I've been a little confused while learning about distribution theory and how the dirac delta function is defined through that lens, so it was really interesting and informative to see how a different branch of mathematics adds intuition and creates a more complete understanding of this object. Thank you for this great video.
One thing that helped me understand distributions a bit better is the fact that most (but not all) of them can actually be represented by integration against a measure, just like with linear functionals acting on continuous functions. So really, once you get an intuitive feel for what measures do, you just need to account for some "extraneous" types of distributions like the Cauchy principal value and derivatives of the Dirac delta in order to completely characterize distributions.
As a former engineering student I have never thought of the Dirac function as anything more than a normalized ideal pulse input. It is really nice to know about the math it originated from. Great video.
This was the clearest interpretation of measure theory I've seen in a while. I feel like your video could really use a better thumbnail to get more views.
As a physics student whose had way more than his share of suffering from major mathematical itchiness and headache when it comes to the Dirac delta function, I really appreciate your effort and enjoyed the simplicity of the presentation and also think I actually did learn a number of new things; namely, the Riesz-Markov-Kakutani thing. Truth is, even with my preliminary trainings in real analysis, abstract and linear algebra, and even some rudimentary knowledge of basic topology and measure theory, this particular subject is way too advanced to be readily grasped, even by the expert folk with a healthy background in math. I also watched a lot of content under #SoME2 to understand how “renormalization” works and learn its basic principles, but unfortunately it’s one of those difficult, inaccessible materials. Nevertheless, kudos on a great video and hope your channel grows BIG👍🏻❤️
@@carlaparla2717 You could either take it in the topology of measures, or in the topology of distributions (these are the respective weak* topologies for the space of continuous functions, and the space of smooth compactly supported functions). Basically, a sequence of measures mu_n converges weak* to mu if for all continuous functions, integral(f dmu_n) converges to integral(f dmu). So in this case, the dmu_n would be [n/sqrt(2pi)]*e^{-(nx)^2/2} dx, where the standard deviation is 1/n, and they converge to the Dirac measure.
Thank you for this video. It honestly made quite a few of my holes in math knowledge filled. Not just the dirac delta function. I wish you success in this youtube adventure.
This was really well done! I can't believe I just found your channel - as a video creator myself, I understand how much time this must have taken. Liked and subscribed 💛
So I‘ve actually been thinking about the idea of integrating with dx as an input and not just as the final term, and the Dirac delta seems like it’d be 1/dx at 0 and 0 everywhere else based on the introduction. This would satisfy both properties (no I didn’t accidentally recreate the Dirac delta I just realized how something I was thinking about could be used to create something like it)
I've been (ab)using the delta function quite a bit lately. Great Video 👌, interpreting delta in the context of measure theory demystifies quite a few things
This was absolutely fantastic. As somebody doing Many-Body QFT with interactions I always just took it as a given and even though I did some measure theory but never Functional Analysis I was always surprised and bothered by some integrals (for example a rigorous calculation of a particle-hole bubble) and the relationship of things like the Heaviside step “function” and the Dirac “function”. This helped a lot for my intuition and I hope you will gain subscribers soon, your way of explaining is superb
THANK YOU FOR MENTIONING THE RIESZ MARKOV KAKUTANI THEOREM I have been having analytical mechanics and have been messing around with variational principles, and I've always wondered why we're extremizing one type of functional (namely one where we are integrating) and now I know why! Thank you, I searched everywhere for this!
The Dirac Delta is absolutely a function. It's a set of ordered pairs such that if (a,b) and (a,c) are elements, then b=c. That's the definition of a function, and it fulfills it.
I quit my carrer in formal mathematics last year, i remember concepts from this video in a course i took on the second year. there's always so much to learn in math i will always love it for this, as for me i will be shifting gears into electrical engineering next year i recon it suits me better and truth be told the formalisms and rigor got the best of me but you keep at it shit pays off, or so I've been told.
