Originally I wanted to upload this on 14th July, George Green's birthday, but this took longer than expected, so here we are. Correction: in 19:11, the Green's function lacks a factor of 1/m. The omega^2 in the oscillator equation should be replaced by k. Technically I didn't say that omega has to be the angular frequency, but since it normally does, people do point that out, so I'm also pointing it out as well. If you are interested in how to actually find the Green's functions, you can see here (but only if you are comfortable with normal ODE solving and/or multivariable calculus): drive.google.com/file/d/1D6E857eTvqM1CQgS1vYwcLqhLeGFL-aV/view?usp=sharing
what an unhelpful video. it doesn't even show how to find what green's functions are. "look them up on wikipedia bro". You might as well have shortened the whole video to "how to solve linear differential equations : look them up on wolfram alpha bro".
please make vedio on practical applications of green functions, like wave equation, transfer functions etc., this will help an engineer to teach lesson to physicist friends!!
For those familiar with linear systems analysis, there is a useful analogy: Green's function corresponds to the system's response when the input is an impulse function (Dirac's delta). Thus, to obtain the solution for a different excitation, we use the convolution integral of the impulse response (Green's function) with the input to the system.
@@winstonvpeloso Think again. If I tell you that an operator can be represented by a matrix, you shouldn't conclude that you know everything about Functional Analysis if you have just taken a course in Linear Algebra. It's exactly the same here. Green functions are differential operators acting on distributions. There are *many* more nuances than you think.
@@Evan490BC you misread my comment (or it was more nuanced than you think). all i meant was that the video and walter’s comment contain a surprisingly similar amount of information given the difference in length. how much of that detail did the video cover?
can I say Green's function is system's response when the Forcing term is Dirac's Delta? Forcing term as explained at ruclips.net/video/ism2SfZgFJg/видео.html
It is great to have different perspectives to understand complex things like Green's function. Having some signal processing background, explanation by Prof. Walter made some sense on the concept of Green's function. Explanation of Green's function in this video also makes sense from different perspective. Great video all in all.
Thanks for the kind words! However, the form of submission says that the video / blog post should have a length suitable for a 5-10 minute view, and this video is well over the time limit. Will have to make another video if I would enter the contest!
@Tim Wagemann it does have to be a new video though, not currently out there on the internet. But I agree that if you have another cool topic to talk about, you should go ahead and submit, so many people could find you and you’re great at explaining maths.
I just fell on my head and checked youtube for something that I could watch without having to concentrate to hard. I didn't know about greens function. I managed to follow the video almost to the end :-) I return happily tomorrow when my head is better. thank you for your work.
you know dude even if you do not say it out loud but having been through college maths I can tell everyone that making this video is not easy. For such a crazy high level topic being explained so simply there is easily multiple hours of work put in to generate every minute of video, from scripting, conceptualizing, text and sketches, animation, voicing, music, and ensuring at each stage it is making sense to a newcomer and adding all the required bits in a predigested easy to follow way requires tremendous hard work as well as tremendous effort. He has summarized 6ish hours of maths in 20 minutes and made it accessible to every single person who has even a basic math foundation. Serious hats off dude. You are amazing. Absolutely amazing!
I think this is the most approachable video on Green's functions I've ever seen. Thanks for making this! It's going to take a few watches to sink in, but already it's starting to make more sense. Your videos are always super interesting, and extremely helpful!
Thanks so much for the kind words! Indeed the video is made with the intention that people without much advanced knowledge could understand, so I'm glad that people find it approachable!
I just finished my third year of a maths degree and the intuition that I had gathered for Green's functions was that it was an "infinitesimal amount of solution to the DE" that is integrated over the region. Of course they don't explain anything at all in this aspect so it's nice to see it explained with animations
Great stuff man, when I was at the university I found tons of resources for lower division math and physics, but once I started my upper divisions things like these were harder to find, and made in such a comprehensive way at that. Thank you and may you prosper
I really appreciate the fact that you make these videos interesting to those who already know a little bit of math and wish to go a bit deeper in. Thank you.
This is by far the best video on Green's functions I could find. I'm currently taking Electrodynamics at uni and it helped me finally understand this topic. Thank you!
I've never seen a video giving us such an AMAZING both introduction to green functions and using them. When our teacher for theoretical physics explained us this years ago I only slept in during the lecture. Many, MANY thanks! This video is PERFECT. No more words to say.
This is the best explanation of Green's functions I've seen, thank you! And the applications are limitless: the propagators in Feynman diagrams are based on Green's functions for example, so if you get this video, you're well set to learn quantum field theory
I think from the responses you have received it's clear that many of the viewers ,if not all, want advance topics to be covered as well. So we hope you will not let us down.BTW, Keep going sir, you are doing a great job 🙂🙂🙂.
I get to learn a way more (at least geometrically) than from my instructors. Propagators can be a real nuisance in QM, without understanding what is a green function. Great Explanation! ❤️
Thanks! Glad it helps! I only know its more classical applications, but not quite how it could be applied in QM, but glad that it is useful for much more areas than I imagined!
@@mathemaniac The propagator is arguably the key component in any quantum mechanical system when you do any kind of scattering process with particle collisions. There's so many ways to visualize it that its hard to keep track of, lol.
@@mathemaniac Usually scattering is formulated in momentum space "q" (as in the Fourier conjugate to position space), so the 1/r^2 force is replaced by 1/q^2 for massless photons. Add mass and it's 1/(q^2+m^2)..which is why you hear ppl talk about "scattering poles". Since it's all done in perturbation theory, at higher order you get nested Green's Functions and the divergent integrals of renormalization. Thankfully, Feynman came up with a bookkeeping method that is squiggly pictures. Most virtual particles are really just Green's functions.
This was...far beyond what i expected. You guys could have gotten away with much, much less of an effort without any pushback. Instead, we are left with this... An absolutely beautiful, visually pleasing, simple yet concise explanations which work hand in hand with the animations to bring us an intuitive, entry-level walk-through of the green's function I'm honestly awe-struck. I can confidently say this is easily one of the best videos on function I have yet had the privilege to enjoy here on RUclips. (and I watch nothing but science and physics docs on RUclips etc) What an absolutely superb masterpiece, what an incredibly engaging tool that undoubtedly will benefit thousands and thousands of inquisitive minds. Thank you so much for everyone responsible for this labor of love. It truly shows your passion for your field, and hoo boy what a treat the whole video was. It is insanely rare that animations, live explanations, and facts all come together so brilliant and organically organized in such a way that the end product comes together to create something much, much greater than each part on it's own. What an honor.
