When you said Heaviside function in example 2, I think you meant sgn() function. A Heaviside function is basically a unit step. That said - phenomenal lecture! Just like this, now I understand the big idea of generalized derivatives of distributions. Thanks, Dr. Peyam! 🙏🏽😊🎊
Woah... does this bring back a ton of memories... Having to deal with the derivative of |x| (the absolute value) tormented me daily... Generalized derivatives.. a great topic (although I didn't end up using them). Heaviside functions is one of my favourite math inventions - I just find this theory very elegant. A great little video but I really can't read anything off the blackboard.
Thank you for keeping inspiring me! Do you have any recomended book for learning distribution theory on a graduate level? I have some books, but im always looking for more ways to learn this :)
Dear Dr. Peyman, you're the only one on youtube I've seen giving "tutorials" on functional analysis. Is there any chance you would consider something related to Banach Algebra and Spectral analysis for bounded/unbounded operators?
@@MathsatBondiBeach Pretty sure he could potentially give a decent overview of the main theorems and some simple applications as well. But this is my opinion.
@@MathsatBondiBeach I can agree for the coherent and comprehensive way. But Peyman already made youtube tutorials for some simple case of calculus with distributions, and if I remember correctly there was also a video with the open-mapping theorem, Banach-Steinhaus, Hahn-Banach as well, this is functional analysis. There wasn't any proof (if I remember correctly) and this is totally fine because at least he gave the idea of what the main theorems are. It's a lot of work if you want to give an actual class on the subject, but for the main ideas I think be shallow is fine.
I have had a look at some of the comments on this video and it is clear to me that the level in knowledge is WAY (and I mean WAY) below what I experienced at Cambridge in the early 1970s. I would not attempt to try to explain the sweep of functional analysis to this motley crew of viewers.
@@ItsLukkio Analysis courses (let alone advanced ones) generally have low student numbers across the world (there are a few exceptions I know of). It is a really small market. It could be done I guess but you would be preaching to a very small audience. There are topics which cross over with engineering and physics which attract bigger audiences and still have a big analysis content which can be modified for the audience. That would be one possible approach. "Wavelets for beginners......"
I'm not sure I completely understood the technique. I'll watch it again later when I have time, but I'd like to see the technique applied to some more interesting non-differentiable (or even, merely "hard-to-differentiate") function. Nothing too wild and crazy, but something more interesting. Also, it would be interesting if there was some discussion of graphical interpretations of this technique (if there exists some sensible way to visualize it).
Dr. Peyam, could you do a video about the derivative of x! ? I have been curious about this. I’m sure that using the gamma function would help but I don’t know how to do it.
GreenMeansGO I used the capital pi function in terms of x defined by the integral from 0 to inf. of t^x * e^-t * dt. I partially diferentiated in terms of x and I get the integral from 0 to inf. of e^-t* t^x* ln(t)* dt . Maybe you can manipulate it a little with integration by parts, sorry but I think I'll leave that up to you, haha.
If one studies generalized functions and integral transforms (Laplace's or Fourier's) and convolutions, then the similar thing is done with Heaviside and Dirac functions. Does the `distribution theory' claim that we can choose any compact support infinitely differentiable kernel with zero boundaries we can have the similar effect, moreover, it has relation to intergral transforms?
Hi Dr. Peyam, I am student at UCI! I have not taken a class with you but I recently found out you started teaching here too. Was this filled on the UCI campus, and if so which classroom? I ask because UCI does not have many classrooms with blackboards to work on.. most of them are whiteboards and I like chalkboards better :( Any tips for finding blackboards on campus?
Yep, it was filmed at UCI! I think that particular video was filmed somewhere in 440 Rowland. Before 306 Rowland had blackboards too, but they replaced them with whiteboards!
Calculus is the math you usually learn in high school (derivatives and integrals), analysis is the theory of it, which you usually learn at universities here
So, im pretty sure im off into the woods when peyam is the only video I can find on this. im gonna try to understand this first without the video. As a hobbiest I need to learn how to not be reliant on others and use my head. But for sure you gave me the first idea of how to do this by the description. Now my question is what test function should i use? Do test functions like from the Finite Element Method do well in these kinds of situations? or do I have to create unique test functions?
Ok do I didn't watch this video yet but this is something I was thinking about for some time and had an idea that I didn't do anything with. For normal differentiation you make one point be the point you want to differentiate and then let an other point go towards that and then get the fraction of the x and y difference when their difference approaches 0. Now I though that if instead of having one point stationary you let that one approach the point from the other side then undifferentiable points in a function should also become undifferentiable.
