Note: At 4:10 that’s supposed to be the integral of x d(x^2), not dx. I am NOT calculating the integral of x dx, that’s why the answer is 2/3, not 1/2.
Dr. Peyam, you have a wonderful style. I wish more professors could have just a little of your enthusiasm. Glad I stumbled across your Stieltjes integral lecture!
We actually learned this integral instead of a reimann integral in my analysis class, quite challenging but very satisfying when you actually understand it.
Interesting. I've always done this trick called "put smth under the differential". And it has always made perfect sense because it's the same as applying the chain rule backwards. And now I am surprised that there is some additional definition to this integral.
The German word Matjes comes from the Dutch word maatjes and means small hearings. Via adding the suffix "je" you can make a diminutive of a word in Dutch and the last s makes it a plural. But "Stieltjes" is a strange word/name even for Dutch people. The Dutch word stiel means craft but it is strange to make a diminutive of that.
I love your videos! I saw this in an undergraduate-course-level Introduction to Measure Theory. Hardest course I've taken as an undergrad, and I remember the professor calculating this integral, as well as Lebesgue integrals, in the midst of Statistics and Probability. It was hard to get approved for this one! I will check your "Lebesgue Integral Overview" video.
Well, the name of the integral seems weird at first, but it turns out that, it reminds me the way my teacher taught me to avoid using u-sub. For example, let's say integral (ln x / x dx). Instead of u-sub, my teacher taught me to turn it into integral (ln x d(ln x)) = (ln x)^2/2. I really did not think there is a proper (and awkward) name for such an integral!
Guilherme Guimarães not really. The premise is still the same as substitution, just less explicit. By letting u=ln x, you're trying to integrate with respect to d(u) (normal substitution) = d(ln x) (the above technique).
Haven't got what's difference between this nicely-named integral and making simple substitution. I was tought that some intergrals can be much easier solved by making impicit substitution kind of: f(u(x))u'(x)dx = f(u(x))d(u(x)) We named it "put under differential" Have using this for a long time without any think that it has its own name, and moreover, is more generilized then Riemann intergal
So cool both the video and the integral hehe Other videos never give me this kind of intuition as much as your videos give Dr.P And now that I have made it a ritual: noice
Hey Dr. Peyam, great video as usual! But i was wondering why you said the Stieltjes Integral was less general than the Lebesgue Integral. In our lecture we defined the lebesgue integral via first defnining a Lebesgue pre-measure, then extending that to the Lebesgue-measure and then defning an Integral by any measure. We also defnied a "Stieltjes-pre-measure", so i would imagine if you would extend that to a measure in the same way you could define the Stieltjes-integral with that and you would have something thats definitely more general, because the Lebesgue-measure is just the special case alpha = x. Or is that going to lead to problems in some of the nice proofs?
Since a Stieltjes pre-measure is a special kind of pre-measure, the Lebesgue integral is more general, since it works for any kind of pre-measure, not just the Stieltjes one! Also by Stieltjes Integral I’m referring to the one in this video, where I’m presenting it the Riemann way (it’s sometimes called the Riemann-Stieltjes integral)
It’s very useful in statistics, but I don’t think it’s used much in math, since the Lebesgue integral is much better! But still, a nice generalization of the Riemann integral
In Statistics we use it for ignoring the difference between continuous and discrete random variables. It allows us to express in a close simple form distribution functions and functions of distribution functions (namely Expected values, variances,...etc) that otherwise would be cumbersome. Lebesgue Integral is far more beautiful, elegant,...but things there get far more complicated. You can see Doctor Peyam´s videos about that. He has done a great job for us.
Hi, im 2 years late. I've just discovered this integral when studying QM. It is used in the spectral decomposition of general self-adjoint operators, the measure being an orthogonal projector, function of the eigenvalue of the operator.
nullplan01 Who claimed that he is German? I don't see anyone? To my knowledge, I believe to have heard he was born in Austria and was educated in a French school. What citizenship he holds is unknown and irrelevant to me. I'm just asking if he would feel like making some Math Videos of this kind in German because: 1. there aren't a lot of good German math channels and 2. His German is very good and I'm sure it could be fun.