Awesomevideo, always had trouble with Green's Functions in undergrad, felt too abstract. Now that they're coming up again in graduate E/m this video was a life saver for me.
This is, among your videos, the one I could least follow. I don't know physics and, to me, the examples only obfuscated the subject. In general I still love your videos thou, just felt the urge to, once again, modulate your knowledge of our background. So go on! Maybe one day I will come back to this one.
I had completely forgotten about the method of images, and as I reach the end of the video I couldn't do anything for 2-3 minutes as the flashbacks started coming up in my mind.
Mathemaniac, you are one of the best teachers I've ever seen. Those animations, a visual interpretation of maths could be a key tool for anyone's comprehension capabilities. I may test out if someone from my family without maths background can understand this. This could be awesome. Wish me luck.
An interesting analogy I always make is to consider the delta function to be the identity matrix and green function is the inverse matrix of the linear operator.
Yes! In fact it's not uncommon to see convolutions with Green's functions written as 1/L, where L is some linear differential operator. Because it's exactly what it is, L(1/L)f(x) = f(x)
@@david203 Sorry, for some reason I assumed you had more math background than is likely the case. Maybe I was right but I still poorly expressed myself, which would be sad but also very likely. XD Long story short, it can be a function of two arguments if you want it to. Just write G(x-y) instead of G(x).
Absolutely amazing work! The explanations and visuals are stunning. The exercise really helped with my engagement and ensuring I understood. Forgot to mention in the form, but adding more questions throughout, if possible, would be awesome
Great introduction into this topic. I never managed to get a hang on Green's functions as I expected them to be something totally different. Black math magic basically. Your changed that. Thank you!
Thanks! This was already in my video idea list, but note that I am not specifically gearing my content towards engineering or any other direction, so I can't guarantee anything specific to engineering (especially since I am not an engineer myself).
Excellent video, and excellent channel. Does a good job unlocking intuition for Green's functions. I didn't come across this concept (explicitly anyway) until I started studying QFT and propagators. This video would have helped accelerate my learning! I also use electrostatic potential as my 'toy model' to get a handle on what Green's functions represent. I.e. to conceptualize the Green's function as the analogue of the electric potential of a point charge, which of course must be integrated to get the potential of a charge distribution, the latter just being a 'sum' of point charges. I like to think of it this way: since the source is a sum of point sources, the solution will be a sum (due to linearity) of 'point solutions', which are the Green's functions. Thanks for the great content!
Thanks for the kind words! The electric potential is the easiest one to visualize, which is why I chose it. It is possible, although more difficult, to visualize Green's functions using the oscillator example, but definitely the electrostatics is a lot more intuitive.
18:25 the field dropping to 0 at infinity is so "obviously" correct when talking about the electrical charge potential that I didn't even consider that depending on the problem, that might not be the boundary condition we want for our differential equation. Dropping to 0 is just so nice it wasn't until you pointed out otherwise that I realized other possibilities might exist. Thankfully a lot of the times we want to solve, the boundary condition is a nice 0 or something like it.
This is such a great video to gain some intuition for Green's Funktion. Thank you for all the effort you put into this. It gave me some good help to understand my electrodynamics class.
A few comments. First, it may be a typo but for the oscillator, bc the acceleration term includes the mass, the governing DE should not include the frequency but rather the stiffness. One arrives at the frequency when the system is mass normalized. Second, the inhomogenous BCs can be turned into a linear combination of symmetirc and anti-symmetric BCs that are easier to solve. In the case of the cube, consider the symmetric BC of +2 on the top and bottom face and zero elsewhere and the anti-symmetric case of +2 on the top face and -2 on the bottom face and zero elsewhere. Not necessarily easy to solve but easier. The solution is then the linear combination of these two cases. But still a great video!
Nice video! I appreciate the well-chosen visuals and your clear and relaxed voice (the sound quality and the pace are great!). As for the Dirac delta, I would say that calling it a "function" and referring to distribution (or just measure) theory for further reading should be satisfying for those who like rigour (as I do). In any case, what you described tells us exactly what the Dirac delta *is* as a functional on the space of continuous functions, and also how it appears as a weak-* limit of functions. Good work!
beautiful explanation, I had difficulties understanding the idea behind those functions, but you put it very simply together. Thank you! I immediately subbed :)
Thank you very much for Green's function beautiful explanation. I was looking for it during several years. I discovered your interesting and deeply mathematical channel due to this function. I watched other your videos and they are also very interesting. I am happy to find your channel. Thank you again and go on in such way.
Most people call it a function, even if it is not a function, as I said in the video. Just search Dirac delta function online: both Wikipedia and Mathworld call it delta function, while specifically saying it's not a function.
@@jojo_jo2212 we know that. It is a limit of a normalized distribution, and has historically been called the delta-function, and least it was back in my day.