So we know the derivative of a function that is -1 for negative numbers and 1 for positive numbers. How about a function that outputs 1 for irrational and -1 for rational numbers?
There’s the Lebesgue integral for that :P But other than that, not sure! Because the integral replaces the derivative, but not sure what would replace the integral!
a hyper integral? or maybe an integral over an area... but i always thought the way was to cheat the DI method, just integrate 1 and differentiate the inside, given that is possible, until you either get a term directly proportional to the original or a zero line. like the integral of ln(x)= x ln(x) - 1/x integral(x/x)
I also think that most people would call H(x) the function that indicates the non-negative numbers. H: R->{0,1}, H(x)=0 if x=0, but that's just convention.
Yes. I also once saw a definition of a (I'M quite sure not really existing) limit of some kind of: H(x) = 1/(1+exp(-x/k)) for k going to zero. But that is a physicist type of definition^^
Another suggestion on differentiating abs(x): Split the function up (just how you did) and of course the derivative =-1 for x=0, thus it is of course the function you introduced (H(x)). And then you might know (or have seen it once) the principle behind that function, and namely abs(x)' = abs(x)/x. And all of this with neither calculating and playing around with function nor rather counterintuitive (by that I mean that it is not obvious why exactly that function was used) introductions; also would have shaved some time off the video ;] edit: In addition abs(x)/x would have been simpler to differentiate again, just use the quotient rule and you are done (H'(x) = 0/x^2, which also spikes at x=0 and otherwise is 0)
Here's one for you to stir the damn pot. I have come to believe the best way to define 1/0 and 0/0 is this: 1/0 = +/-oo. and 0/0 = anything. Yep, you read that right. 0/0 is any and every real number (perhaps even with also +/-oo added), and can be substituted for such arbitrarily. And 1/0 has _two_ values, both equally good - the two ends of the extended real number line. There isn't really any more honest I believe you can get than that. Yes, that makes division a "multivalued function" in perhaps, at least at a point, the "worst" possible way, but we know those things exist, and there's nothing wrong with them. Something has to give, so why not make it the simplest "something" of all: the functionality of division? It's no different from saying that (-1)^(1/2) = +/-i, and being reasonable to substitute either without fault or, if you are trying to solve some problem, you have to figure which one leads to a sensible solution. Likewise, you get a 0/0 - you have to figure out now what the number is that you should best substitute there, but any substitute is mathematically valid. If I have 0/0 + 60, saying 0/0 + 60 = 61 (i.e. 0/0 = 1) is as legit as saying (-1)^(1/2) = -i. No difference at all. Just a lot more options to pick from. They otherwise have identical "ontological" status. a/b is a two-input multivalued function just as z^(1/2) (complex) or even x^(1/2) (real) if we want to get really general. It's that simple.
When you use integrals to calculate derivatives you have achieved true transcendence.
Any day that Dr. Peyam uploads a video is a great day.
one of the coolest dudes on youtube! i've become addicted to your videos
Please differentiate something like the Weierstrass function: continuous but non-differentiable everywhere :D
I remember you as a graduate student at Berkeley, and I was also jealous when people would walk out of your section with cake before my class.
*sees video title*
*mind explodes*
*clicks immediately*
When you said Heaviside function in example 2, I think you meant sgn() function. A Heaviside function is basically a unit step. That said - phenomenal lecture! Just like this, now I understand the big idea of generalized derivatives of distributions. Thanks, Dr. Peyam! 🙏🏽😊🎊
Woah... does this bring back a ton of memories... Having to deal with the derivative of |x| (the absolute value) tormented me daily... Generalized derivatives.. a great topic (although I didn't end up using them). Heaviside functions is one of my favourite math inventions - I just find this theory very elegant.
A great little video but I really can't read anything off the blackboard.
Thank you for keeping inspiring me! Do you have any recomended book for learning distribution theory on a graduate level? I have some books, but im always looking for more ways to learn this :)
I like the one by friedlander and Joshi
@@drpeyam Thank you!
"how to get an integer quotient in n/m for integers 0
7:30 The GEN DER of f, I see what you did there :)
OMGGGG, I didn’t even notice that!!! 😃😃😃
Now, that's the kind of GEN DER studies I can get behind xD
Well there's also a "soul" in maths so...
It has something to do with manifolds, if I recall correctly...