Would a substitution also work? For example, integral of x d(x^2) from 0 to 1. We set y=x^2, so x=y^1/2. That means we have the integral of y^1/2 dy, from 0^1/2 wich is 0, to 1^1/2 wich is also 1. Integrated that would be (2/3)y^3/2 evaluated from 0 to 1, wich would also be 2/3. Am i right, or is it a coincidence?
Aber das ist doch garnicht gleich dem Riemann - Integral oder wie muss ich das sehen? int_0^1 x dx = 1/2. Riemann and Lebesque calculate for the area under the curve 1/2, which area is given by Stieltjes with 2/3?
If I understand what is happening correctly, more than bending this is stretching (or "modulating" - given alpha could be anything, not just a monotonic function!) the x axis itself. [This is almost (again if I understand this correctly) like a double integral, except that rather than integrating the same function twice with respect to two variables we integrate once with respect to two "simultaneous" functions in the same variable. But I may have misunderstood the concept completely!]
Have a question : what is this useful for because normaly the answer to this integral with dx is 1/2 but you got 2/3 so how could you solve some integral with this technique to get the normal dx answer ? I feel kind of uncomfortable with this
It’s not very useful in math, but apparently more useful in statistics. I don’t think you can use Stieltjes integrals with alpha to solve integrals with dx.
Very nice! A geometric interpretation would have been very nice, or why we would want alpha to be something else other than just x. You mentioned statistics and blah, maybe it's just not so easy to give easier, practical examples.
Yeah, I can’t really think of a more practical application, since mathematicians mainly use the Lebesgue integral anyway. But I’m guessing that if you want your integral to emphasize the point 0 more, you’d use alpha(x) = x^2 instead of x, but I agree, it’s more of a statistical thing
Sometimes it can be a nice way to write substitutions. For example in spherical coordinates you integrate over sin(theta) dtheta or even nicer dcos(theta)
Great video! But, in 9:40 you said smooth, ain't it enough for alpha to be Differentiable? or it has to be smooth? also in the last example alpha is called RELU(rectified linear unit). Maybe you should do a follow up video and do integration by parts for Stieltjes Integral :). P.S. Do you know any good pure math books(for third year of uni or so)? preferably something with differential equations
Once differentiable is enough! I use smooth in a broad sense, as in “take as many derivatives as you need” (which may be 1 or infinity :P). Good idea, but I think that might just follow from the product rule (for once differentiable alpha). Oh, and I highly recommend the 4 books by Stein and Shakarchi, they’re a great introduction to post real analysis topics. And I also like the differential equation book by Hirsch/Smale/Devaney, and the PDE book by Evans
The books are "Princeton Lectures in Analysis" series and "differential equation dynamical systems and an introduction to chaos"? I failed to find the last one. I'll look into them thanks very much, I'll probably start with the third book of "Princeton Lectures in Analysis"
Estoy aquí porque es el único video con un ejemplo así, con una integral de Riemann Stieltjes a partir de la definición de sumatoria (casi todos usan el teorema del cambio de variable y el del cambio de la derivada de la función integrante). Y es que hice el ejercicio de integrar x^2 con alfa x^3 en el intervalo [0,1]... Y si, me salió como en el vídeo (aunque hice diferente el "prework", pero igual llegué al resultado)... Por cierto, sale 3/5!!!!!!
![Seungmin "그렇게, 천천히, 우리(As we are)" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/kAzmhLHePqU/mqdefault.jpg)
Note: At 4:10 that’s supposed to be the integral of x d(x^2), not dx. I am NOT calculating the integral of x dx, that’s why the answer is 2/3, not 1/2.
Omer Lublin Wow, that’s a great way of putting it!!!
Ah now I see it ... I am so sorry 😐
xd(x^2) = x*2xdx = 2x^2dx = 2/3 x^3
What's the difference?