I must admit, I am not the target audience for this video. If I had just received a clumsy explanation of the Dirac Delta in some class, this would certainly be a great video! But since this is the first I have heard of this equation, this all felt a bit disconnected from any context. It reminds me of a guide I once read on building linked lists. I understood it, but without any implementation shown, I didn’t know how to use anything I had read or when I would even want to. I don’t really know what the video is missing since I know nothing about this subject you didn’t just tell me, but I think this needs to be a bit more grounded rather than being some small piece of a puzzle I have never seen. For context, I took 4 calculus classes in college. Otherwise, the video is very clear and the visuals are good. Thanks for making it.
Thanks for the feedback! In practice, physicists and engineers use the Dirac delta to model phenomena that occur over a very short period of time or in a very small region of space. If you vibrate a mechanical system, the input to that system is a sine wave. But if you hit the system with a hammer (imparting a force over a very short period of time), you can model the input as a Dirac delta. Similarly, a distribution of charges create an electric field, and you can solve for the electric field by solving a differential equation. If you take the charge distribution to be a Dirac delta, that models a charge localized to a point, such as an electron, and can therefore solve for the electric field created by an electron.
@@kieransquared Ok, thanks for the explanation. I actually have a bachelors in mechanical engineering, but every time a class mentioned impulse, they kind of glossed over it and never mentioned any of this.
@@kieransquared in a sense it seems physicists are always ahead of mathematicians in using mathematical concepts. Only after some time mathematicians catch up and make it rigorous, but somehow it always works. It strongly reminds me of calculus early days. When Newton used it concepts like limits and derivatives weren't well defined. He essentially divided 0 over 0, but without any problems or paradoxes emerging. Then only in the 19th century an actual rigorous underbelly emerged with the epsilon delta definition, riemann sums, etc showing it was alright after all.
Your video is not just about understanding the Dirac Delta, it also gave an intuitive way to understand Lebesgue integral, which is pretty hard to visualize while reading the theory of it. Now I can understand why some engineers approximated area of a region by drawing it on a piece of cardboard, then measure its weight and use the "weight to volume" formula to determine the area since volume = height x base-area 🤣Turn out they are just doing what Lebesgue integral did
I think saying that delta itself is a functional is bit misleading. It's more accurate to say that delta is the kernel of the evaluation functional E_0(f) = f(0). Similarly, shifting delta gives the kernel for E_y(f) = f(y) in general.
But the kernel of a functional (in the sense of the Schwarz kernel theorem, if we're talking about distributions) is itself a functional - it's perfectly rigorous to say that the Dirac delta is precisely the evaluation functional, and it's how plenty of books define it.
@@kieransquared Schwartz asserts a one-to-one correspondence between distributions and maps by the kernel relationship, but I've never seen anyone make a full identification between them as such. Not to say I don't believe you, I'm sure there are references like that. It's not too far off from making an identification between a linear map and its matrix. Still, I would honestly give the same critique to those authors as I have here.
@@alxjones I think pretty much every text on distributions I've seen (notably Hormander's tome on distribution theory) defines the delta distribution as the map delta_a(f) = f(a) (and also wikipedia, Wolfram mathworld, etc). Now that I think more about it, I'm also not even sure what it means for a functional to have a kernel - I'm only familiar with the notion of a kernel for "distribution-valued" functionals, i.e. bounded linear maps from test functions to distributions.
Isn’t a linear functional a linear map from a vector space V to its field F?* Is the vector space in this case a certain set of functions like C([a,b])?
Yup, in this case the vector space is C([a,b]), which is a vector space over R. Here we also need the linear functionals to be continuous, since C([a,b]) is infinite dimensional. (linear functionals on finite dimensional spaces are always continuous)
The only critique I would have is, to my mind the usual "working mathematician" way of thinking about the dirac delta is as a type of generalized function called a Schwartz distribution, rather than the purely measure theoretical way it is presented here. But I have a difficult time imagining how to make that an approachable topic for a video like this. You hinted at this briefly by mentioning that it is a linear functional, but then pretty much just abandoned this train of thought.
