It's the end of the day on Christmas day, and I'm watching a maths video. It is honestly so incredible that I've finally reached a point where I can understand all the interesting stuff I've been seeing for months on the maths side of UA-cam, now that I've properly studied it. Thank you for videos like these, they're the reason I love maths so much.
I find you very expressive, like my maths teacher never used to move the hands, speak in an enthusiastic way, rather he had a machine voice and was dead serious . I discovered you today and I really admire your style of teaching
I think this is one of the first math result where my eyes legitimately widened when the final box was drawn around the n! equation (and I'm a final year engineering student so I've seen plenty of neat results!) Thanks for the crystal clear explanation Tom - I had seen this formula before but the derivation really made it click. All the best from McMaster University in Canada.
the way he makes calculus interesting and not boring to a high school kid that hasn't done anything past the required algebra classes is amazing. I literally understand nothing that is happening but it must be really cool if you understand it!
This is great. I feel like he talks about this very briefly in 'Surely You're Joking, Mr Feynman' but I've never seen an explanation of it that I could particularly get my head around until now. Thanks!
What a way to link teaching a new integration method with something interesting in math. I had heard of the Gamma function and its sibling before but had never quite understood how they were exactly connected to factorials.
One of my favorite "differentiation under the integral sign" problems is sin(x)/x, from 0 to infinity. You do it by defining I(a) = int sin(x) exp(-ax) / x. Then differentiate w.r.t. a, which gives a closed-form for I'(a), which you then integrate to find I(a). Finally set a = 0.
My teacher of my tution during my school actually taught this and never told us it was feynmann's technique, he just said it's a better of doing integral that are similar to integrals you know already. He also said that it won't be asked in exams but still learning it would be helpful.
I loved your the explanation to this example. I’m sure any electrical engineers or maths majors watching will immediately think of how similar this example is to the Laplace transform, which uses similar proofs for some of the standard transforms.
Here's an even slicker way to integrate x^n exp(-x): Let a be a parameter with -1 < a < 1. First, integrate exp((a-1)x) dx from 0 to infinity, and expand the result as a power series in a (it's a geometric series). Then, expand exp((a-1)x) as a power series in a, and integrate the result term-by-term dx. Now compare coefficients!
This is beyond my understanding and education level, but you're such a good presenter that I'm happily watching nonetheless. Have to say, I greatly admire you and your enthusiasm in communicating maths! While I struggle to follow along with maybe even most of your videos, some of them have nonetheless made it click for me and I've come away with more understanding than I had before, and that is deeply appreciated!
I've seen quite a few uses of Feyman's integration technique. What is never discussed is the conditions under which this method works. I.e it is never stated and it is never verified.
I tried integrating with IBP and skipping the times one in n!. Basically I tried integrating n!*(0-infinity)(x*e^-x). Using IBP you get that the answer is infinity*n!. I might have done something wrong but I don't think I did. I think this result comes about because the n! in the problem I gave wasn't actually n!, it was n! without the times 1 term.
I think it’s important to note, which I think Tom did in this video, that the ‘trick’ wasn’t invented by Feynman. He just famously used it in many important applications, having read about it in a calculus book.
try this: elliptic integral of the 2nd kind, integrate sqrt(1+c*sin^2(z) ] dz = (2/3)*csc^2(z)*(c*sin^2(z)+1)^(3/2), the c-value is negative, csc has a zero-point issue
I could only see repeated IBP to get the reduction formula - this is standard method found in literally every textbook. I don't see Feynman's method anywhere. Did I miss something?
At 1:33 you forgot the closing parenthesis in the integral equation for 1/(1+x^3)! (That's an exclamation, not a factorial lol.) When I realized it I made that same face as the bad drawing.
This is well cool, loved your new video on the atomic bomb equation on numberphile too! Really nice trick now we know how to do it, integration is more knowing all these tricks and how to apply them eh? Cool stuff though.