It is a good video but I advise everyon watch Ali Hajimiri series on signals. I am pretty sure I remember him dealing with Green's functions as well (I don't remember him calling it green's functions, I watched that series a couple of years ago maybe he did but I forgot it, but now that I have seen the explanation of what green's functions are, yes, he explained this concept). In that series he also explains Dirac delta very well. When it comes to dirac delta, he explained using Dirac delta in an integral something like this: think of dirac delta as a way to escape the integral, if you have dirac(x-alpha) in the integral that just pulls the value when x = alpha. Remove the integral and calculate the expression at x=alpha basically. That way of thinking about dirac delta is very useful in this video as well. He obviously explained how dirac delta has to have area of 1 and he showed the limit for dirac delta. Great video BTW, you gave an intuition that greens function in your example is electric potential by a point like particle and then if you want electric potential you just sum up electric potentials from point like particles and you can do this switch of order of operations (first calculating electric potential from point like particles and then adding them up) because you are dealing with a linear operator. That's nice. Ali Hajimir's videos also a great intuition for Green's function. He talks about signals, so he presents a device which has an input signal and it outputs a signal (device can be thought of as representive an operator, while the signals are functions). He asks the question - what signal should we input into our device, so that the output signal is the dirac delta. That input signal is the green's function for our device, If I understood your video correctly. Once we have this function we can use it to construct any output signal that we want. How? Dirac delta will be our building block of our output signal. Remember the property of dirac delta that when we put dirac delta in an integral and multiply it with a function f, the integral will evaluate to function f when the argument of dirac delta is zero. So, if we have dirac(x) times some function f(x) in the integral (which includes x=0), the result will be function f evaluated at 0. Ok, so, we can now construct one point of our output function, if we plug in the output function to be f, then at f(0) we will get the correct result. How do we get our entire output function? We need a way to slide dirac delta, so we can write dirac(x - dummyVariable) and let's say we integrate with the respect to dummyVariable from minus to plus infinity. Then, we will get our entire output function, because the integral evaluates when dirac's argument is 0, meaning that when x = dummyVariable that's the result of the integral. Since we are integrating from negative to positive infinity, we are going to get the entire output function. That's what I remember from his class that I watched a couple of years ago, I don't remember that this was called the green's function, but upon seeing your video I realized that that was called the green function. I like both explanations and I reccomend everyone watch that entire series, it is a lot of fun and explanations are good. Here is the 1st video in the series, there are 40 videos but trust me, they are worth it. ruclips.net/video/i9WixHfiZPU/видео.html
Excellent video with compelling visuals. I wish science teachers could explain these functions early on to help students get motivated. Thank you so much for beautiful explanation of green function.
im so lucky that this came out recently, im having my exam in "mathematical methods for physics" 3 weeks from now (currently studying for it!). I hope you do a video about eigenvalue-expansion and other methods for solving DE:s!
Hope your exams will go well, But since the other methods you mentioned like eigenfunctuon decomposition, it is too similar to a textbook that I wouldn't want to put on the channel, unless I can find a unique enough perspective on it.
How do you learn this by yourself? Does reading the textbook and struggling with the exercises work? Is there any other technique? I never was able to get this. Thank you for this video. It is greatly appreciated. Ten years ago, I was struggling with this so badly that I may have become crazy and am still recovering from this topic.
Glad it helps! For me, it is more about having a great lecturer on this topic, but honestly with maths, you just have to do the exercises to get familiar with a concept, no exception.
@@mathemaniac Thank you for your work. Your videos are a great supplement and complement for understanding these topics. I always worked on math and physics exercises and am a machine in terms of solving problems, I don't mind slogging through a ton of exercises just for practice.... however it got to a point where I would not be able to solve exercises anymore(proofs, PDEs, etc) no matter how much I read the text book section or re-reading articles. It just sucked staring at an exercise for weeks with no progress, guidance, nor understanding of what to do. Competent AND friendly lecturers are rare. That combination doesn't exist from my experience. I'm now grateful for RUclips for allowing people like you to exist in my life. Thank you from the bottom of my heart.
@@lambdamax I really do believe that advanced math is exactly like this. So many professional physicists, including the author of this video, will admit to areas of mathematics they just don't understand. Something is wrong when it is almost impossible to use all the fundamental tools of a profession or academic discipline.
Something that really needs a good video explanation (because I've never seen one) is the connection between Green's functions and particles in quantum physics.
While watching this video, the next video in the recommendations was (A Swift introduction to Geometric algebra) and it changed the way I think about vectors forever!
What's the advantage to using Green's function to solve the mass spring system rather than Laplace transforms? Are there situations where it's more difficult to work with Laplace transforms than Green's functions?
The Laplace Transform is an Special Limit Case of Greens Fuctions...Is like "compare" a M-16 With a Howitzer roughly ...SomeTimes there is not any advantage .....An even above exist the Power of Harmonic Analysis methods...
Good. I have an important question, which is defined as a Green-function in $V3$, I currently do things in Hodge-theories. And the only possible functions in $V3$ are the ones bounded by a theoretical Greens-Griffiths function, which you can see as the algebraic-closure of $X_{1}\mathfrak{n}$ or also written as $X= \mathfrak{H}_{n} \{1- 2\}$ where the Greens-Griffiths function is induced as $X:= \mathfrak{V3}$ which admits continuous-variations on a compact-oriented manifold.
>not even requiring that much calculus I can _barely_ keep up with my calc 3 knowledge, the only reason why I know this so well is from some other RUclipsr who went over electrostatics
George Green, largely self-educated, attended Gonville and Caius College, Cambridge as an undergraduate at the age of 40. He was regarded as a brilliant mathematician and sometime after graduating (4th in his year) was given a fellowship - but not long before he died. He wasn't appreciated in his day. It was only after his death that others (Lord Kelvin, in particular) appreciated exactly what he had done. As I am sure many know, his work has widespread application thoughout physics, including quantum physics. As an aside Caius has had other "famous" mathematicians: John Venn (his portrait was in the dining hall the last time I was there); John Horton Conway; Stephen Hawking. As a College it is more known for its medics than its maths. Its alumni include 14 Nobel Prize winners.
One thing I like about the Green's Functions method is that you're solving a DE (calculus) but the method is purely and completely Linear Algebra, with little input from calculus itself apart from the boundary/initial conditions
Great video. Clear, concise and effective. I have a suggestion, if you can, leave the lower space of the video empty as it gets used up by subtitles for people who have it on. Thanks for the video!
Video is great! However I wish you left us "answers" for oscillator exercise. How I'd answer (could be totally wrong). 1. F(t) is external force applied to our oscillator, therefore F(tau) gives weight to functions, and the delta tells if perturbation is applied at that time. 2. I briefly mentioned it in 1 3. :/ 4. G(t,tau)*F(tau) over all d(tau) should give us solution. So G(t,tau) should somehow tell us, how much of force F felt during time tau? I'm really confused on this one. 5. Linearity mean it doesnt matter if we add forces to spring independently and then sum displacement of oscillations, or we add all forces and then measure displacement. If someone could help me, that would be great, but I'm already thankful for the video, and with some other resources, solving these problems is less of a mystery, but I don't fully grasp the intuition.