This is like Riemann integrals but u-subbed.
Will You No difference for smooth alpha, but the approach I gave works as well for alpha that is not differentiable
@@drpeyam Thank you...
I saw this Dr Peyem video in my search for "reimann stieltjes integrals" and immediately understood it would be exactly the video i was searching for.
What does "reimann" mean?
Dr. Peyam, you have a wonderful style. I wish more professors could have just a little of your enthusiasm. Glad I stumbled across your Stieltjes integral lecture!
Your videos just keep becoming more interesting... this last month or so has been crazy! Keep up with your work!
Thank you very much. For us, statisticians, this video is very important. We know you go to great lengths to produce this videos and we appreciate it.
We actually learned this integral instead of a reimann integral in my analysis class, quite challenging but very satisfying when you actually understand it.
I really wanna see an introduction to contour integration. Complex analysis is always so much fun to me.
is it me or is this guy very VERY happy to introduce us this integral ?
0:20
Interesting. I've always done this trick called "put smth under the differential". And it has always made perfect sense because it's the same as applying the chain rule backwards. And now I am surprised that there is some additional definition to this integral.
if only my teachers sound happy like you do :-)
I love how math excites you, great video! Please keep with the great work!
Your enthusiasm made me really happy to study RS integrals
The German word Matjes comes from the Dutch word maatjes and means small hearings. Via adding the suffix "je" you can make a diminutive of a word in Dutch and the last s makes it a plural. But "Stieltjes" is a strange word/name even for Dutch people. The Dutch word stiel means craft but it is strange to make a diminutive of that.
This is awesome, thank you!
What is "hearings"? Nonsense!
Use the CHEN-LUUUH!!!
Riky Agazzi yeah great. It's from black pen red pen ISN'T IT?!
No, Dr. P started it! He's the original!
blackpenredpen ok good to know, thanks!
blackpenredpen Technically Xuemin Tu started it 😂😂😂😂
Dr. Peyam's Show ahhhh yes!!!!
Awesome job. The Stieltjes integral hits the dab
thanks for post it, dear Dr Peyam, for some students is unknown, today i've learned from you... greetings for you.
integral x d(x^2) = integral sqrt(x^2) d(x^2) = integral sqrt(t) dt = t^1.5*2/3 + C = x^3*2/3+C
So answer is 2/3
Does this substitution work due to us being in the positive domain? Or would we be able to get complex solutions?
It will be nice if you would like to have a talk on stochastic integral!
I have yet the doubt about the procedure in 9:06 in treating de differentials like numbers or fractions. I know the chain rule, though
I love your videos! I saw this in an undergraduate-course-level Introduction to Measure Theory. Hardest course I've taken as an undergrad, and I remember the professor calculating this integral, as well as Lebesgue integrals, in the midst of Statistics and Probability. It was hard to get approved for this one! I will check your "Lebesgue Integral Overview" video.
Thank you!!
Well, the name of the integral seems weird at first, but it turns out that, it reminds me the way my teacher taught me to avoid using u-sub. For example, let's say integral (ln x / x dx). Instead of u-sub, my teacher taught me to turn it into integral (ln x d(ln x)) = (ln x)^2/2. I really did not think there is a proper (and awkward) name for such an integral!
How did u do that ? How do u change dx to d(ln x) can u please explain me ?
d(ln x)/dx = 1/x, so d(ln x) = 1/x dx. That's how my teacher taught me.
WOw dude ! TYVM ! This is amazing, can be quite helpful with some trick Riemann Integral !...
Guilherme Guimarães not really. The premise is still the same as substitution, just less explicit. By letting u=ln x, you're trying to integrate with respect to d(u) (normal substitution) = d(ln x) (the above technique).
My teacher did the same thing. But, by then I already knew substitution because of youtube videos so I never liked that method
Excellent video as always! could you make one about stochastic calculation please?
The question is: How many of your viewers know what Matjes are?
