The Gaussian Integral is DESTROYED by Feynman’s Technique
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- Опубликовано: 11 июн 2024
- In this video I demonstrate the method used to solve the Gaussian integral using Feynman’s integration technique, I was very excited to present this video as it combines 2 of the math world’s favourite internet concepts, the Gaussian integral and Feynman’s integration technique.
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why is feynman zesty in all your video?
Good. But as a musician i suggest to turn off music. I cannot resist to pay attention to how Chopin Is played..
Fully agree with your observations. So did i
Thank you for naming Chopin.
Who told/suggested Feynman to use exactly THAT particular f(a)? Of course, he used that function because he knew already the result of the integral. Definitely a tricky technique (like most of Feynman's ones).
He must have had the plan/idea of solving for I2; the integral chosen looks very similar to other integrals for Feynman’s trick involving exponentials-with the exception of the extra term (1 + ..), which was used to solve for I2
I find the calculation of this integral by using polar coordinates much more elegant. Debussy's Arabesque as music in the background is nice.
It uses more substantial theory though: 2 dimensional transformation rule.
@@winstongludovatz111 You just have to consider the square of the Gaussian integral
and then use polar coordinates to get an elementary integral.
@@renesperb Polar coordinates operate in two dimensions and the corresponding integral is two dimensional only its value is the square of a one dimensional integral.
@@winstongludovatz111 The square of the Gaussian Integral can be written as 4*integral over (0 , inf ) x (0, inf) of Exp[- ( x^2 + y^2)] = π / 2* integral of Exp[ - r^2 ] * r , where 0 < r < inf. , but this integral is just 1/2 of - Exp[- r^2 ] from 0 to inf , =1.
@@renesperb That's only half the argument. The other half is the evaluation of the two dimensional Gaussian integral where you switch from Cartesian to polar coordinates and that uses the two dimensional transformation formula which is a whole lot less trivial than the Fundamental Theorem of Calculus.
You sound so honest and at the same time hilarious making the video worth to watch
A da is missing from the left-hand side of several of the steps. Apart from this, it’s pleasurable to follow the process.
I am glad you made the effort to write out every step! Awesome!!!
I personally would like this video without music, as a musician i find it annoying. my brain keeps telling me to listen to the music.
I think for non musicians the music makes the video much more enjoyable; dead silence as he thinks would be pretty awkward. If it truly bothers you, you can download the video and use a background music isolation AI tool online to remove it which should only take a couple mins.
Not a serious musician but I also find the piece too "rich" and the volume too high. Maybe something less complex like 1600 slow pieces instead of Listz-like stuff and a little less loud.
@@blabberblabbing8935 It's Chopin Ballade No.1. I guess the creator likes Chopin. Maybe he could choose something like Nocturne or Mazurka from him which is also fascinating.
@@catfromlothal8506 Oh my bad. Didn't sound at all like a ballad... or maybe I just acknowledged it when it went all crazy distracting fast tempo...
If anything I would rather have simple stuff like Pachelbel canon and things that don't get in the way... or my way...😅
Noted @@catfromlothal8506
Very nice job, nice alternative to the polar coordinate technique.
I realized that I reached the end of the video...Feynman/Chopin - worked well! Many thanks!
Nice application of the Feynman technique.
The background music sounds strange and is a distraction under accelerated playback, so maybe it can be omitted for future videos.
I don't hear any music
Omitted "from" future videos. Why does "for" suddenly have to be the all-purpose preposition?
@@TimKozlowski-bp5tg It's in the background
@@ericnorwood652 intended meaning is the same as "...so maybe for future videos it can be omitted."
Hi! That was a great video. I had a question @ 5:19, How should one go about selecting what function to use if they're trying to solve an integral for the first time with feynman's technique?
I quite enjoyed that. Well done 👍.
Beautiful!
It is amazing that someone would keep playing with that until you get to the answer. I'm impressed. I think the 3D version is much easier to grasp, using infinitesimal rings, but this is more impressive in some ways.
That Feynman was one clever dude. 😀
Totally agree!
Time very well utilized watching your program. Now to put it on paper and see how far I understood u.😮
I agree. One time for maths, one time for Chopin's ballades.
I did this for a school project, I found the solution in a paper by Keith Conrad if anyone is wondering where
I think this is the best method of solving the Gaussian integral!!
Great work 👌👏💯
Thanks 🔥
ballade no 1!
Can Feynman's technique be applied to any integral? if not, what are the conditions for it to be applied, please?
If you can define the function of the paramater to be differentiable, then you can use it. Feynmans technique it's just differentiable under the integral sign, also know as Leibniz rule for differentiation under integral sign: If you have a function f(x,t), any differential/integral operation and their composition commute.
@@rajinfootonchuriquen I think the question is how you can find an auxiliary function like f(a) that will help to calculate the integral.