Thanks for your appreciation! It was only meant to be "reiterating what this video says, but in a different context", and I didn't receive any real attempts in the exercise, so you are the first one doing it :) Essentially, what I intended was that using that momentum change = integral of force over small period of time, you can obtain the first answer (by a similar definition of delta function in 1D), and I am expecting "point impulse / impulse" on Q2, but I guess that's what you mean by "perturbation". For Q3: It is supposed to be that "applied force can be thought of as a 'continuous sum' of point impulses". For Q4: the Green's function describes the displacement of the oscillator after we apply an impulse. For this reason, Green's function is usually called the "impulse response". For Q5: Yes, the idea is right: exactly copying the "adding different charge distributions --> adding up the electric potential", so in this case, "adding different forces --> adding up the displacement" For Q6: From the formula that x(t) = int G(t, tau)*F(tau) d(tau), we can interpret that the displacement is a continuous sum of the impulse responses. Have updated the description in case anyone wants to do the same, and don't see this comment.
Great video! Slight typo at ~19:30. The harmonic oscillator ode shouldn't still have an m on the second order time derivative, if you already have ω^2 as your coefficient on x(t). This propagates into your solution as well, which is why there's a weird, dimension-ful argument in the sine function. But awesome video nonetheless!
Differentiation with respect to ROA (for lack of the symbol) will describe a smooth downward deceleration of the springing pendulum in a linear representation, absV by T you might say.
Nice job. I hated the way Green's functions were taught to us. Maybe I've forgotten it since I did it, but just the simple illustration showing how the function is a series of functions of x and xi helped.
I understand that we're taking the limit at 7:04 . I also understand that the volume V never actually equals zero because that's essentially what a limit means. But the sight of (0/0)=0 is jaw-dropping. Especially that in that specific line, the limit isn't explicitly written before the expression, but it's implicitly understood that it's a limiting process. I know it's only an intuitive video but I couldn't not say that *confused emoji* Other than that, great video! Keep up the good work
Thanks for the compliment! The computer did take approximations of the volume, but it is so small that it returns 0. Sorry for the confusion this generates.
Julian Schwinger , the legendary theoretical physicist, found the true value of Green's function, I think. So, please make a video about Julian Schwinger, who is my hero, if you have time.
Nice video. In addition to the unbounded or semi-bounded cases, which can be solved by pure integral transforms, it would be nice to have a discussion of solutions in bounded regions where the eigenfunction expansion method is used.
Thanks! The video is supposed to be geared towards a more general audience - I specifically don't require too much knowledge of calculus to begin with, so the eigenfunction expansion method or the Fourier or Laplace transforms definitely go beyond the scope.
@@mathemaniac Is the difference between Fourier and Laplace transforms that Fourier applies only to periodic functions while Laplace applies to functions having infinite domains?
@@david203 The Fourier *Series* is used for periodic functions. The Fourier *transform* and Laplace transform are for functions where their domains are unbounded: FT for functions f:Rd->C, and LaplaceT for functions f:[0,inf)->C
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Originally I wanted to upload this on 14th July, George Green's birthday, but this took longer than expected, so here we are.
Correction: in 19:11, the Green's function lacks a factor of 1/m.
The omega^2 in the oscillator equation should be replaced by k. Technically I didn't say that omega has to be the angular frequency, but since it normally does, people do point that out, so I'm also pointing it out as well.
If you are interested in how to actually find the Green's functions, you can see here (but only if you are comfortable with normal ODE solving and/or multivariable calculus): drive.google.com/file/d/1D6E857eTvqM1CQgS1vYwcLqhLeGFL-aV/view?usp=sharing
what an unhelpful video. it doesn't even show how to find what green's functions are. "look them up on wikipedia bro". You might as well have shortened the whole video to "how to solve linear differential equations : look them up on wolfram alpha bro".
@@frankdimeglio8216 I made $1 billion while in a coma, using this one weird trick...
please make vedio on practical applications of green functions, like wave equation, transfer functions etc., this will help an engineer to teach lesson to physicist friends!!
For those familiar with linear systems analysis, there is a useful analogy: Green's function corresponds to the system's response when the input is an impulse function (Dirac's delta). Thus, to obtain the solution for a different excitation, we use the convolution integral of the impulse response (Green's function) with the input to the system.
damn. this comment made the video obsolete from my pov.
@@winstonvpeloso Think again. If I tell you that an operator can be represented by a matrix, you shouldn't conclude that you know everything about Functional Analysis if you have just taken a course in Linear Algebra. It's exactly the same here. Green functions are differential operators acting on distributions. There are *many* more nuances than you think.
@@Evan490BC you misread my comment (or it was more nuanced than you think). all i meant was that the video and walter’s comment contain a surprisingly similar amount of information given the difference in length. how much of that detail did the video cover?
can I say Green's function is system's response when the Forcing term is Dirac's Delta?
Forcing term as explained at ruclips.net/video/ism2SfZgFJg/видео.html
It is great to have different perspectives to understand complex things like Green's function. Having some signal processing background, explanation by Prof. Walter made some sense on the concept of Green's function.
Explanation of Green's function in this video also makes sense from different perspective. Great video all in all.
3Blue1Brown is having a video contest this summer. You should submit this! It's great!
Thanks for the kind words! However, the form of submission says that the video / blog post should have a length suitable for a 5-10 minute view, and this video is well over the time limit. Will have to make another video if I would enter the contest!
@@mathemaniac please do enter, more people need to know about this amazing channel!
@Tim Wagemann it does have to be a new video though, not currently out there on the internet.
But I agree that if you have another cool topic to talk about, you should go ahead and submit, so many people could find you and you’re great at explaining maths.
Nah, I don't think he has the voice for it.
10 minutes through the video. I love it. Wonderful.
I came to this channel for the first time.
Solve DEs nuts
2 years late but this comment sent me. The internet always wins 😂
Nice
@@addisonkirtley1691 Glad I could make you smile. :^)
Gottem
I just fell on my head and checked youtube for something that I could watch without having to concentrate to hard. I didn't know about greens function. I managed to follow the video almost to the end :-) I return happily tomorrow when my head is better. thank you for your work.