Hahahaha, I was hoping someone would get the reference 😂😂😂😂
Well you have quite some german viewers, so q few would, I think^^
i dont...
You are good at math, but your german could use some polish^^
dahlhoff.de/wp-content/uploads/167_edlesmatjesfiletinoel.png
I feel lucky to come across your videos on youtube!
I love saying “Stieltjes” too! 😂
What's the purpose of this integral? The result is different from the riemman integral, what does that 2/3 mean?
Awesome video, thanks! Your performance is much better when you don't use any notes.
Haven't got what's difference between this nicely-named integral and making simple substitution. I was tought that some intergrals can be much easier solved by making impicit substitution kind of:
f(u(x))u'(x)dx = f(u(x))d(u(x))
We named it "put under differential"
Have using this for a long time without any think that it has its own name, and moreover, is more generilized then Riemann intergal
So cool both the video and the integral hehe
Other videos never give me this kind of intuition as much as your videos give Dr.P
And now that I have made it a ritual: noice
Dr. Peyam you and math are amazing
Thank you very much! You are very talented at transferring knowledge.
How delightful lecture it is
Wow! fantastic! It's very good examples to understand this integral calculate.
Thank you!! 😊
Like integration means the area under the curve what does riemann stieljes integral means physically
Hey Dr. Peyam, great video as usual! But i was wondering why you said the Stieltjes Integral was less general than the Lebesgue Integral. In our lecture we defined the lebesgue integral via first defnining a Lebesgue pre-measure, then extending that to the Lebesgue-measure and then defning an Integral by any measure. We also defnied a "Stieltjes-pre-measure", so i would imagine if you would extend that to a measure in the same way you could define the Stieltjes-integral with that and you would have something thats definitely more general, because the Lebesgue-measure is just the special case alpha = x. Or is that going to lead to problems in some of the nice proofs?
Since a Stieltjes pre-measure is a special kind of pre-measure, the Lebesgue integral is more general, since it works for any kind of pre-measure, not just the Stieltjes one! Also by Stieltjes Integral I’m referring to the one in this video, where I’m presenting it the Riemann way (it’s sometimes called the Riemann-Stieltjes integral)
Ah that makes sense. Thanks :)
Thanks for the video. What is the difference between the Stieltjes integral and the Riemann-Stieltjes Integral?
Same I think!
can you please make me understand what do you mean by "taking x and stretching out with x^2". I am trying to understand the picture of this integral
Thanks for this video. I understand how this kind of integral works but I fail to understand how it could be useful.
It’s very useful in statistics, but I don’t think it’s used much in math, since the Lebesgue integral is much better! But still, a nice generalization of the Riemann integral
Thanks for your fast answer Mr. πm :) Understood!
Dr. Peyam's Show And Lebesgue-Stieljes integral (instead of (b-a) as the measure of interval (a,b) take (alpha(b)-alpha(a))) is the best
In Statistics we use it for ignoring the difference between continuous and discrete random variables. It allows us to express in a close simple form distribution functions and functions of distribution functions (namely Expected values, variances,...etc) that otherwise would be cumbersome.
Lebesgue Integral is far more beautiful, elegant,...but things there get far more complicated. You can see Doctor Peyam´s videos about that. He has done a great job for us.
Hi, im 2 years late. I've just discovered this integral when studying QM. It is used in the spectral decomposition of general self-adjoint operators, the measure being an orthogonal projector, function of the eigenvalue of the operator.
Where did you get the square for i instead of just i? Also how did one become i?
11:06 I think the upper bound should be 0(LHS)
It is, though, no?
Now I get it...
The name was Rumple stieltjes skin
What is the motivation for doing this type of integration?
Why do you multiply i/n times the squared terms?