True, I considered presenting the Dirac delta as a distribution instead, but I realized it would almost be overkill, as it’s a lot harder to conceptualize distributions compared to measures. Maybe I’ll make another video about distributions at some point. In any case, since measures are a subset of the distributions, I think it’s easier to understand the measure-theoretic approach first before distributions, even if it’s less powerful.
Underrated & underappreciated video. A must watch for anyone planning on going down the measure theoretic probability route.
Also enjoyed the subtle but powerful introduction to things like radon-nikodym without explicitly mentioning their names :D
Thanks! I feel like there’s lots of parts of measure theory that seem really technical but have nice concrete intuition behind them.
I've been a little confused while learning about distribution theory and how the dirac delta function is defined through that lens, so it was really interesting and informative to see how a different branch of mathematics adds intuition and creates a more complete understanding of this object.
Thank you for this great video.
One thing that helped me understand distributions a bit better is the fact that most (but not all) of them can actually be represented by integration against a measure, just like with linear functionals acting on continuous functions. So really, once you get an intuitive feel for what measures do, you just need to account for some "extraneous" types of distributions like the Cauchy principal value and derivatives of the Dirac delta in order to completely characterize distributions.
As a former engineering student I have never thought of the Dirac function as anything more than a normalized ideal pulse input. It is really nice to know about the math it originated from. Great video.
If I remember correctly it was the other way around. Dirac had used it in physics before mathematicians sorted everything out with it.
This was the clearest interpretation of measure theory I've seen in a while.
I feel like your video could really use a better thumbnail to get more views.
Thanks! Made a quick updated thumbnail, I’m not very artistic but hopefully it’s a bit more eye-catching.
@@kieransquared looks like your video blew up since you changed it haha
As a physics student whose had way more than his share of suffering from major mathematical itchiness and headache when it comes to the Dirac delta function, I really appreciate your effort and enjoyed the simplicity of the presentation and also think I actually did learn a number of new things; namely, the Riesz-Markov-Kakutani thing. Truth is, even with my preliminary trainings in real analysis, abstract and linear algebra, and even some rudimentary knowledge of basic topology and measure theory, this particular subject is way too advanced to be readily grasped, even by the expert folk with a healthy background in math. I also watched a lot of content under #SoME2 to understand how “renormalization” works and learn its basic principles, but unfortunately it’s one of those difficult, inaccessible materials. Nevertheless, kudos on a great video and hope your channel grows BIG👍🏻❤️
The delta function can also be defined as the limit of Gaussians with mean 0, as variance -> 0.
Yup! I actually have another video that touches upon this :)
In what topological space is this limit taken?
@@carlaparla2717 You could either take it in the topology of measures, or in the topology of distributions (these are the respective weak* topologies for the space of continuous functions, and the space of smooth compactly supported functions). Basically, a sequence of measures mu_n converges weak* to mu if for all continuous functions, integral(f dmu_n) converges to integral(f dmu). So in this case, the dmu_n would be [n/sqrt(2pi)]*e^{-(nx)^2/2} dx, where the standard deviation is 1/n, and they converge to the Dirac measure.
@@kieransquared couldn't you just use the uniform distribution to make life easier?
@@10babiscar That’s true, a uniform probability measure over [-h,h] also converges in the weak* topology to the Dirac measure as h goes to zero
Summer of math expedition #some2 is the GOAT of RUclips content, hashtags, collabs, or whatever this awesome thing is. Wow!!
Thank you for this video. It honestly made quite a few of my holes in math knowledge filled. Not just the dirac delta function. I wish you success in this youtube adventure.