Thanks for the kind words!
you know dude even if you do not say it out loud but having been through college maths I can tell everyone that making this video is not easy. For such a crazy high level topic being explained so simply there is easily multiple hours of work put in to generate every minute of video, from scripting, conceptualizing, text and sketches, animation, voicing, music, and ensuring at each stage it is making sense to a newcomer and adding all the required bits in a predigested easy to follow way requires tremendous hard work as well as tremendous effort. He has summarized 6ish hours of maths in 20 minutes and made it accessible to every single person who has even a basic math foundation. Serious hats off dude. You are amazing. Absolutely amazing!
Thank you so so much for the kind words! This video did take a long time to put together!
I think this is the most approachable video on Green's functions I've ever seen. Thanks for making this! It's going to take a few watches to sink in, but already it's starting to make more sense. Your videos are always super interesting, and extremely helpful!
Thanks so much for the kind words! Indeed the video is made with the intention that people without much advanced knowledge could understand, so I'm glad that people find it approachable!
I just finished my third year of a maths degree and the intuition that I had gathered for Green's functions was that it was an "infinitesimal amount of solution to the DE" that is integrated over the region. Of course they don't explain anything at all in this aspect so it's nice to see it explained with animations
Thanks!
Great stuff man, when I was at the university I found tons of resources for lower division math and physics, but once I started my upper divisions things like these were harder to find, and made in such a comprehensive way at that. Thank you and may you prosper
Glad to help!
I really appreciate the fact that you make these videos interesting to those who already know a little bit of math and wish to go a bit deeper in. Thank you.
Thank you very much!
This is by far the best video on Green's functions I could find. I'm currently taking Electrodynamics at uni and it helped me finally understand this topic. Thank you!
I've never seen a video giving us such an AMAZING both introduction to green functions and using them. When our teacher for theoretical physics explained us this years ago I only slept in during the lecture. Many, MANY thanks! This video is PERFECT. No more words to say.
Thanks so much! Glad to help!
VERY POWERFUL. When learning Green’s Functions (long forgotten) - after you do enough - you can basically just write down the answer.
This is the best explanation of Green's functions I've seen, thank you! And the applications are limitless: the propagators in Feynman diagrams are based on Green's functions for example, so if you get this video, you're well set to learn quantum field theory
Thanks for the kind words!
yeah, green's functions have like 8 different names, from propagators, to correlation functions, to response functions,
Definitely see these everywhere in higher level physics. Great to see the E&M examples!
Thank you for this flash back to my theoretical electrodynamics lecture. Back when studying physics was kinda fun...
I hated electrodynamics so much I dropped out of my PhD program entirely. I wish I had found it fun.
I think from the responses you have received it's clear that many of the viewers ,if not all, want advance topics to be covered as well. So we hope you will not let us down.BTW, Keep going sir, you are doing a great job 🙂🙂🙂.
Thanks! Will consider more advanced topics in the future!
You have done it! You have taught what my lecturers have failed to teach for the whole semester in 23 minutes!
I get to learn a way more (at least geometrically) than from my instructors.
Propagators can be a real nuisance in QM, without understanding what is a green function. Great Explanation! ❤️
Thanks! Glad it helps! I only know its more classical applications, but not quite how it could be applied in QM, but glad that it is useful for much more areas than I imagined!
@@mathemaniac The propagator is arguably the key component in any quantum mechanical system when you do any kind of scattering process with particle collisions. There's so many ways to visualize it that its hard to keep track of, lol.
@@mathemaniac Usually scattering is formulated in momentum space "q" (as in the Fourier conjugate to position space), so the 1/r^2 force is replaced by 1/q^2 for massless photons. Add mass and it's 1/(q^2+m^2)..which is why you hear ppl talk about "scattering poles". Since it's all done in perturbation theory, at higher order you get nested Green's Functions and the divergent integrals of renormalization. Thankfully, Feynman came up with a bookkeeping method that is squiggly pictures. Most virtual particles are really just Green's functions.
This should've existed 7 months ago for my exams.
Haha sorry about that! Hope that your exams went well nonetheless.
For me, 11 years ago. 🤣
What a wonderful explanation! I hope every university teaches this way!
Thanks so much for the kind words!
This was...far beyond what i expected. You guys could have gotten away with much, much less of an effort without any pushback. Instead, we are left with this...
An absolutely beautiful, visually pleasing, simple yet concise explanations which work hand in hand with the animations to bring us an intuitive, entry-level walk-through of the green's function
I'm honestly awe-struck. I can confidently say this is easily one of the best videos on function I have yet had the privilege to enjoy here on RUclips. (and I watch nothing but science and physics docs on RUclips etc)
What an absolutely superb masterpiece, what an incredibly engaging tool that undoubtedly will benefit thousands and thousands of inquisitive minds.
Thank you so much for everyone responsible for this labor of love. It truly shows your passion for your field, and hoo boy what a treat the whole video was. It is insanely rare that animations, live explanations, and facts all come together so brilliant and organically organized in such a way that the end product comes together to create something much, much greater than each part on it's own.
What an honor.
That's so kind! Thank you!
Awesomevideo, always had trouble with Green's Functions in undergrad, felt too abstract. Now that they're coming up again in graduate E/m this video was a life saver for me.
This is, among your videos, the one I could least follow. I don't know physics and, to me, the examples only obfuscated the subject. In general I still love your videos thou, just felt the urge to, once again, modulate your knowledge of our background. So go on! Maybe one day I will come back to this one.
I graduated in physics, yet could not follow much of this video. Too many basics not understood, I guess.
I had completely forgotten about the method of images, and as I reach the end of the video I couldn't do anything for 2-3 minutes as the flashbacks started coming up in my mind.
This is a fantastic explanation! A lot of pieces suddenly fell into place after watching this.
Glad it helps!
Mathemaniac, you are one of the best teachers I've ever seen.
Those animations, a visual interpretation of maths could be a key tool for anyone's comprehension capabilities. I may test out if someone from my family without maths background can understand this.
This could be awesome. Wish me luck.
