as well as details that may be useful for your next promised video) see B. Gyenis (2017) "Maxwell and the normal distribution" in Studies in History and Philosophy of Modern Physics vol 57, doi: 10.1016/j.shpsb.2017.01.001 .
Just wanna Say, Thanks for doing whatever you are doing. Never stop 3B1B
You know I actually learned the proof about spheres you mentioned by hand proving it for my harmonic analysis class last semester! I was hoping to be able to learn more about abstract harmonic analysis and/or wavelets when I set out in that course, but it actually wound up being a very classical class, and featured many diversions on historical analytic number theory, special functions, and computational techniques. It was not at all what I was expecting at the outset, but it was fascinating!
Also, I'll mention that I was one of the people who got to meet you at JMM, and got a picture! I've been following your content for years, and you're still such an inspiration to the way that I communicate math to non-mathematicians, even if I know most of the content that you post these days! I still find it very interesting and always a good review though :)
I love the pieces of art that aren’t normally part of the “aesthetic” of 3b1b but somehow still fit right in.
i was gonna say... also the light in that piece of art is something beautiful
I wonder if it's AI generated. Looks so to me, cause the hands are bad 😅
I thoroughly dislike the part where it says "aided by Midjourney"... yuck.
@@KernelLeak Midjourney is just like any other artist, it learns by looking at 1000s of examples
@@tobiascornille Well, I doubt there are too many paintings of statisticians explaining the Central Limit Theorem to their friends while at lunch...
It always blows me away not only how good your animations look, but also how well they underline the concept you are teaching!
YES YES YES! Thank you so much! When I took statistics, I always had this sense that I was missing something, because I never had the same intuition for it that I did for other areas of math. Something about this video just made everything click. Keep up the good work!
how do we know that we can factor the function f2(x,y) into the form of g(x)h(x) if we know that these variables are independent from each others?
This video is breathtakingly marvelous! You elegantly answered several lingering questions that I thought I held separately and brought it all together in a visually stunning package. It was like a gift from heaven. You remind me of why I love mathematics so much. I'll be rehearsing this lesson in my mind for years to come. You've earned every penny of Patreon support I've given to date on this one video alone, and I have enjoyed so many others. Many, many thanks for sharing your extraordinary gifts with us.
Thank you Grant
I live in a time period where I can see this in my hand from the comfort of my couch. World class, exceptional
I'm grateful!
you are part of the reason why the world will be smarter and achieve more. the impact you and your channel will have on people. I wish I had this growing up
I have done engineering from a very lowly college but still, my engineering math teacher was so succinct in teaching exactly how you have taught with animation at that time I didn't care enough but now watching your video just made me realise there are good teachers in every corner but we just pass them and don't appreciate their hard work.
Your channel always helps in learning new things and re-learning what is hidden inside our minds. Thank you so much for your contribution.
how do we know that we can factor the function f2(x,y) into the form of g(x)h(x) if we know that these variables are independent from each others?
These gaussian integrals are all over the place in quantum field theory! My QFT homework also taugt me a fun relationship between the area of the unit sphere in D dimensions and the Gamma function. 2 pi^(D/2) / \Gamma (D/2) . Would love to hear your thoughts on renormalization sometime that stuff is WILD. The Zeta function even shows up sometimes. I've started to make my own visualizations for physics and I find them so helpful. Great video!
Looks like the footnote at the end of the video can be generalized to non-integer dimensions
The problem at the end was a delight to solve honestly. Part 2 was a really fun extension of the initial trick in the video. Thanks for sharing it. The video itself was also equally great.
I think this is the best video you've ever made. I absolutely love the distinction you make between proofs that are beautiful (slick, elegant, clean) and proofs that provide intuition (clearly use the assumptions that they start with). I can't wait for the finale!
I absolutely love you and your videos. I'm a math and computer science major. Between the statistics of building neural networks and the math of fourier analysis that I've been studying in detail in school, these last three videos have given me such a synthesis and crystallization. Everything is connected, and math is the language that describes those patterns!
Great video. I love the storytelling in this video. To me, one of the fun parts of working in the field of math, is the social aspect of sharing knowledge and the explaining and reasoning which gives me joy, and the pictures of the people talking in a café adds to that feeling. I have a freind, who everytime we meet, demands me to explain some math to him, so that he can get his mind blown. This video made me think of that.