This was really well done! I can't believe I just found your channel - as a video creator myself, I understand how much time this must have taken. Liked and subscribed 💛
So I‘ve actually been thinking about the idea of integrating with dx as an input and not just as the final term, and the Dirac delta seems like it’d be 1/dx at 0 and 0 everywhere else based on the introduction. This would satisfy both properties (no I didn’t accidentally recreate the Dirac delta I just realized how something I was thinking about could be used to create something like it)
I've been (ab)using the delta function quite a bit lately. Great Video 👌, interpreting delta in the context of measure theory demystifies quite a few things
This was absolutely fantastic. As somebody doing Many-Body QFT with interactions I always just took it as a given and even though I did some measure theory but never Functional Analysis I was always surprised and bothered by some integrals (for example a rigorous calculation of a particle-hole bubble) and the relationship of things like the Heaviside step “function” and the Dirac “function”.
This helped a lot for my intuition and I hope you will gain subscribers soon, your way of explaining is superb
THANK YOU FOR MENTIONING THE RIESZ MARKOV KAKUTANI THEOREM
I have been having analytical mechanics and have been messing around with variational principles, and I've always wondered why we're extremizing one type of functional (namely one where we are integrating) and now I know why! Thank you, I searched everywhere for this!
The Dirac Delta is absolutely a function. It's a set of ordered pairs such that if (a,b) and (a,c) are elements, then b=c. That's the definition of a function, and it fulfills it.
Excellent video, enjoyed every second of it! Also didn’t know the Riesz-Markov-Kakatai theorem, so cool!
Oh my god can you do more on measure theory, this is a great video. I did not appreciate measure theory (as an engineer) until I watch your video
I quit my carrer in formal mathematics last year, i remember concepts from this video in a course i took on the second year. there's always so much to learn in math i will always love it for this, as for me i will be shifting gears into electrical engineering next year i recon it suits me better and truth be told the formalisms and rigor got the best of me but you keep at it shit pays off, or so I've been told.
Yessss buddy this is the best SoME2 video so far, though I might be biased as a physicist.
I must admit, I am not the target audience for this video. If I had just received a clumsy explanation of the Dirac Delta in some class, this would certainly be a great video! But since this is the first I have heard of this equation, this all felt a bit disconnected from any context. It reminds me of a guide I once read on building linked lists. I understood it, but without any implementation shown, I didn’t know how to use anything I had read or when I would even want to.
I don’t really know what the video is missing since I know nothing about this subject you didn’t just tell me, but I think this needs to be a bit more grounded rather than being some small piece of a puzzle I have never seen.
For context, I took 4 calculus classes in college.
Otherwise, the video is very clear and the visuals are good. Thanks for making it.
Thanks for the feedback! In practice, physicists and engineers use the Dirac delta to model phenomena that occur over a very short period of time or in a very small region of space. If you vibrate a mechanical system, the input to that system is a sine wave. But if you hit the system with a hammer (imparting a force over a very short period of time), you can model the input as a Dirac delta. Similarly, a distribution of charges create an electric field, and you can solve for the electric field by solving a differential equation. If you take the charge distribution to be a Dirac delta, that models a charge localized to a point, such as an electron, and can therefore solve for the electric field created by an electron.
@@kieransquared Ok, thanks for the explanation. I actually have a bachelors in mechanical engineering, but every time a class mentioned impulse, they kind of glossed over it and never mentioned any of this.
@@kieransquared in a sense it seems physicists are always ahead of mathematicians in using mathematical concepts. Only after some time mathematicians catch up and make it rigorous, but somehow it always works.
It strongly reminds me of calculus early days. When Newton used it concepts like limits and derivatives weren't well defined. He essentially divided 0 over 0, but without any problems or paradoxes emerging. Then only in the 19th century an actual rigorous underbelly emerged with the epsilon delta definition, riemann sums, etc showing it was alright after all.
@@dekippiesip because physicists invente things they need, and then come mathematicians to explain "you all are wrong and you can't do like this!" :)
I love this and really appreciate your explanation.
Do you have any sources that explains this further by any chance?
Thanks for the wonderful video - very clear and concise.
Very clear and intuitive 👌👌
Very nice! Some true rigorous definitions and explanations!
