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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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.
Don't forget debussy tho
By the way, the pieces played were chopin's ballade n1 and debussy's arabesque n1 if anyone was wondering
I don’t mind the music but a softer volume would be nice.
Me too, I am a musician and a mathematician and it is hard to do both.
why is feynman zesty in all your video?
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.
One thing I never liked about this integration technique is that it is not obvious how to choose the auxiliary function. So it is difficult to teach to students, as they would just have to guess at the auxiliary function (or memorize examples) and hope for the best! Feynman has written that he learned the method from a 1926 math book by Frederick Woods (Advanced Calculus: A Course Arranged with Special Reference to the Needs of Students of Applied Mathematics) that was given to him by his high school physics teacher. Perhaps there were enough examples in that text that Feynman knew a whole whack of sample integrals to solve with this method.
_"One thing I never liked about this integration technique is that it is not obvious how to choose the auxiliary function."_
Lol. One thing which is inherent to mathematics is the fact that there is no known technique or algorithm to find a proof, or, appropriately, a disproof, of any mathematical statement. Moreover, it was shown (by Goedel, Turing and others) that such an algorithm can by no means exist.
Feynmann bases this trick off of intergratong factors for differential equations. Look up a graph
A da is missing from the left-hand side of several of the steps. Apart from this, it’s pleasurable to follow the process.
This is brilliant! However it probably takes nobody less than Feynman himself to come up with the idea of introducing 1 + x^2 into the equation.
You sound so honest and at the same time hilarious making the video worth to watch
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
Terrific -- Clearn and natural and benign and adult manner and tone -- This 'style' -- (which cannot be just 'put on' for the occasion -- It comes from being completely comfortable with one's subject matter) -- is so important for the simple quality of 'effective communication' -- i.e. necessary for 'communication', as such. Puts the pupil at ease -- but also enables complicated ideas to be easily absorbed by a beginning pupil -- whether the pupil is learning mathematics or the staff of a corporation or voters going into an election or the troops going into battle. (As David Hilbert said -- the leading research mathematician of the XX century, "You don't really understand a concept until you are able to explain it to the layman.")
Delete the background piano Concerto
lose the music.
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."
It would be much better if you turn that noise in the background off.
That 'noise' is Chopin's first ballade. It is very distracting for someone who loves music to watch the video at the same time.
I am glad you made the effort to write out every step! Awesome!!!
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?
Gave it a try, but the music is very distracting. Using closed captioning helped, but not really worth the effort when I wanted to actually learn something Feynman today
I did this for a school project, I found the solution in a paper by Keith Conrad if anyone is wondering where
I quite enjoyed that. Well done 👍.
Yes, Sergio. The Chopin is perhaps too powerful. I'm guessing the pianist is Cyprien Katsaris. Sounds like him... love it.
I'll have to go through this is detail, but one thing is for sure, the polar coordinate method is far simpler for solving this particular integral.
I realized that I reached the end of the video...Feynman/Chopin - worked well! Many thanks!
Very nice job, nice alternative to the polar coordinate technique.
Tricks on the blackboard are performed by a professional mathematician. Don't try to repeat them on the exam.
Have a small question, isnt -a² (1+x²) = -a² -a²x² ?
He wrote -a²x²
Pls let me know if im right or wrong
so u are right even he is right
like
and he dint only take -a²x²
e^a*e^b=e^a+b
laws of exponent
join -2ae^-a^2*e^-a^2x^2
after joing u get what u wrote -2ae^-a² -a²x²
Fascinating! Thank you
Double integration is my favorite method...
Is it possible to get an analytic solution for arbitrary limits of integration i.e.: other than - to + infinity? I'm aware of numerical methods that converge quickly.
Very well done and presented!!
Lose the background music, it is very distracting.
Great, thanks!
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
Great work 👌👏💯
Thanks 🔥
Explained with the ease of a natural teacher. My Year 13 Further Maths students could follow this. Well done for making every step so clear. Bravo!
Wow, thanks!
I don't agree with that. A decent knowledge of integration is necessary to understand this, but the details of the easy calculations are shown step by step by step by step. Boring. Would have been a very nice 5 minute video.
@@zaphodbeeblebrox-fz5fh The comment was about whether my students could follow it. I doubt you know them as well as I do.
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The exercise demonstrates how i-reflection rotation reduces 0-1-2-ness in 3-ness to the method of reverse-inverted Pi-bifurcation function.
Physics combines nodal-vibrational point-line-circle strings and drum head mass-energy-momentum continuous unity-connection categorizations.
From self-defining experience of embodiment, we extract the information of In-form-ation substantiation holography vanishing-into-no-thing Perspective Principle.
Good practices.
Ok
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.
@18:31 some people use a and b and some people use s and t. Depends on whether the lower or upper limit is undefined
Beautiful!