0:41 I like to think of derivatives and integrals like this taking the derivative of a function is like taking something apart, you can use anything and go to any extent, using a screwdriver, a hammer, or a bomb. But taking the integral of a function is like perfectly putting something back together, for the former you don't need much skill, for the latter you need to be much more skilled.
This is such a cool trick, but it seems to be tailor made to work only in quite specific cases, does anyone know if there are general integral types for which this technique always works? similar to an identity of sorts I guess
But in this new definition of factorials, if 'n' is not an integer, we will never get x^0 while doing repeated by parts, thus the first term never vanishes. On the other side, in the fynmans method, n seems to be integer always. So, how do this make sense anyway for non integer numbers
I'm confused - why when e^-ax is differentiated, how can you differentiate with respect to a, as seen at 10:21, even though the integral says dx at the end? Apologies if this is a stupid question.
We are doing a partial derivative which means that we are saying the function e^(-ax) is in fact a function of two variables so f(x,a) = e^(-ax), and then we can differentiate with respect to a only. This doesn't affect the integral because it is with respect to the variable x as you say. I'd suggest taking a look ay my video on partial differentiation if this idea is new to you: ua-cam.com/video/RVwcBGzQcT8/v-deo.html
I don’t like how loosely the improper is treated. It’s important to emphasize that an improper integral is a limiting process where convergence is key! However, that was clearly not the point of this video. Fun exposition, nonetheless. For those of you wanting to know, generally, when and why this works, look up Leibniz Integral Rule. In a nutshell, continuity of a function of two variables and a certain partial derivative explains when you can interchange the order of partial differentiation and integration.
I know it's not the most important part, but i think the first proof could be cleaner, by defining Un as the integral. Then you've shown that for any n, Un = n U(n-1). Then no need to do another differentiation by parts, and the reasonning by recurrence is easy if people are not satisfied with the fact that this is basically the definition of the factorial (and avoid having to wave hands cause the reasonning by recurrence is already implied in it). But that's just my two cents. Nice stuff though. Do we know what is the equation feynman solved in 30 minutes ?
Nice spot. The iterative formula is indeed another way to do this one. I don't know what the Feynman integral was I'm afraid, but maybe Ben mentions it in his book??
Thats a very cool trick but the trade off is that you need some good insight to alter the integrand such that you can get something which is the general case of your integrand that when differentiated generates your result.
I love your videos, but missing a dx there buddy, around the 4 min mark. And then again close to the 5 minute mark. I think you will carry it all the way through the end
Sounds like Γ(x) with extra steps Brilliantly explained though. I wonder what kind of problem were those scientists having that took them 3 months to think about this trick, figured it out myself in a Laguerre Polynomials problem in about an hour and a half. I'm just learning now that this method is the Feynman Integral Trick
It's very clearly explained but I am not sure how to chose an integral that would lead you to the integral that you want to calculate. is it always int(e^(-ax))?
Ben's cartoons are excellent, but don't just take my word for it, check them out for yourself here: mathwithbaddrawings.com/
Never expected machine gun kelly to explain Feynman integration technique, but here we are
I saved pop punk, so now its time to save maths...
@@TomRocksMaths ratio
@@zzzzzzzjsjyue2175 insanely massive L
@@zzzzzzzjsjyue2175 YOOOOOO SICK RATIO
@@zzzzzzzjsjyue2175 lol
Gud moanin' :3
Well hello there...
@@TomRocksMaths I like seeing Flammy here:) To be expected I guess.
It's the end of the day on Christmas day, and I'm watching a maths video. It is honestly so incredible that I've finally reached a point where I can understand all the interesting stuff I've been seeing for months on the maths side of UA-cam, now that I've properly studied it. Thank you for videos like these, they're the reason I love maths so much.
This is amazing to read - well done and keep it up :)
I find you very expressive, like my maths teacher never used to move the hands, speak in an enthusiastic way, rather he had a machine voice and was dead serious . I discovered you today and I really admire your style of teaching
Thanks and welcome to the channel :)
I think this is one of the first math result where my eyes legitimately widened when the final box was drawn around the n! equation (and I'm a final year engineering student so I've seen plenty of neat results!) Thanks for the crystal clear explanation Tom - I had seen this formula before but the derivation really made it click. All the best from McMaster University in Canada.