Great presentation! Just a note, at 2:47 you should have written m([a,b])=|b-a| . Thanks
Awesome, and a great intro to measure theory too
Very nice video!
thanks! do i know you from somewhere?
@@kieransquared do I know YOU from somewhere?
Really really good video.
Beautiful
Very interesting video !
Thanks a lot
This was really really cool! Thank you so much!
Your video is not just about understanding the Dirac Delta, it also gave an intuitive way to understand Lebesgue integral, which is pretty hard to visualize while reading the theory of it.
Now I can understand why some engineers approximated area of a region by drawing it on a piece of cardboard, then measure its weight and use the "weight to volume" formula to determine the area since volume = height x base-area 🤣Turn out they are just doing what Lebesgue integral did
Phenomenal
Well explained. Subscribed 👍🏻
I think saying that delta itself is a functional is bit misleading. It's more accurate to say that delta is the kernel of the evaluation functional E_0(f) = f(0). Similarly, shifting delta gives the kernel for E_y(f) = f(y) in general.
But the kernel of a functional (in the sense of the Schwarz kernel theorem, if we're talking about distributions) is itself a functional - it's perfectly rigorous to say that the Dirac delta is precisely the evaluation functional, and it's how plenty of books define it.
@@kieransquared Schwartz asserts a one-to-one correspondence between distributions and maps by the kernel relationship, but I've never seen anyone make a full identification between them as such.
Not to say I don't believe you, I'm sure there are references like that. It's not too far off from making an identification between a linear map and its matrix. Still, I would honestly give the same critique to those authors as I have here.
@@alxjones I think pretty much every text on distributions I've seen (notably Hormander's tome on distribution theory) defines the delta distribution as the map delta_a(f) = f(a) (and also wikipedia, Wolfram mathworld, etc). Now that I think more about it, I'm also not even sure what it means for a functional to have a kernel - I'm only familiar with the notion of a kernel for "distribution-valued" functionals, i.e. bounded linear maps from test functions to distributions.
more on measure theory for noobs? that would be awesome. suscribed.
Isn’t a linear functional a linear map from a vector space V to its field F?* Is the vector space in this case a certain set of functions like C([a,b])?
Yup, in this case the vector space is C([a,b]), which is a vector space over R. Here we also need the linear functionals to be continuous, since C([a,b]) is infinite dimensional. (linear functionals on finite dimensional spaces are always continuous)
can you 'actually' have it be = infinity if you use surreal numbers? because then omega * eta = 1
Great video 😁
The only critique I would have is, to my mind the usual "working mathematician" way of thinking about the dirac delta is as a type of generalized function called a Schwartz distribution, rather than the purely measure theoretical way it is presented here. But I have a difficult time imagining how to make that an approachable topic for a video like this. You hinted at this briefly by mentioning that it is a linear functional, but then pretty much just abandoned this train of thought.
True, I considered presenting the Dirac delta as a distribution instead, but I realized it would almost be overkill, as it’s a lot harder to conceptualize distributions compared to measures. Maybe I’ll make another video about distributions at some point. In any case, since measures are a subset of the distributions, I think it’s easier to understand the measure-theoretic approach first before distributions, even if it’s less powerful.
Very nifty!
Flashbacks to "Optimization in Vector Spaces"
Neat!
Good video although there was a video by ThatMathThing showing how you could interpret it as an actual function
Can you send the link and/or video title on RUclips? I can't seem to find it.
@@WindsorMason ruclips.net/video/kA3r4Td2E3E/видео.html&lc=UgwPJEk0ilR56Sz0Unp4AaABAg
3:00 Measures obey subadditivity, the = sign of the boxed equation should be
The union symbol with a flat bottom denotes disjoint union; also subadditive set functions need not be measures so there needs to be equality there.
@@kieransquared Oh ok thx. I did not know the flat bottom means disjoint union.
My brother I would like to buy you lunch or a coffee or something.. anything.
i think dirac had a very bad case of the flu during the week that caused him to dream up this nonsense