@@Mrkol_ do you think the other gender studies is not something you can get behind? Can you elaborate because it sounds very anti-intellectual
he differentiatt
he integratt
but most importantly
he infinitt
Dear Dr. Peyman, you're the only one on youtube I've seen giving "tutorials" on functional analysis. Is there any chance you would consider something related to Banach Algebra and Spectral analysis for bounded/unbounded operators?
@@MathsatBondiBeach Pretty sure he could potentially give a decent overview of the main theorems and some simple applications as well. But this is my opinion.
@@MathsatBondiBeach I can agree for the coherent and comprehensive way. But Peyman already made youtube tutorials for some simple case of calculus with distributions, and if I remember correctly there was also a video with the open-mapping theorem, Banach-Steinhaus, Hahn-Banach as well, this is functional analysis. There wasn't any proof (if I remember correctly) and this is totally fine because at least he gave the idea of what the main theorems are. It's a lot of work if you want to give an actual class on the subject, but for the main ideas I think be shallow is fine.
I have had a look at some of the comments on this video and it is clear to me that the level in knowledge is WAY (and I mean WAY) below what I experienced at Cambridge in the early 1970s. I would not attempt to try to explain the sweep of functional analysis to this motley crew of viewers.
@@peterhall6656 I'll still keep hoping.
@@ItsLukkio Analysis courses (let alone advanced ones) generally have low student numbers across the world (there are a few exceptions I know of). It is a really small market. It could be done I guess but you would be preaching to a very small audience. There are topics which cross over with engineering and physics which attract bigger audiences and still have a big analysis content which can be modified for the audience. That would be one possible approach. "Wavelets for beginners......"
I'm not sure I completely understood the technique. I'll watch it again later when I have time, but I'd like to see the technique applied to some more interesting non-differentiable (or even, merely "hard-to-differentiate") function. Nothing too wild and crazy, but something more interesting. Also, it would be interesting if there was some discussion of graphical interpretations of this technique (if there exists some sensible way to visualize it).
Do I see a Dirac delta distribution coming? Great video!
Already up haha
Dr. Peyam, could you do a video about the derivative of x! ? I have been curious about this. I’m sure that using the gamma function would help but I don’t know how to do it.
GreenMeansGO I used the capital pi function in terms of x defined by the integral from 0 to inf. of t^x * e^-t * dt. I partially diferentiated in terms of x and I get the integral from 0 to inf. of e^-t* t^x* ln(t)* dt . Maybe you can manipulate it a little with integration by parts, sorry but I think I'll leave that up to you, haha.
Your presentation is pretty cool :)
If one studies generalized functions and integral transforms (Laplace's or Fourier's) and convolutions, then the similar thing is done with Heaviside and Dirac functions. Does the `distribution theory' claim that we can choose any compact support infinitely differentiable kernel with zero boundaries we can have the similar effect, moreover, it has relation to intergral transforms?
This one was a but beyond my level of mathematical education
Nice video! Does anyone know what the 'motivation' means at around the one minute mark? Thanks.
Hi Dr. Peyam, I am student at UCI! I have not taken a class with you but I recently found out you started teaching here too.
Was this filled on the UCI campus, and if so which classroom? I ask because UCI does not have many classrooms with blackboards to work on.. most of them are whiteboards and I like chalkboards better :( Any tips for finding blackboards on campus?
Yep, it was filmed at UCI! I think that particular video was filmed somewhere in 440 Rowland. Before 306 Rowland had blackboards too, but they replaced them with whiteboards!
you looked like you dressed up for a lecture
thanks, Dr. Peyam. great lecture
Me: it's |x|/x. Everyone: the true intellectual
is there a difference between calculus and analysis in the us? here in germany, it's just called "analysis"
Calculus is the math you usually learn in high school (derivatives and integrals), analysis is the theory of it, which you usually learn at universities here
So, im pretty sure im off into the woods when peyam is the only video I can find on this.
im gonna try to understand this first without the video. As a hobbiest I need to learn how to not be reliant on others and use my head. But for sure you gave me the first idea of how to do this by the description.
Now my question is what test function should i use? Do test functions like from the Finite Element Method do well in these kinds of situations? or do I have to create unique test functions?
Would you care to try to differentiate Weierstrass’ function using this approach dr Peyam? 😉
Ok do I didn't watch this video yet but this is something I was thinking about for some time and had an idea that I didn't do anything with. For normal differentiation you make one point be the point you want to differentiate and then let an other point go towards that and then get the fraction of the x and y difference when their difference approaches 0. Now I though that if instead of having one point stationary you let that one approach the point from the other side then undifferentiable points in a function should also become undifferentiable.