Thanks Patrick :)
I am a first year engineering student at McMaster rn. I found it interesting as well
refreshingly different. loved it. wondering why you did not utter the word “granmmmaaa” function.
Thanks!! I talk about the Gamma function in great detail here in fact: ua-cam.com/video/7y-XTrfNvCs/v-deo.html
his disrespect of dx's hurts my feelings
I was screaming 🤣
He's sloppy
Exactly. I was eagerly waiting for him to realize that he just put an equal sign between a differential and a function.
@@DaveJ6515 What you mean
@@Bozzigmupp 7:41 integration by parts
the way he makes calculus interesting and not boring to a high school kid that hasn't done anything past the required algebra classes is amazing. I literally understand nothing that is happening but it must be really cool if you understand it!
As long as you have the excitment about learning I guarentee you'll become a master in anything you wanna do!
This is great. I feel like he talks about this very briefly in 'Surely You're Joking, Mr Feynman' but I've never seen an explanation of it that I could particularly get my head around until now. Thanks!
Glad you enjoyed it Gav!
What a way to link teaching a new integration method with something interesting in math. I had heard of the Gamma function and its sibling before but had never quite understood how they were exactly connected to factorials.
Glad you enjoyed it :)
One of my favorite "differentiation under the integral sign" problems is sin(x)/x, from 0 to infinity. You do it by defining
I(a) = int sin(x) exp(-ax) / x.
Then differentiate w.r.t. a, which gives a closed-form for I'(a), which you then integrate to find I(a). Finally set a = 0.
Love it!
Wow, such a simple and amazing trick. Thanks for such a great explanation
My teacher of my tution during my school actually taught this and never told us it was feynmann's technique, he just said it's a better of doing integral that are similar to integrals you know already. He also said that it won't be asked in exams but still learning it would be helpful.
this was soooooo satisfying im literally bursting with happiness
My mind is blown....this deserves a follow!
I loved your the explanation to this example. I’m sure any electrical engineers or maths majors watching will immediately think of how similar this example is to the Laplace transform, which uses similar proofs for some of the standard transforms.
This is also called the laplace transform of x^n when s=1
Respected Dr. Tom! You always remind me of my one friend who left me after the graduation. 🥺
you can also solve this integral with induction by starting at n=0 and proving the induction step.
Nice spot.
Here's an even slicker way to integrate x^n exp(-x): Let a be a parameter with -1 < a < 1. First, integrate exp((a-1)x) dx from 0 to infinity, and expand the result as a power series in a (it's a geometric series). Then, expand exp((a-1)x) as a power series in a, and integrate the result term-by-term dx. Now compare coefficients!
These are great John - thanks for sharing :)
This is beyond my understanding and education level, but you're such a good presenter that I'm happily watching nonetheless. Have to say, I greatly admire you and your enthusiasm in communicating maths! While I struggle to follow along with maybe even most of your videos, some of them have nonetheless made it click for me and I've come away with more understanding than I had before, and that is deeply appreciated!
This is amazing to hear - thank you
Someone better put out an APB for all those missing differentials.
Great explanation .... interesting story , Feynman itself is sure special kinda genius 👍
I've seen quite a few uses of Feyman's integration technique. What is never discussed is the conditions under which this method works. I.e it is never stated and it is never verified.
Exactly It would be really useful to know straight away when this is applicable
Feynman trick is the best
Interesting and worthwhile video.
i am jealous of that buttery smooth handwriting.
Feynman did invent this method. He popularized it.
best video i've watched in august !
Beautifully done....clearly explained
Glad you liked it Suhas :)
This is somehow incredibly beautiful 😌
Sir I am from india and I like your work and love it
I am from Ireland and I love Indian dishes. 😋
the drawings made me smile , what fun
Ben's drawings are awesome
This is beautiful
I tried integrating with IBP and skipping the times one in n!. Basically I tried integrating n!*(0-infinity)(x*e^-x). Using IBP you get that the answer is infinity*n!. I might have done something wrong but I don't think I did. I think this result comes about because the n! in the problem I gave wasn't actually n!, it was n! without the times 1 term.
differentiating that function with respect to a is crazy
6:09, daammm I’m feeling a bit attacked right now 😂
nice video, but actually the infinitesimal dv =e ^-x dx... these little things are important, they keep things clear.