This is simply beautiful. I studied mathematics and physics. I started studying on my own analysis. These videos explains a lot of intuition.
A crystal clear introduction to the idea behind Green's functions!
Excited to learn something thanks to you!
Glad to hear it!
An interesting analogy I always make is to consider the delta function to be the identity matrix and green function is the inverse matrix of the linear operator.
That's an interesting perspective!
Yes! In fact it's not uncommon to see convolutions with Green's functions written as 1/L, where L is some linear differential operator. Because it's exactly what it is,
L(1/L)f(x) = f(x)
Only wish I understood how Green's function could be an inverse matrix, seeing that Green's function is a function of one scalar variable.
@@Ricocossa1 I could not follow this. I was lost with the very first sentence, sorry.
@@david203 Sorry, for some reason I assumed you had more math background than is likely the case. Maybe I was right but I still poorly expressed myself, which would be sad but also very likely. XD
Long story short, it can be a function of two arguments if you want it to. Just write G(x-y) instead of G(x).
The animation, the background music, the tone, the words are exaclty match the 3b1b style. Nice job.
This is incredible! A fascinating look at Green's functions. Amazing job
Thanks!
Absolutely amazing work! The explanations and visuals are stunning. The exercise really helped with my engagement and ensuring I understood. Forgot to mention in the form, but adding more questions throughout, if possible, would be awesome
Thanks so much! Glad that the exercise is useful!
Wow - what a fabulously instructive and interesting video! I learned more about solving PDEs from it than an entire college course!
Amazing video. The best yet on Green's functions on the internet in my opinion. Thanks a bunch man!
Wow, thanks!
This is a very nice video! Glad that the algorithm recommended me this!
Great introduction into this topic. I never managed to get a hang on Green's functions as I expected them to be something totally different. Black math magic basically. Your changed that. Thank you!
Brilliant work. Would love to see videos on integration of complex functions and their applications in engineering someday.
Thanks! This was already in my video idea list, but note that I am not specifically gearing my content towards engineering or any other direction, so I can't guarantee anything specific to engineering (especially since I am not an engineer myself).
Excellent video, and excellent channel. Does a good job unlocking intuition for Green's functions. I didn't come across this concept (explicitly anyway) until I started studying QFT and propagators. This video would have helped accelerate my learning! I also use electrostatic potential as my 'toy model' to get a handle on what Green's functions represent. I.e. to conceptualize the Green's function as the analogue of the electric potential of a point charge, which of course must be integrated to get the potential of a charge distribution, the latter just being a 'sum' of point charges. I like to think of it this way: since the source is a sum of point sources, the solution will be a sum (due to linearity) of 'point solutions', which are the Green's functions. Thanks for the great content!
Thanks for the kind words! The electric potential is the easiest one to visualize, which is why I chose it. It is possible, although more difficult, to visualize Green's functions using the oscillator example, but definitely the electrostatics is a lot more intuitive.
18:25 the field dropping to 0 at infinity is so "obviously" correct when talking about the electrical charge potential that I didn't even consider that depending on the problem, that might not be the boundary condition we want for our differential equation. Dropping to 0 is just so nice it wasn't until you pointed out otherwise that I realized other possibilities might exist. Thankfully a lot of the times we want to solve, the boundary condition is a nice 0 or something like it.
I would like the phase for 4 as 4295
This is such a great video to gain some intuition for Green's Funktion. Thank you for all the effort you put into this. It gave me some good help to understand my electrodynamics class.
Hope it helps!
Thanks for the video. I first came across this Green chap when I was taught a bit of physics long ago and they introduced us to Green's Lemma.
I found your channel today! Your videos are great and I hope/expect you will reach a larger audience soon
Thanks so much for the kind words!
A few comments. First, it may be a typo but for the oscillator, bc the acceleration term includes the mass, the governing DE should not include the frequency but rather the stiffness. One arrives at the frequency when the system is mass normalized. Second, the inhomogenous BCs can be turned into a linear combination of symmetirc and anti-symmetric BCs that are easier to solve. In the case of the cube, consider the symmetric BC of +2 on the top and bottom face and zero elsewhere and the anti-symmetric case of +2 on the top face and -2 on the bottom face and zero elsewhere. Not necessarily easy to solve but easier. The solution is then the linear combination of these two cases.
But still a great video!
Yes, your first point has already been covered by another commenter as well, but I can't edit the video on RUclips.
Good video about the introduction of green's functions. Thanks a lot
Nice video! I appreciate the well-chosen visuals and your clear and relaxed voice (the sound quality and the pace are great!).
As for the Dirac delta, I would say that calling it a "function" and referring to distribution (or just measure) theory for further reading should be satisfying for those who like rigour (as I do). In any case, what you described tells us exactly what the Dirac delta *is* as a functional on the space of continuous functions, and also how it appears as a weak-* limit of functions.
Good work!
Thanks so much for the compliment!
Cool video ! Thanks so much! This is a cool visualisation! Exited to learn something new !
Glad you liked it!
beautiful explanation, I had difficulties understanding the idea behind those functions, but you put it very simply together. Thank you! I immediately subbed :)
Glad it helped!
omg- THE BEST MATH EXPLANATIONS EVER. Thanks.
Happy to help!
Very nice introduction to Green's functions, Thank you!
Thanks so much!
Excellent video I was struggling to understand green functions in quantum field theory
Thank you very much for Green's function beautiful explanation. I was looking for it during several years. I discovered your interesting and deeply mathematical channel due to this function. I watched other your videos and they are also very interesting. I am happy to find your channel. Thank you again and go on in such way.
Thanks so much for the kind words!
I love Green's functions! Delta function rock!
Indeed!
Delta what you said
I think you spelled "distribution" wrong
Most people call it a function, even if it is not a function, as I said in the video. Just search Dirac delta function online: both Wikipedia and Mathworld call it delta function, while specifically saying it's not a function.
@@jojo_jo2212 we know that. It is a limit of a normalized distribution, and has historically been called the delta-function, and least it was back in my day.