@@tommyrjensen that's only guessing. It's like asking Which technique of integration should be use? Integration it's not like differentiation, doesn't has a algorithmic "fit all" solution.
@@rajinfootonchuriquen It does not always seem like guessing. Like if an integrand is a product of two functions of which one is easy to differentiate and the other is easy to integrate, then you use integration by parts. If the integrand is a composition of functions, you use substitution. And so on. If "Feynman's technique" is useful at all, how would it not be possible to determine when and how to apply it? Doesn't seem to make sense.
@@rajinfootonchuriquen But there are conditions, you can't switch differentiation and integration for any f(t,x). I'm not familiar with the Leibniz rule, but the similar theorem in measure theory requires differentiability for a.a. x, measurability in every t and the existence of an integrable g(x) s.t. |d/dt f(t,x)|
Double integration is my favorite method...
Great, thanks!
The technic is fantastic
What application is being used to write on?
Goodnotes on Ipad
You 'only' need to guess the right auxliary function to integrate and 'just know' that (arctan x)' = 1/(1 + x^2). Yes, yes, differentiating inverse trig functions is nothing compared to guessing convenient auxliary problems to solve. I'd call it: Gaussian integral made even more difficult. 😁But hey, a very nice video.
Then what method do you think is easier
Knowing the derivative of arctan is a standard result so yeah you're supposed to just know it or at the very least look it up in an integral results table. It's like integrating 1/(x+1) for example, you could waste time going the long way around or just say its Ln|x+1|. If you want to integrate the 1/(x² +1) function you use a tan trig substitution, it's just long so I skipped over it. Also nearly every method I've seen on solving the gaussian relys on "just knowing" to do certain steps, I understand it can be frustrating if certain steps aren't intuitive
@@lol1991 If you are a mathematician the result is obvious to you. If you are a physicist you'd probably prefer polar coordinates trick. Changing coordinates is bread and butter for physicists. If you are a student you're always screwed.
@@Jagoalexander Right, if the steps were intuitive we wouldn't be talking about Gaussian integral, so frustration has no place here. I am not complaining. Some people surely enjoy it more when they are taken deep into the woods and suddenly arrive at a solution.
@@WielkiKaleson
Yep, physicist here, I much prefer polar coordinates. Feels very natural compared to this mess.
This method is NOT called "Feynman Integration" , IT'S CALLED *Leibniz Integral Rule* .
Gottfried Leibniz DISCOVERED THE RULE,
Feynman POPULARISED IT.
THIS IS Leibniz's technique, NOT FEYNMAN'S.
GIVE THE CREDIT TO THE RIGHT PERSON FOR GOODNESS SAKE.
Cry more nerd
This is true, I mean come on... Feynman was a physicist, we don't invent integration techniques.
Very easy to follow. Good job! Keep em coming!
Awesome, thank you!
I like the music 😭😭
Let's use the feynman technique : you explain the problem to anyone and then wooooaaaaa, you manage to solve it
Please make videos on sieve theory
Bravo
HELLO DUDE GUD VID I ALSO LIKE THE MUSIC KEEP GOING
Sergio that IS NOT Chopin's music, it is Debusy. Good choise, good taste.
It s Chopin’s Ballade No.1 bro
It's both. It changes in the middle. I find it very annoying.
I personally loved the background music, helped me concentrate.😊
Best video I’ve ever seen
Although logical It could be confusing for students .
I know from your accent you did with Maple ?
Next e^((-x^2)/2)
That isn't any different. You just have a constant factor of ½ to correct for.
It’s beautiful. A small correction, you need to state a>0, otherwise it does not follow the limit of u as infinity.
If I’m not mistaken, even if a 0 or
Yes this is correct, but then the working must be amended, you cannot just say u=a*x limit is infinity.
@@kostaskostas2470ohhh I see thank you
Quantum Field Infinities
Contradictory:
Quantum Field Theory
Feynman Diagrams with infinite terms like:
∫ d4k / (k2 - m2) = ∞
Perturbative quantum field theories rely on renormalization to subtract infinite quantities from equations, which is an ad-hoc procedure lacking conceptual justification.
Non-Contradictory:
Infinitesimal Regulator QFT
∫ d4k / [(k2 - m2 + ε2)1/2] < ∞
Using infinitesimals ε as regulators instead of adhoc renormalization avoids true mathematical infinities while preserving empirical results.
So true man
With all due respects, kill the music. Maths is good.
Why mention Feyman? Differentiation under the integral sign was well known long before Feyman
Feynman didn't invent it, but he was known for using this method a lot.
Right but it became popular due to Feynman using it, if I remember correctly he discovered it in a class textbook and couldn’t understand why no one was using it as it is very powerful for certain problems
I find the music refreshing
Interesting, but before destroying anything about Gauss they must first get near to him.
Sorry for the music being a bit loud 😅
No worries, Ballade No.1, one of my favourites
The music is horrible, what the hey. Had to stop watching.