I love how this channel is a gateway for amateur mathematics enjoyers (like me) to gain an intuition on a topic and be able to contextualize and explain these (admittedly rather complex) concepts. I, for example, am a high school student who loves learning about math but my teachers refuse to talk about anything other than basic high school math and give us hundreds of pages of excersises to mechanically compute. On the other hand, I cannot really understand the dense mathematical textbooks for university-level students as firstly, I do not even know the motivation or reasoning behind the proofs and derivations therein, but I also do not understand what the notation means in practice. However, your videos provide the bridge that lets an uninitiated, amateur mathematician understand and learn about these things. For example, after watching your calculus series like 3 times I finally understand the stuff in calculus textbooks and I could follow the proofs and the rigorous definitions there. Therefore, I think it’s no underestimation that you have changed the course of the most mathematics enthusiasts’ lives, including me, of any person ever. Thank you for your hard work!
It's teacher appreciation week, and I must name you one of the best teachers I have ever had, Mr Grant! Your calculus and linear algebra series opened doors for me, thank you so so much!!
AI인걸 못 느낄정도로 어마어마하게 자연스럽네요.
억양이 살짝 단조롭다는 느낌은 들지만, 대본을 읽는 사람들은 보통 이렇게 말하거든요.
Visualization is without doubt, the tool for learning mathematics ...enhanced understanding ,Intuition Development ,Improved Retention and engagement and motivation...outstanding Job !!!
This was such a great video! Well paced, framed and explained. I only hope that the last part comes out soon. Excited!
I keep finding in math that learning the history of how an idea was discovered serves to illuminate the idea in general, and makes the "modern" version of that idea so much more satisfying as well as easier to work with. Thank you as always.
If anyone is wondering how b^x can be written as e^(cx), it can be done because b^x = e^(ln(b^x)) = e^(x*ln(b)) = e^(x*c) = e^(cx), where c = ln(b)
*Edit:* If c is negative, it implies that b lies between 0 and 1.
what's the time stamp
@@im-Anarchy Around 20:05 is when b^x is written as e^(cx)
@@nothinginteresting1662 actually that was pretty easy but I am grateful If you are okay with solving my doubt which is.
14:44 how the hell f(x,y)=g(x)h(y)
@@im-Anarchy g(x) and h(y) are distributions of x and y respectively. Then it is shown that g and h are equivalent because there is no difference after switching the axes. The reason for having g(x)h(y) is that x and y are independent of each other. If they were not, it would have reduced to a single variable function in either x or y.
@@nothinginteresting1662 that's ok now let me ask a personal question are you a student or working profesinal , entrepreneur. are you successful and happy, because I want some career guidence
The Herschel-Maxwell derivation is also a very nice justification for why Gaussian convolutions are separable (i.e. you can apply a gaussian blur to an image by blurring it only in the horizontal direction and then blurring that vertically).
how do we know that we can factor the function f2(x,y) into the form of g(x)h(x) if we know that these variables are independent from each others?
I love how you explained intuitively the Jacobi determinant for the polar transformation, you’re way of explaining math is amazing
Yeah that was fun. I remember me skipping class a lot in Uni and just cramming double integrals without going to class just trying to solve the exercises.
One of them was triple integral for sphere volume and i had no good parametrisation with nice bounds in mind. And then i remembered a book about a tank crew positioning the turret by rotating the canon with 2 angles for azimuth and zenith.
That was when i naively discovered polar transforms and even funnier I just naively represented x y and z as terms of R alfa and beta then pretended I did a variable change and just calculated dx dy and dz plugged in and somehow got the right result.
Many years had passed until i found out the jacobian from algebra was actually not put there to torture us😂
i wish I could be a mathematician like you. The ability to intuitively explain these stuff is very uncommon and it really stems out of your admiration and curiosity towards the subject. Thank You!!
Amazing video. Your explain things soo well. Maths seems so much fun with your videos
I don't know how this is possible, I was thinking about this thing right today, thank you 3blue1brown!
I go to this channel whenever I want to feel elevated intellectually! Great stuff on math!
This has actually made me understand statistics better than my university when discussing Radiation measurements. It wasn't alone as I went out of my way to look up further information beyond the supplied reading material, but it definitely helped and motivated for me to do such, thanks for that.
"the question raised by the hypothetical statistician's friend" - You heard it, even 3b1b thinks statisticians have no real friends.
Bro the Korean sounds so good. It is slightly low volume and muted, but overall very promising!
I’m glad you put “beyond integral tricks” in the title, otherwise I would have scrolled past lol. I didn’t know about Herschel’s derivation.
This is a great video, thanks for making it!
Your presentation style and graphics are absolutely outstanding!!! A true pleasure to watch and learn from! Thank you!!!!
Ohhh i just realised that that's why the maxwell boltzmann distribution is what it is. I like this a lot more than the proof we saw in uni, which iirc used quantum mechanics (which maxwell didn't know about). Very cool thanks
Love the user name
Again, I'm blown away with how Grant is all about how someone could rediscover maths and is suspicious of tricks in doing maths. Without doubt, the best maths channel on UA-cam. He just gets what learning maths is all about: how would someone rediscover for themselves why something is the way it is- to feel the maths in their bones, not just remember tricks. Just brilliant.