So we know the derivative of a function that is -1 for negative numbers and 1 for positive numbers.
How about a function that outputs 1 for irrational and -1 for rational numbers?
It should be the sum of a bunch of diracs at each rational number!
Special but interesting. Thanks.
I would be pretty interest to see this with the integer part function
Oh, this just becomes the sum of dirac deltas at each integer! :)
And the proof is pretty much similar, since the integer part is just a sum of a bunch of Heaviside functions!
how about integrating non integrable functions?
There’s the Lebesgue integral for that :P But other than that, not sure! Because the integral replaces the derivative, but not sure what would replace the integral!
a hyper integral? or maybe an integral over an area... but i always thought the way was to cheat the DI method, just integrate 1 and differentiate the inside, given that is possible, until you either get a term directly proportional to the original or a zero line.
like the integral of ln(x)= x ln(x) - 1/x integral(x/x)
Thankyou. Nice
Bravo!
You could define (|x|)' as |x|/x.
Then (|x|)''=(|x|/x)'=((|x|/x)x-|x|*1)/x^2=(|x|-|x|)/x^2=0.
Except at 0 because |x|/x is not defined at 0.
This is a great formula, especially in higher dimensions! What I like about my approach is that it explains what happens at 0 :)
You can also define |x| as sqrt(x^2) and differentiate
Isn't H(x) just sign(x)?
Aaron Quitta yup
Der Justus Then what's up with all of the weird discussion and notation?
I also think that most people would call H(x) the function that indicates the non-negative numbers.
H: R->{0,1}, H(x)=0 if x=0, but that's just convention.
H(x) is sometimes defined a little bit different, as Peyam stated e.g. H(x)=0 for x0
Yes. I also once saw a definition of a (I'M quite sure not really existing) limit of some kind of:
H(x) = 1/(1+exp(-x/k)) for k going to zero. But that is a physicist type of definition^^
Lol the GEN-DER of f
I thought math was safe from identity politics XD
general derivative shortened to gen der LOL
...the Dirac Delta MEASURE!
In all honesty, I'm finding it hard to see the crayon writing against the dirty blackboard...
Yeah, I know, sorry :(
justcarcrazy It's called 'chalk'
Hilbert Black In French it's "craie", which is more similar to crayon :p I don't know what justcarcrazy's first language is so it might be irrelevant
Destroy an undesttroyable shield with a sword wich can not destroy anything.
WOW!!!
31:19 jihad!
Another suggestion on differentiating abs(x):
Split the function up (just how you did) and of course the derivative =-1 for x=0, thus it is of course the function you introduced (H(x)).
And then you might know (or have seen it once) the principle behind that function, and namely abs(x)' = abs(x)/x.
And all of this with neither calculating and playing around with function nor rather counterintuitive (by that I mean that it is not obvious why exactly that function was used) introductions; also would have shaved some time off the video ;]
edit: In addition abs(x)/x would have been simpler to differentiate again, just use the quotient rule and you are done (H'(x) = 0/x^2, which also spikes at x=0 and otherwise is 0)
That’s one way of doing it, but the point of this video is to introduce a more general method that might potentially work for more general functions!
Чё за хэ это МКН?
how to sin:
Dirac delta function
Here's one for you to stir the damn pot. I have come to believe the best way to define 1/0 and 0/0 is this: 1/0 = +/-oo. and 0/0 = anything. Yep, you read that right. 0/0 is any and every real number (perhaps even with also +/-oo added), and can be substituted for such arbitrarily. And 1/0 has _two_ values, both equally good - the two ends of the extended real number line.
There isn't really any more honest I believe you can get than that. Yes, that makes division a "multivalued function" in perhaps, at least at a point, the "worst" possible way, but we know those things exist, and there's nothing wrong with them. Something has to give, so why not make it the simplest "something" of all: the functionality of division? It's no different from saying that (-1)^(1/2) = +/-i, and being reasonable to substitute either without fault or, if you are trying to solve some problem, you have to figure which one leads to a sensible solution. Likewise, you get a 0/0 - you have to figure out now what the number is that you should best substitute there, but any substitute is mathematically valid. If I have 0/0 + 60, saying 0/0 + 60 = 61 (i.e. 0/0 = 1) is as legit as saying (-1)^(1/2) = -i. No difference at all. Just a lot more options to pick from. They otherwise have identical "ontological" status. a/b is a two-input multivalued function just as z^(1/2) (complex) or even x^(1/2) (real) if we want to get really general.
It's that simple.
General derivative = gender