@@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.
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.")
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.
When I heard Ballade 1 I thought I had another tab open, but it turns out that you have executed the amazing idea of allowing us to ascend even more than when we follow along with the maths...
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 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...😅
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 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.
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.
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.
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?
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.
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.
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.
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
What a beautifull proof - thanks for showing it. From a retired Danish Physicist. Bravo. If we could only solve the serious problems in the political world today the same way.
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.
4 місяці тому
from morocco thank you for this complete clear solution
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.
@@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)|
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.
When learning any practical skills by repeating the rules harmonically to position cause-effect functions in numberness dominance sequences of positioning, (One Electron Theory Wave-packaging guesstimate), it is absolutely always NOW, e-Pi-i sync-duration instantaneously and the Universal state-ment of 1-0-infinity conformity to the Singularity-point Logarithmic Centre of Time Duration Timing, aka QM-TIME Completeness cause-effect Actuality. 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.
Is basically a convenient value that cancels out when you take the derivative with respect to a, also when f(a) at a=0 is equal to a integral form known 1/(1+x²)
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
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.
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.
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.
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.
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.
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.
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.)
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?
_"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.
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.
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.
A da is missing from the left-hand side of several of the steps. Apart from this, it’s pleasurable to follow the process.
You sound so honest and at the same time hilarious making the video worth to watch
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.")
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
I am glad you made the effort to write out every step! Awesome!!!
why is feynman zesty in all your video?
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.
I realized that I reached the end of the video...Feynman/Chopin - worked well! Many thanks!
I did this for a school project, I found the solution in a paper by Keith Conrad if anyone is wondering where
When I heard Ballade 1 I thought I had another tab open, but it turns out that you have executed the amazing idea of allowing us to ascend even more than when we follow along with the maths...
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."
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.
Double integration is my favorite method...
Yes, Sergio. The Chopin is perhaps too powerful. I'm guessing the pianist is Cyprien Katsaris. Sounds like him... love it.
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.
@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
I think this is the best method of solving the Gaussian integral!!
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.
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
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.
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.
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.
I quite enjoyed that. Well done 👍.
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!
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?
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.
Tricks on the blackboard are performed by a professional mathematician. Don't try to repeat them on the exam.
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.
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
Liked the music but perhaps a little quieter. I really enjoyed this presentation. Thanks.
What a beautifull proof - thanks for showing it. From a retired Danish Physicist. Bravo. If we could only solve the serious problems in the political world today the same way.
Delete the background piano Concerto
lose the music.
the piano background makes it hard to listen
Fascinating! Thank you
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.
from morocco thank you for this complete clear solution
Very easy to follow. Good job! Keep em coming!
Awesome, thank you!
ballade no 1!
feminist gaussian integral gets DESTROYED by based feynman's technique
Very well done and presented!!
Great lecture
Beautiful!
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)|
The technic is fantastic
You can just square and change it into polar coordinates
Lose the background music, it is very distracting.
I agree totally.
Regrettably, I can't listen because the background noise fries my brain.
Great work 👌👏💯
Thanks 🔥
Let's use the feynman technique : you explain the problem to anyone and then wooooaaaaa, you manage to solve it
I mean the other method is so much simpler. But this method is useful in other areas (in stats for example).
Would you not agree it's fun to see it solved by feynmans method compared to others?
I personally loved the background music, helped me concentrate.😊
Great, 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.
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.
Although logical It could be confusing for students .
What application is being used to write on?
Goodnotes on Ipad
Work of art
When learning any practical skills by repeating the rules harmonically to position cause-effect functions in numberness dominance sequences of positioning, (One Electron Theory Wave-packaging guesstimate), it is absolutely always NOW, e-Pi-i sync-duration instantaneously and the Universal state-ment of 1-0-infinity conformity to the Singularity-point Logarithmic Centre of Time Duration Timing, aka QM-TIME Completeness cause-effect Actuality.
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
Awesome vid. I don't understand how you arrive at the equation found @5:43. Like where does the (1+x^2) on the denominator come from
Is basically a convenient value that cancels out when you take the derivative with respect to a, also when f(a) at a=0 is equal to a integral form known 1/(1+x²)
5'8"????
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
Which becomes omega minus
do the limits not flip when we take out the -2*e^(-a^2) at 10:48
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
HELLO DUDE GUD VID I ALSO LIKE THE MUSIC KEEP GOING
Feynman's technique is nice and powerfull but this is not the best example in wich it is usefull ... 🖖
What would you say is?
Next e^((-x^2)/2)
That isn't any different. You just have a constant factor of ½ to correct for.
I think you have to use infinity because how can you truly know the square root of the irrational number π.
Best video I’ve ever seen
Please make videos on sieve theory
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
I like the music 😭😭
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
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.
By using addition -subtractaction -multiplication -division :the proton is quatum qubit as a effect of all msth
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.
So 1/1+x square is the connection
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.
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, but before destroying anything about Gauss they must first get near to him.
Feynman’a technique seems like it should be illegal.
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 suggesred that ∫e(-x²)dx=Constant 🌵🌵🌿
Brilliant.
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.
Bravo
Please take no offense, but there are moments where you sound like Joker from The Dark Knight trilogy
Good music
Fun fact that is the answer for (1/2)! Which is (√π)/2
Sorry for the music being a bit loud 😅
No worries, Ballade No.1, one of my favourites
Not ast all!
polar coordinate is much easier.
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.
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.
My brain hurts
I know from your accent you did with Maple ?
Haha what do you mean?
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.
intéressant, mais il y a beaucoup plus simple.