I LOVE THIS CHANNEL!! I have always wondered what pi was doing there. Thank you so much Grant for throwing back the covers. You truly have reawakened my wonder for Math
5:32 One way I’ve solved the integral of e^(-x^2) is to use the Taylor expansion of e^x (this was during my university days). As mentioned, the anti derivative is non-elementary
I was this week reading a paper on how to create neural networks that outputted a measurement and a covariance. And that pi was really something weird in the loss function that I could not understand. Thanks for making math more bearable!
Hi Miguel, that sounds like a very interesting paper, would you mind sharing the name?
@@yuhanmao6512 yes, please share
This is incredible. Thank you for explaining with such amazing clarity. It will be hard to top this.
Love the video! I really like how you explain such complex things so simply and in such short time!
This is weirdly relevant for me, cause I'm doing a course on Statistical Mechanics and we're dealing with this exact problem of deriving the gaussian distribution. Thanks a lot 3b1b!
Im here for the Korean AI dub
Man hold up, that's AI??
@@smirnoff_the_best_vodkayes, he said it in the post
wow. amazing.. so realistic
@3Blue1Brown please post a video about the different areas of math and how someone with a basic understanding of calculus can proceed to self-learn various advanced concepts in mathematics like real analysis, etc
I'm guessing Grant is building up to a 23 minute video explaining a function that describes everything everywhere
God.
all at once
@@BB-yi1oqalmost!
how do we know that we can factor the function f2(x,y) into the form of g(x)h(x) if we know that these variables are independent from each others?
From the moment I first learned about the fascinating concept of the Gaussian Integral, I had hoped that 3Blue1Brown would produce a video on the subject. And true to form, the video they created was truly exceptional. Grant's expertise and presentation style were truly captivating, and I am grateful for the opportunity to have learned from such a talented educator.
Thanks Grant.
These videos are timeless and will be used for generations to come
Hands-down perfect video. I am patiently waiting for the next one, it is going to be a revelation!
The Unreasonable Ineffectivenss of Spell Checkers (1:04)
Always wondered why the normal distribution is the way it is. The explanation in this video is extremely satisfying. Thank you!
Man it's so unfair that you didn't get a mathematical prize and recognition for creating this historical channel
I really appreciate that you elaborate some trivial things, as, for example in 8:40. You spell out exactly, what taking an antiderivative value at point \inf means. Your manner of spelling things out in concise yet meaningful way is extremely helpful
Keep up your incredible work, Grant!
The artwork is really good. I hope I can one day draw like that...and also the math is beautiful too
Sorry to say that it’s, at least in part, ai generated
@@l3gacyb3ta21 Actually, it isn't. Purely human.
. @sanderson84060 What leads you to that conclsion? The description claims it was made with aid from Midjourney
Great series!
This md art is an eyesore ngl
I love this,
I made a project about volumes of spheres in higher dimensions in the past, and watching this video added a lot more clarity and understanding! Ty for the great content!
Great video! Can you do a video on Bessel's correction and "degrees of freedom" in statistics in general?
Thanks are just a little word to describe how helpful your videos are. I learnt mathematics when I started following you. Now, I am blessed to be able to understand this video, which is more than 25 mins. While in the past, I was not even able to understand why we sum or take a root of a variable.
Thank you, Grant.
I'm glad you enjoyed, thanks so much!
The truly eye-pleasing art style at the beginning fits nicely with the glibness of the rotational symmetry of the function under consideration.
As a Korean, I deeply feel thank you. I found primal theory about integrating which continuing in my univ.😊
23:25 "The Central Limit Theorem which is all about adding together many different independent variables." One interesting artifact of the CLT is the requirement of adding the variables. If, for instance, you multiply the variables together, you get a lognormal distribution instead. You can see this for yourself by calculating the expected probabilities of two dice whose result is multiplied.
Therefore, the CLT is most applicable for data that varies by only a single order of magnitude. If you add two dice, the largest result is only 6X the smallest result which is inside a single order of magnitude, but if you multiply the dice, the largest result is 36X the smallest result. So one way to determining whether your data really follows a normal distribution is to calculate the "order of magnitude" present in the data. Because the normal distribution decays so quickly it can't model widely varying data very well and you will likely need a fat-tailed distribution. This has bitten many "experts" when the probability of extreme events is much larger than what is implied by a normal distribution.
Link for examples please
One of the best videos I have seen in quite some time. Loved the curious and pedagogical approach.