It is a good video but I advise everyon watch Ali Hajimiri series on signals. I am pretty sure I remember him dealing with Green's functions as well (I don't remember him calling it green's functions, I watched that series a couple of years ago maybe he did but I forgot it, but now that I have seen the explanation of what green's functions are, yes, he explained this concept). In that series he also explains Dirac delta very well. When it comes to dirac delta, he explained using Dirac delta in an integral something like this: think of dirac delta as a way to escape the integral, if you have dirac(x-alpha) in the integral that just pulls the value when x = alpha. Remove the integral and calculate the expression at x=alpha basically.
That way of thinking about dirac delta is very useful in this video as well. He obviously explained how dirac delta has to have area of 1 and he showed the limit for dirac delta. Great video BTW, you gave an intuition that greens function in your example is electric potential by a point like particle and then if you want electric potential you just sum up electric potentials from point like particles and you can do this switch of order of operations (first calculating electric potential from point like particles and then adding them up) because you are dealing with a linear operator. That's nice. Ali Hajimir's videos also a great intuition for Green's function.
He talks about signals, so he presents a device which has an input signal and it outputs a signal (device can be thought of as representive an operator, while the signals are functions). He asks the question - what signal should we input into our device, so that the output signal is the dirac delta. That input signal is the green's function for our device, If I understood your video correctly. Once we have this function we can use it to construct any output signal that we want. How? Dirac delta will be our building block of our output signal. Remember the property of dirac delta that when we put dirac delta in an integral and multiply it with a function f, the integral will evaluate to function f when the argument of dirac delta is zero. So, if we have dirac(x) times some function f(x) in the integral (which includes x=0), the result will be function f evaluated at 0. Ok, so, we can now construct one point of our output function, if we plug in the output function to be f, then at f(0) we will get the correct result. How do we get our entire output function? We need a way to slide dirac delta, so we can write dirac(x - dummyVariable) and let's say we integrate with the respect to dummyVariable from minus to plus infinity. Then, we will get our entire output function, because the integral evaluates when dirac's argument is 0, meaning that when x = dummyVariable that's the result of the integral. Since we are integrating from negative to positive infinity, we are going to get the entire output function. That's what I remember from his class that I watched a couple of years ago, I don't remember that this was called the green's function, but upon seeing your video I realized that that was called the green function. I like both explanations and I reccomend everyone watch that entire series, it is a lot of fun and explanations are good. Here is the 1st video in the series, there are 40 videos but trust me, they are worth it. ruclips.net/video/i9WixHfiZPU/видео.html
Great channel for those wanting to quickly understand a topic ... thanks ... just subscribed.
Thanks so much!
Excellent video with compelling visuals. I wish science teachers could explain these functions early on to help students get motivated. Thank you so much for beautiful explanation of green function.
Thanks so much for the compliment!
im so lucky that this came out recently, im having my exam in "mathematical methods for physics" 3 weeks from now (currently studying for it!). I hope you do a video about eigenvalue-expansion and other methods for solving DE:s!
Hope your exams will go well, But since the other methods you mentioned like eigenfunctuon decomposition, it is too similar to a textbook that I wouldn't want to put on the channel, unless I can find a unique enough perspective on it.
Good luck 🤞
Beautiful and just the right timing as well! I'm about to start MATH2100 which covers PDEs.
Hope that it will help!
Your visuals are incredible
Thanks so much for the kind words!
How do you learn this by yourself? Does reading the textbook and struggling with the exercises work? Is there any other technique? I never was able to get this. Thank you for this video. It is greatly appreciated. Ten years ago, I was struggling with this so badly that I may have become crazy and am still recovering from this topic.
Glad it helps! For me, it is more about having a great lecturer on this topic, but honestly with maths, you just have to do the exercises to get familiar with a concept, no exception.
@@mathemaniac Thank you for your work. Your videos are a great supplement and complement for understanding these topics. I always worked on math and physics exercises and am a machine in terms of solving problems, I don't mind slogging through a ton of exercises just for practice.... however it got to a point where I would not be able to solve exercises anymore(proofs, PDEs, etc) no matter how much I read the text book section or re-reading articles. It just sucked staring at an exercise for weeks with no progress, guidance, nor understanding of what to do. Competent AND friendly lecturers are rare. That combination doesn't exist from my experience. I'm now grateful for RUclips for allowing people like you to exist in my life. Thank you from the bottom of my heart.
Awww really glad to help! This melts my heart :)
@@lambdamax I really do believe that advanced math is exactly like this. So many professional physicists, including the author of this video, will admit to areas of mathematics they just don't understand. Something is wrong when it is almost impossible to use all the fundamental tools of a profession or academic discipline.
Something that really needs a good video explanation (because I've never seen one) is the connection between Green's functions and particles in quantum physics.
Absolutely amazing video!
Thanks!
While watching this video, the next video in the recommendations was (A Swift introduction to Geometric algebra) and it changed the way I think about vectors forever!
wow dude, the quality of video is amazing!!
Thanks for the kind words!
3:01 Linear property means that the operation is symmetrical and goes both ways. From a source set onto the target set and the other way around.
Make every function depend on time and you would have the subject of my exam next thursday for my control theory class. What a happy coincidence!
What's the advantage to using Green's function to solve the mass spring system rather than Laplace transforms? Are there situations where it's more difficult to work with Laplace transforms than Green's functions?
The Laplace Transform is an Special Limit Case of Greens Fuctions...Is like "compare" a M-16 With a Howitzer roughly ...SomeTimes there is not any advantage .....An even above exist the Power of Harmonic Analysis methods...
Actually very well explained, congrats!
Glad it was helpful!
Great explanation! My only regret is that I didn't get to watch this video before taking Jackson EM. Would have saved a bit of my soul lol
Thanks!
Good.
I have an important question, which is defined as a Green-function in $V3$, I currently do things in Hodge-theories. And the only possible functions in $V3$ are the ones bounded by a theoretical Greens-Griffiths function, which you can see as the algebraic-closure of $X_{1}\mathfrak{n}$ or also written as $X= \mathfrak{H}_{n} \{1- 2\}$ where the Greens-Griffiths function is induced as $X:= \mathfrak{V3}$ which admits continuous-variations on a compact-oriented manifold.