I think it’s important to note, which I think Tom did in this video, that the ‘trick’ wasn’t invented by Feynman. He just famously used it in many important applications, having read about it in a calculus book.
try this: elliptic integral of the 2nd kind, integrate sqrt(1+c*sin^2(z) ] dz = (2/3)*csc^2(z)*(c*sin^2(z)+1)^(3/2), the c-value is negative, csc has a zero-point issue
try standard integration in parts
same goes for the inverted ^-1 version integral
This was great. Your teaching style resonates with me more than 3Blue1Brown, ngl
very nice explanation. good job 👍
Thank you so much for your work :)
My pleasure Filip :)
wow that was amazing thank you
The little drawn guy missed many chances to remind you to put in the missing "dx" as in "dv = e^(-x) dx", but nonetheless I loved this presentation
This is a very important and correct comment. I agree 100%. If you rock maths, you should do it right, since maths won't enjoy it.😀
That drawing is in my school workbook
Well explained.
This is nitpicky, but shouldn't dv = e^(-x) * dx ? (also du should have a dx)
had the same issue haha
I guess only needed to do parts in its entirety for one step because, you found a formula right away
I smell the papa flammy, but also the papa tom :D - nice video on the Feynmann technique.
😋
Absolutely brilliant video
How do we know that it is correct to push differentiation under the integral sign? The hypotheses of Leibniz rule (finite bounds) are not met here.
I could only see repeated IBP to get the reduction formula - this is standard method found in literally every textbook. I don't see Feynman's method anywhere. Did I miss something?
I’m 37 and bad in math, but i’m still watching this 😂
As long as you're having fun :)
37 is prime
:|
Very well explained man!
this has an incredible symmetry at first glance too long to post here
good explanation
Hi, I hope you can still see this comment. I’ve got probably the dumbest question here.
What IF we assume a ain’t equal 1? Would it make any point?
Hey Tom, nice Viedo ! Could you maybe explain when it is formally okay to change the Integral and the differentiation?
It's called 'Leibniz Rule' and tells you that a 'full' derivative outside of the integral becomes a 'partial' derivative inside.
At 1:33 you forgot the closing parenthesis in the integral equation for 1/(1+x^3)! (That's an exclamation, not a factorial lol.) When I realized it I made that same face as the bad drawing.
This looks very interest8ng but i have no clue what any of this means
Well as long as you're having fun!
@@TomRocksMaths thanks
15 year old me: Ewww maths.
20 year old me: Fascinating, I understand nothing, but by god is this amazing!
From Colombia, I admire you :').
Amazing thanks Pardo :)
This is well cool, loved your new video on the atomic bomb equation on numberphile too! Really nice trick now we know how to do it, integration is more knowing all these tricks and how to apply them eh? Cool stuff though.
Absolutely - integration is all about having as many tools as possible in your toolbox.
Tom, If you are going to insult Americans, at least take your shirt off.
No homo
I really enjoyed
Awesome
0:41 I like to think of derivatives and integrals like this
taking the derivative of a function is like taking something apart, you can use anything and go to any extent, using a screwdriver, a hammer, or a bomb. But taking the integral of a function is like perfectly putting something back together, for the former you don't need much skill, for the latter you need to be much more skilled.
nice metaphor!
This is such a cool trick, but it seems to be tailor made to work only in quite specific cases, does anyone know if there are general integral types for which this technique always works? similar to an identity of sorts I guess
only correct if n is a positive integer
good
Hi presenter, give pls cases of indefinite integrals using Feynman method.