A vaguely interesting little thing I realized a while back is you can write that as e^(-x)^(x) ... So there's a sense in which there's an exponential going in one direction, and then another going in the opposite direction. In this way, it feels to me to reflect 2 opposing exponential forces... Another way to look at it that is a little more circle-y is e^(i*pi)^(i*pi)... It works for matrices too (applying the linear transformation to rotate by 90 degrees twice yields something gaussian), and of course the infinite sums, but I guess that's all just kind of just recapitulating definitions. I spent way more time than I should have trying to figure out if there's anything deeper there and I couldn't find anything though
how do we know that we can factor the function f2(x,y) into the form of g(x)h(x) if we know that these variables are independent from each others?
@@ct---cp8li because if they are independent we can say that function f(x,y) is basically some function in x and some function in y clubbed together...really i too dont know exactly why but its kinda feel like it should be true will be glad hear if got any other satisfying answer...
@@ct---cp8li I couldn't figure out how f2 turned into g * h . Then I remembered this is probability, and that g and h are the probability of two independent events, so of course it turns into a product.
Hi Grant! I'm an Industrial Engineering junior who's interested in pure math and statistics and altough I'm comfortable with their applications, I lack most of the technical knowledge and cenceptual understanding to fully absorb the essence of the notions' origins and proofs in these fields. I was surfing on the web about natural logarithms of complex numbers, which somehow lead me back to the central limit theorem and I pondered upon the very questions you examine in this video-essay. It's so refreshing to find such an expert source with a very fluid and graphic teaching method, explaining all the miniscule details that I was specifically curious of and many more like them about such sneak and elusive concepts. You're a real gem! Thank you for broadening the minds and enthusiasms of millions like me!
I couldn't figure out how f2 turned into g * h at 14:58. Then I remembered this is probability, and that g and h are the probability of two independent events, so of course it turns into a product.
Haha I paused at that point too before realising.
24:56 that's a few seconds of results that I never knew I needed - screen printed for later.
Yes, I knew all this from err... 50 years ago but I never saw how it fitted together, so satisfying - thanks.
1:30 I remember when I saw for the first time that you could derive the formula for higher dimensional spheres in that way. It was in Peskin&Schroeder's book on QFT. They have this two-liner in the middle of a computation where they show that result, and I remember thinking why I'd never thought of that before.
It also pops up in introduction to statistical mechanics in the derivation of Boltzmann entropy of an ideal gas. In the microcanonical ensemble we coun't all the states that have the same energy or put otherwise are in the same shell of a multidimensional sphere.
I like that this video exemplified that the Gaussian stems from radially symmetric and uncorrelated, which is just the ideal gas.
My gosh, this is just mind blowing. It just gives you a totally in-depth way to look into things.
My compliments to the artist and the inspiration to use their beautiful watercolors to merge abstract mathematics and humanity. Beautiful artwork and wonderfully complimentary to the elegant graphics and insights.
I don't know if you have read the description, seems like it was mostly Ai generated, which isn't a bad thing but seems things are changing bit too fast
The best explanation I've seen anywhere. Thank you for making this video.
This matches Chapter 7.2 of Probability Theory: The Logic of Science by E T Jaynes. The chapter provides several motivations of Gaussians. So if anyone wants to anticipate the next video, I recommend reading it.
Aside of the beautiful animations and very well-explained math, the art in this one was especially nice!
@3blue1brown As you’ve already talked about normal distribution, can you also talk about estimation theory (ex. maximum likelihood estimation, bayesian estimation) and hypothesis testing such as likelihood ratio test or wald test?
Really wonderful, both demonstration and your manner to explain is such a real beauty !
Thanks for that
please continue the differential equations series
I watch those videos mostly because or the beauty of your explanations, I understand a bit of them but the holes in basic mathematics concepts I have after growing in a country with poor basic education will never let me fully appreciate the full extent of your message
Eugene Wigner's philosophical approach was very profound. The "unreasonable effectiveness of mathematics..." is an amazing paper to say the least ! Whenever I discuss math and the universe, the conversation always tends towards this result
Love what you do! It's great to have a mathematician ask the "Huh?" and why are these connected questions . In my studies the sense of wonder and sublime beauty of the connectedness (as well as original joy of the how of discovery by the original mathematicians) was wrung out of the process. You ask the "meta" questions!
Grant, you should do a video about the new proof that 2 high schoolers came up with for the Pythagorean Theorem! It’s in the news, easy to find and a great discover for to young mathematicians.
Beautiful, elegant and easy to understand... congratulations 🎉
Listened to in Korean audio, can't understand it, but sounds sweet.
Good initiative 3b1b
Grant, I want to thank you wholeheartly for this. Your previous video and this one made it click for me.
Just beautiful....
I'm so incredibly happy you're doing the probability videos. Been waiting for this series since first discovering your channel 5 year ago! The linear algebra one boosted my gpa by probably a full point lol