>not even requiring that much calculus
I can _barely_ keep up with my calc 3 knowledge, the only reason why I know this so well is from some other RUclipsr who went over electrostatics
George Green, largely self-educated, attended Gonville and Caius College, Cambridge as an undergraduate at the age of 40. He was regarded as a brilliant mathematician and sometime after graduating (4th in his year) was given a fellowship - but not long before he died. He wasn't appreciated in his day. It was only after his death that others (Lord Kelvin, in particular) appreciated exactly what he had done. As I am sure many know, his work has widespread application thoughout physics, including quantum physics.
As an aside Caius has had other "famous" mathematicians: John Venn (his portrait was in the dining hall the last time I was there); John Horton Conway; Stephen Hawking. As a College it is more known for its medics than its maths. Its alumni include 14 Nobel Prize winners.
The great explanation I have ever seen!
Thanks!
Great video! looking forward to seeing more!
Thanks!
One thing I like about the Green's Functions method is that you're solving a DE (calculus) but the method is purely and completely Linear Algebra, with little input from calculus itself apart from the boundary/initial conditions
Thanks for putting efforts and making this video
Thanks for the kind words!
Thank you for making this video! It is very useful & friendly!
Thanks for the compliment!
Many thanks for the video. Well done!
Great video. Clear, concise and effective.
I have a suggestion, if you can, leave the lower space of the video empty as it gets used up by subtitles for people who have it on. Thanks for the video!
Thanks! I will take more care of this in the future.
Getting the hang of Green's now, thanks.
Glad to help!
i had these exact problems in my electrodynamics exam last week! wish i had this sooner
Thank you, I just learn Electrodynamics, perfect video, greetings from Berlin
Very clear explanation!
Glad you liked it!
Very well done video! My only complaint is that it left me wanting more.
There will be a video tomorrow! Actually a video series even!
Video is great! However I wish you left us "answers" for oscillator exercise.
How I'd answer (could be totally wrong).
1. F(t) is external force applied to our oscillator, therefore F(tau) gives weight to functions, and the delta tells if perturbation is applied at that time.
2. I briefly mentioned it in 1
3. :/
4. G(t,tau)*F(tau) over all d(tau) should give us solution. So G(t,tau) should somehow tell us, how much of force F felt during time tau? I'm really confused on this one.
5. Linearity mean it doesnt matter if we add forces to spring independently and then sum displacement of oscillations, or we add all forces and then measure displacement.
If someone could help me, that would be great, but I'm already thankful for the video, and with some other resources, solving these problems is less of a mystery, but I don't fully grasp the intuition.
Thanks for your appreciation!
It was only meant to be "reiterating what this video says, but in a different context", and I didn't receive any real attempts in the exercise, so you are the first one doing it :)
Essentially, what I intended was that using that momentum change = integral of force over small period of time, you can obtain the first answer (by a similar definition of delta function in 1D), and I am expecting "point impulse / impulse" on Q2, but I guess that's what you mean by "perturbation".
For Q3: It is supposed to be that "applied force can be thought of as a 'continuous sum' of point impulses".
For Q4: the Green's function describes the displacement of the oscillator after we apply an impulse. For this reason, Green's function is usually called the "impulse response".
For Q5: Yes, the idea is right: exactly copying the "adding different charge distributions --> adding up the electric potential", so in this case, "adding different forces --> adding up the displacement"
For Q6: From the formula that x(t) = int G(t, tau)*F(tau) d(tau), we can interpret that the displacement is a continuous sum of the impulse responses.
Have updated the description in case anyone wants to do the same, and don't see this comment.
Great video, your explanations are really good, and the visuals are super pleasent. I hope your channel because more popular - you deserve it! (:
Thanks so much for the kind words!
@@mathemaniac thank you for the amazing content!
Great video!
Slight typo at ~19:30. The harmonic oscillator ode shouldn't still have an m on the second order time derivative, if you already have ω^2 as your coefficient on x(t). This propagates into your solution as well, which is why there's a weird, dimension-ful argument in the sine function.
But awesome video nonetheless!
Thanks for bringing this
Good explanation. Thanks
Understandable explanations!!! Just too many ads to go through...
So the Green's function is basically the impulse response of the system
Differentiation with respect to ROA (for lack of the symbol) will describe a smooth downward deceleration of the springing pendulum in a linear representation, absV by T you might say.
Nice job. I hated the way Green's functions were taught to us. Maybe I've forgotten it since I did it, but just the simple illustration showing how the function is a series of functions of x and xi helped.
you are so good at this thank you
Happy to help!
Love this, keep going
I understand that we're taking the limit at 7:04 . I also understand that the volume V never actually equals zero because that's essentially what a limit means. But the sight of (0/0)=0 is jaw-dropping. Especially that in that specific line, the limit isn't explicitly written before the expression, but it's implicitly understood that it's a limiting process. I know it's only an intuitive video but I couldn't not say that *confused emoji*
Other than that, great video! Keep up the good work
Thanks for the compliment! The computer did take approximations of the volume, but it is so small that it returns 0. Sorry for the confusion this generates.
Really nice one. Thanks!
Julian Schwinger , the legendary theoretical physicist, found the true value of Green's function, I think.
So, please make a video about Julian Schwinger, who is my hero, if you have time.
Nice video. In addition to the unbounded or semi-bounded cases, which can be solved by pure integral transforms, it would be nice to have a discussion of solutions in bounded regions where the eigenfunction expansion method is used.
Thanks! The video is supposed to be geared towards a more general audience - I specifically don't require too much knowledge of calculus to begin with, so the eigenfunction expansion method or the Fourier or Laplace transforms definitely go beyond the scope.
@@mathemaniac Makes sense. I'm not trying to be a wise guy. Your stuff is great, and the quality of your visuals is outstanding!
@@mathemaniac Is the difference between Fourier and Laplace transforms that Fourier applies only to periodic functions while Laplace applies to functions having infinite domains?
@@david203 The Fourier *Series* is used for periodic functions. The Fourier *transform* and Laplace transform are for functions where their domains are unbounded: FT for functions f:Rd->C, and LaplaceT for functions f:[0,inf)->C
@@strikeemblem2886 So what is the difference? I'm not familiar with Laplace transforms, but I am very familiar with Fourier/Maclaurin transforms.