I'm not exactly opposed to differentiation under the integral sign, but there are conditions that have to be checked if we're doing mathematics (rather than formal manipulations that may or may not make sense). I find it easier to keep track of the conditions for Fubini's theorem (although the conditions for switching integration & differentiation may be equivalent). Here we have a positive integrand (a e^{-a^2(1+x^2)}), so integrating out x & then a or vice versa will give the same answer.
'Destroy' is an over-statement. The approach seems to me no more elegant than the usual polar-coordinates trick.
I also suspect that knowing the answer already helped Feynman to find this path to calculate it. On the other hand, knowing the answer and having seen Feynman's solution to two other integrals (although I had also seen one of those presented using Fubini's theorem), I still had no clue what the trick would be in this case. I tried some ways of writing e^{-x^2} as an integral, but nothing helped.
Sergio that IS NOT Chopin's music, it is Debusy. Good choise, good taste.
It s Chopin’s Ballade No.1 bro
Both composers are played: Chopin Ballade 1 followed by Debussy Arabesque 1
ballade no 1!
I think this is the best method of solving the Gaussian integral!!
Let's use the feynman technique : you explain the problem to anyone and then wooooaaaaa, you manage to solve it
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)|
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!
What application is being used to write on?
Goodnotes on Ipad
the piano background makes it hard to listen
Well done.
Loved the music.
It is slow because explanation are supposed to be slow, you can always fast forward it, but a fast presentation will be difficult to understand for many, with no easy fix.
from morocco thank you for this complete clear solution
The technic is fantastic
Very easy to follow. Good job! Keep em coming!
Awesome, thank you!
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.
Work of art
Brilliant.
Bravo
Liked the music but perhaps a little quieter. I really enjoyed this presentation. Thanks.
You chose this exact substitution because you know the answer in advance. With another substitution you will not get the same result.
Feynman chose it, because he was Feynman. The rest of us are just along for the ride.
So 1/1+x square is the connection
Please make videos on sieve theory
I think you have to use infinity because how can you truly know the square root of the irrational number π.
I'm not a mathematician. Is it appropriate to use height of human adults as an example of a normal distribution which is symmetric about the peak, and extends from negative to positive infinity. Height is restricted to positive numbers, and is not symmetric about the peak. Furthermore, humans have two sub-populations: men and women who have different average heights. Is it appropriate to lump them together? Also, for many of us, it's impossible to learn math and listen to music at the same time, please pick one or the other.
5'8"????
HELLO DUDE GUD VID I ALSO LIKE THE MUSIC KEEP GOING
Although logical It could be confusing for students .
Which becomes omega minus
My brain hurts
20:35
not sure if that step is valid.
the numerator inside the integral is NOT exp[ 0 x constant ], it is exp [ zero x infinity ]
gauss is rolling in his grave, to see mathematical rules treated so cavalierly.
At 19:57, you say f(t) = 0. Obviously as t goes to infinity, e^-(t^2) goes to zero. It appears to me that makes f(t) = improper integral of 0dx which is C, not zero. If I am missing something, please explain. Loved the video and subscribed.
Of course you usually cannot draw the "lim" under the integral, so
lim [t→∞] ∫ f(t, x)dx
will not be equal to
∫ lim [t→∞] f(t, x)dx
in many cases. f(t, x) has to converge _uniformly_ to zero for all x to make this work. And even this might not be sufficient if the integral does not converge absolutely. However, absolute and uniform convergence can easily be confirmed in our present case.
perfect choice of music for me, volume is good too (headphones). But I've been annoyed beyond what is reasonable by other choices 😅
And superb content
You suggesred that ∫e(-x²)dx=Constant 🌵🌵🌿
Feynman's technique is nice and powerfull but this is not the best example in wich it is usefull ... 🖖
What would you say is?
I personally loved the background music, helped me concentrate.😊
Best video I’ve ever seen
Next e^((-x^2)/2)
That isn't any different. You just have a constant factor of ½ to correct for.
I know from your accent you did with Maple ?
Haha what do you mean?
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
Interesting, though seems long winded as presented here. It is understandable. Makes me wonder how this approach was discovered.
I'll check out Leibnitz and other methods.
You differentiate wrt a, x constant. At the same step, you integrate wrt x. But if x is constant, then dx = 0. Therefore, the integral is also 0, which is not how you continue the derivation. Have I misunderstood something?
What's "wrt"?
@miloszforman6270 "with respect to"
_"Have I misunderstood something?"_
Yes. Probably you mean the operation at 6:54. Diffentiating with respect to "a" and integrating with respect to "x" are two different steps. This sequence is reversed at 6:54 ("we're going to bring the derivative inside of the integral"). This reversal is allowed in many cases, that is, if the function in question has a sufficiently benign behaviour. You can construct weird cases where this does not work, so we have to take care.