Mr Peyam First I want to thank you because this video has helped me very much
I want you to talk about Hadamard integral next time
Du bist der BESTE 😀
Danke!!! :D
Ah noch ein deutscher der Mathe süchtig ist 😂
How do you know he's German? Because he speaks the language? Because then I'm a processor!
nullplan01 Who claimed that he is German? I don't see anyone? To my knowledge, I believe to have heard he was born in Austria and was educated in a French school. What citizenship he holds is unknown and irrelevant to me. I'm just asking if he would feel like making some Math Videos of this kind in German because: 1. there aren't a lot of good German math channels and 2. His German is very good and I'm sure it could be fun.
Would a substitution also work? For example, integral of x d(x^2) from 0 to 1. We set y=x^2, so x=y^1/2. That means we have the integral of y^1/2 dy, from 0^1/2 wich is 0, to 1^1/2 wich is also 1. Integrated that would be (2/3)y^3/2 evaluated from 0 to 1, wich would also be 2/3. Am i right, or is it a coincidence?
Yep, if alpha is smooth, then it’s substitution, but this method also works if alpha is not differentiable!
What does this amount of integration means physii
Sooooooooo ... ooo ... thanks dear teacher! I got everything! !!
So could you do int xda(x), where a(x) = e^x?
int x (e^x)’ dx = int x e^x dx and then integrate by parts
Dr. Peyam's Show Thanks!
Absolut capo! Total genius...
Is Stieltjes and Reimann Stieltjes integral the same?
Yes
Sir does alpha(x) need be a continous monotone function for it to work?
Not continuous, I think right continuity is enough
Aber das ist doch garnicht gleich dem Riemann - Integral oder wie muss ich das sehen? int_0^1 x dx = 1/2. Riemann and Lebesque calculate for the area under the curve 1/2, which area is given by Stieltjes with 2/3?
Tut mir Leid, aber in diesem Video rechne ich die Integrale von x d(x^2), nicht x dx, darum ist die Antwort 2/3, nicht 1/2
but what area describes that? or isnt there a visualization?
It’s the area under x, but where your axis becomes x^2, so think like bending your axis to become x^2.
If I understand what is happening correctly, more than bending this is stretching (or "modulating" - given alpha could be anything, not just a monotonic function!) the x axis itself.
[This is almost (again if I understand this correctly) like a double integral, except that rather than integrating the same function twice with respect to two variables we integrate once with respect to two "simultaneous" functions in the same variable. But I may have misunderstood the concept completely!]
el diferencial de la integral no es dx, sino el diferencial de una funcion. en este caso una parabola
Doctor, I remark that Riemann integral is a particular case of Lebesgue integral. Am I right?
More or less, at least for a finite closed interval [a,b]
There’s a video on Riemann vs Lebesgue Integral actually!
Okay thank you doctor Peyam :)
could you please do a tutorial on the inverse Laplace transform?
Flammable Maths Sounds like a job for you :)
Dr. Peyam's Show thanks! amazing content though👍
Have a question : what is this useful for because normaly the answer to this integral with dx is 1/2 but you got 2/3 so how could you solve some integral with this technique to get the normal dx answer ? I feel kind of uncomfortable with this
It’s not very useful in math, but apparently more useful in statistics. I don’t think you can use Stieltjes integrals with alpha to solve integrals with dx.
Dr. Peyam's Show thanks I feel better now 😂 but it's nice to do some math for fun
Sir! Kindly help me find the value of ∫(x^5 d(x^2 ) where x is from -2 t0 3
Integral from -2 to 3 of x^5 times 2x dx where
Convert x^5 into (x^2)^(2.5) so the integral will be in the form t^2.5 dt , use the power rule backwards
In general you can use integration by parts and get:
Integral from 0 to 1 of x da(x)
= a(1) - Integral from 0 to 1 of a(x) dx
I kind of prefer the blackboard, black shirt and the chalk all over it haha! anyway, still awesome :D
thank you so much😭 from Korea
Very nice! A geometric interpretation would have been very nice, or why we would want alpha to be something else other than just x. You mentioned statistics and blah, maybe it's just not so easy to give easier, practical examples.