But in this new definition of factorials, if 'n' is not an integer, we will never get x^0 while doing repeated by parts, thus the first term never vanishes. On the other side, in the fynmans method, n seems to be integer always. So, how do this make sense anyway for non integer numbers
I bet Chris didn't want to explain a lot about this, but this is basically the gamma function: en.wikipedia.org/wiki/Gamma_function#Main_definition
@@PanzerfaustBR Thanks
Nice explanation
Glad you enjoyed it Yusuf :)
Brilliant .
Thanks Bachir!!
I couldn't handle it so I subscribed just to see you more often. 😌
Welcome :)
@@TomRocksMaths :D
I'm confused - why when e^-ax is differentiated, how can you differentiate with respect to a, as seen at 10:21, even though the integral says dx at the end? Apologies if this is a stupid question.
We are doing a partial derivative which means that we are saying the function e^(-ax) is in fact a function of two variables so f(x,a) = e^(-ax), and then we can differentiate with respect to a only. This doesn't affect the integral because it is with respect to the variable x as you say. I'd suggest taking a look ay my video on partial differentiation if this idea is new to you: ua-cam.com/video/RVwcBGzQcT8/v-deo.html
This so neat!
Glad you enjoyed it!
I don’t like how loosely the improper is treated. It’s important to emphasize that an improper integral is a limiting process where convergence is key! However, that was clearly not the point of this video. Fun exposition, nonetheless.
For those of you wanting to know, generally, when and why this works, look up Leibniz Integral Rule. In a nutshell, continuity of a function of two variables and a certain partial derivative explains when you can interchange the order of partial differentiation and integration.
I know it's not the most important part, but i think the first proof could be cleaner, by defining Un as the integral. Then you've shown that for any n, Un = n U(n-1). Then no need to do another differentiation by parts, and the reasonning by recurrence is easy if people are not satisfied with the fact that this is basically the definition of the factorial (and avoid having to wave hands cause the reasonning by recurrence is already implied in it). But that's just my two cents.
Nice stuff though. Do we know what is the equation feynman solved in 30 minutes ?
Nice spot. The iterative formula is indeed another way to do this one. I don't know what the Feynman integral was I'm afraid, but maybe Ben mentions it in his book??
Thats a very cool trick but the trade off is that you need some good insight to alter the integrand such that you can get something which is the general case of your integrand that when differentiated generates your result.
It is indeed only applicable to some integrals, but its always nice to have more techniques to add to your integration toolbox :)
I love your videos, but missing a dx there buddy, around the 4 min mark. And then again close to the 5 minute mark. I think you will carry it all the way through the end
nice spot. sorry about that.
Useful thanks
Happy to help :)
i might be wrong but shouldn't differentian of e^(-ax) = -ae^(-ax) ?? i checked online also it gives this... or am i missing something else?
Sir can you explain integral by parts in geometry tipes can
Sounds like Γ(x) with extra steps
Brilliantly explained though. I wonder what kind of problem were those scientists having that took them 3 months to think about this trick, figured it out myself in a Laguerre Polynomials problem in about an hour and a half. I'm just learning now that this method is the Feynman Integral Trick
I don't get that, when he multiples the left side by n!?
We can simply use Laplace transformation to get answers of these integrals in seconds
nice handwriting
Isn't integration under the integral just a consequence of the Leibniz integral Formula?
Great video!
Thanks Adair!
on other note I understand Gamma function
Great video, thanks a lot ! Does anyone know what was the integral that stopped the guy at Los Alamos ? Who were probably quite bright people !
It's very clearly explained but I am not sure how to chose an integral that would lead you to the integral that you want to calculate. is it always int(e^(-ax))?
Thanks for sharing. At 2:09, that's a gamma function (of n+1), right?
Yes, exactly :)
A closing parenthesis seems to be missing after the arctan at 1:30 :)
I realized that as well
That's why the poorly drawn guy is worried in that image.... misplaced his closer :p
Is this (n!) Solution for integral takes a 3 months from scientist Richard Feynman??
didn't need the ads mate great video tho
Missing right parenthesis at 1:33
We can also use Gamma integration method 🙂
Yes, definitely