By using addition -subtractaction -multiplication -division :the proton is quatum qubit as a effect of all msth
I like the music 😭😭
👏👏💯🔝👋
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
that's how cults work.
the cult identity is either attached to individuals being credited for mundane shit, or attached to complete nonsense. this is an example of the former in mathematics. an example of the latter in mathematics would be the claim that 1+1=2 is universally true despite the fact that fraction addition, polynomials, unit conversions, etc. all exist.
@@sumdumbmick (If you de-press the 'Shift' key -- as you depress an alphabetic character on your keyboard -- you will be able to display a 'block capital'. If you put capitals a the beginning of your sentences -- the man reading your post will be more likely to read the entirety of what you have posted -- because they will be less likely to assume that you are not a naive adolescent without a clue about life.)
Wait, you knew it wasn't his but you falsely credited him on purpose?! Wow.
Slick...
I just wonder about the odds of having zero height o some other height near negative infinity. We were told of the mean being 5 foots and 6 inches but we are not sure of the standard deviation before we proceed with our own calculation. We remain eager to see your follow up video on that normal distribution of height.
You need to prove properly that as a tends to \infty: lim int_0^\infty exp(-a^2(1+x^2))/(1+x^2).dx tends to zero.
Yeah, but that’s easy
this is a demonstration of technique using an example, doesn't need a full rigorous proof - imo
Both the Lebesgues dominated convergence theorem or the monotone convergence theorem does the trick.
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
Please take no offense, but there are moments where you sound like Joker from The Dark Knight trilogy
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
Not ast all!
Fun fact that is the answer for (1/2)! Which is (√π)/2
Ну 2
I find the music refreshing
Wrong. You have to use Leibniz’s rule when you move the derivative to inside the integral.
He is using leibniz’s rule.
suggestion:
the only people who will be interested in this are likely to have at least one year of university calculus. however, your presentation is done at the level of quite introductory calculus.
that's a mismatch. cut out some of the really basic stuff.
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.
@@LactationMan Thanks -- (Mr "Uphold justice" may be 'autistic' -- They lack restraint -- Everything that departs from their habitual way of doing things is an affront.)
ADDING MUSIC IS OBNOXIOUS. THUMBS DOWN, DO NOT RECOMMEND
I just need to write more to make this point. If I want to watch a video about math I can add my own music. Do you not comprehend that some people are incapable of hearing human speech while music is playing? That the music takes over their attention? Why TF do you think you can make such a unilateral choice for all potential viewers of your video? You unbelievable a**h**e and f*** you for wasting my time having to write this comment. F***ing idiots, why do you always just do the same thing. Music is soso so so so easy. I can choose my own music if I want to
@@dashxdr😭😭😭😭oh lord
Thats a lot my guy.
I am sorry that you got such an obnoxious comment, a real offence. Your video ist super, but I have to add that I am an extremely auditive person. I can not help but listen to the Chopin piece. So for me, it is quite demanding to share my attention between your exposition and Chopin. Perhaps it would be better to choose some elevator music .... (?) Best regards and thank you for your excellent presentation.
Infinity is not any specific value, being unknowable unknown. No actual negative quantity nor positive quantity ever exist anywhere in the physical world, and there can never be 2 infinities as how human absurd invented notions of negative infinity and the positive infinity have been used in the invented notions of calculus There is no starting point nor ending point in the infinite space of infinity. Hence, whenever you have a defined starting point, then you will be able to get to only a finite point of value, but not going anywhere beyond it , and you cannot actually use infinity as the boundary of your integral evaluation.
It is a malpractice to use such an unknowable unknown value as infinity as a specific value in calculus, and it is even more absurd to claim that you can evaluate from negative infinity to positive infinity, while human subjective understanding can understand only values within the range of quantities [0, 10^31]. O is the smallest notional or the starting point of every value, but it is not any sort of actual quantity, and 10^31 is the max value which human extremely limited knowledge can understand. All values start at the origin 0 of human -invented notion of co-ordinate systems can only go to some limited or finite points of quantities which are beyond 10^31 or human subjective understanding, but not to infinity as falsely claimed by ignorant human mathematicians.
None of human -invented notions of mathematics can be applied when there is no boundary of a limited volume of space as the infinite space of the universe. Human -invented notions of space dimension and direction are applicable only within a well -defined volume of space. It means that whenever a co-ordinate system is used, there can be only limited finte range of quantities, but not going to infinity, while the starting point is always definable, and there is no boundary, or no begging nor any end in the infinite space of the universe or in infinity. The only form of matter which can be infinity is the infinite space of the universe, and all other forms of matter have to be finite, in order to be within and a part of the universe, but not that this value can go to negative infinity, that value can go to positive infinity as ignorantly claimed in human -invented notions of mathematics
Integral of f'(a) should have been written as Ṣ f'(a) da.!!!@