Yeah, I can’t really think of a more practical application, since mathematicians mainly use the Lebesgue integral anyway. But I’m guessing that if you want your integral to emphasize the point 0 more, you’d use alpha(x) = x^2 instead of x, but I agree, it’s more of a statistical thing
Sometimes it can be a nice way to write substitutions. For example in spherical coordinates you integrate over sin(theta) dtheta or even nicer dcos(theta)
I found this in a text about creep law for uniaxial stress in viscoelastic materials.
anyone want to know applications can refer controlled different equations.
Should alpha(x) at least monotone ?
Yeah something like that, and left continuous
love your energy
Super!!!! Thanks a lot, I really enjoyed it !!!!
i love yur hapiness
Dr.payem... Sir can u make a video on malmsten's integral
Vardi Integral ruclips.net/video/W2QFhyC_BQ8/видео.html
I don't get 7:37. Why "2N³"?
When N approaches infinity, N(N+1)(2N+1) approaches 2N^3
This was so very helpful, thank you!
That is A W E S O M E :)
You are one awesome person.
Great video! But, in 9:40 you said smooth, ain't it enough for alpha to be Differentiable? or it has to be smooth?
also in the last example alpha is called RELU(rectified linear unit).
Maybe you should do a follow up video and do integration by parts for Stieltjes Integral :).
P.S. Do you know any good pure math books(for third year of uni or so)? preferably something with differential equations
Once differentiable is enough! I use smooth in a broad sense, as in “take as many derivatives as you need” (which may be 1 or infinity :P).
Good idea, but I think that might just follow from the product rule (for once differentiable alpha).
Oh, and I highly recommend the 4 books by Stein and Shakarchi, they’re a great introduction to post real analysis topics. And I also like the differential equation book by Hirsch/Smale/Devaney, and the PDE book by Evans
The books are "Princeton Lectures in Analysis" series and "differential equation dynamical systems and an introduction to chaos"? I failed to find the last one.
I'll look into them thanks very much, I'll probably start with the third book of "Princeton Lectures in Analysis"
Partial differential equations by Lawrence C Evans
Thank you professor!
Excellent video! You should do more videos without holding your notes in your hands. The viewers can better connect to you.
Ito integral coming next lol
Estoy aquí porque es el único video con un ejemplo así, con una integral de Riemann Stieltjes a partir de la definición de sumatoria (casi todos usan el teorema del cambio de variable y el del cambio de la derivada de la función integrante).
Y es que hice el ejercicio de integrar x^2 con alfa x^3 en el intervalo [0,1]... Y si, me salió como en el vídeo (aunque hice diferente el "prework", pero igual llegué al resultado)... Por cierto, sale 3/5!!!!!!
Muy bien!!! 😁
Your smile is contagious 😂
Dr Peyam did you read your personal messages? :P I've left one :3
Ooooh this is kind of about what i commented before of the diferential in the argument
Amazing, thank you!
They have this same problem in Advanced Calculus by David V. Widder. I thought I recognised it haha.
Oh wow, really? What a coincidence :O
@@drpeyam If interested out of curiosity see page 153 of the second edition. I just recognised the 2/3 in problems I did in similar content.
congrats max
thank you sir, keep going
You forgot to put d alpha x in the integral at first this kind of confused me
Well yeah, that was on purpose; I wanted to show how the Riemann integral and the Stieltjes integral are similar!
Dr. Peyam's Show i ment at minute 4:10 ... sorry I love your work its no criticism or am I wrong
QuickMath My bad, you’re right! That’s what I get for improvising 😂
11:27 thats not discontinous :P
For me, it is "still tears"
You didn't give any interpretation and illustration about alpha x. Please do that.
It's too good sir
the best! thx
the are a errro in f(x_i) is diferent yo x_i
thanks for the video
Awesome!
Int(x da(x))=x a(x) - int(a(x) dx) is another simmilarity.
goat
genial
Still yes
Very nice problem
Stieltjes was Dutch
Are there any other high school students here?
the meme now be real