Green's functions: the genius way to solve DEs

Green's functions is a very powerful and clever technique to solve many differential equations, and since differential equations are the language of lots of physics, including both classical mechanics, electrodynamics, and even quantum field theory, so it is important to know how it works. Of course, this includes some explanation and perhaps a pretty different motivation for Dirac delta function, which is pretty weird, but also not really when you think about it a different way.
Correction: in 19:11, the Green's function lacks a factor of 1/m.
This video simply aims to introduce the Green's functions, what it is supposed to do, how the motivation of it all comes to be, and why it works. If you do need a lot more than introductory knowledge on Green's functions, and you are comfortable in basic differential equation solving, here are some links:
brilliant.org/wiki/greens-fun... (not sponsored)
www.roe.ac.uk/japwww/teaching...
My further write-up on the examples of Green's functions (how to find): drive.google.com/file/d/1D6E8...
For those who want some answers for the exercise towards the end of Chapter 3, i.e. around 15:47:
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.
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: Exactly copying the "adding different charge distributions (implies) adding up the electric potential", so in this case, "adding different forces (implies) 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.
I stopped saying anything more because (1) the video is already very long, (2) this video assumes only basic knowledge of calculus (it is actually better if you don't know too much of the rigour in real analysis, since this is really hand-wavy - and it has to be! Otherwise this would be a lecture in distribution theory, which I am not quite well-versed in), and (3) this really just aims to provide motivation for Green's functions and doing examples would make this more "textbook-y" than it already is.
Of course, the link to the Wikipedia table of Green's functions:
en.wikipedia.org/wiki/Green%2...
Note: they don't state the boundary / initial conditions explicitly, and they don't even use x and xi, or t and tau, usually just their difference. Usually it is that the Green's functions vanish when the position is far away from the origin, and for those involving time, 0 before time tau, assuming that tau is greater than 0 (the so-called "advanced" Green's function)
A little bit of remark after viewing the video once again: in some places, it is a little bit quick, so please treat yourself by pausing if necessary: KZbin allows you to do so! In my defense, different people require different time to pause, and also I don't want too much of dead air, so... that is probably also how a lot of other math videos on KZbin are doing right now.
Video chapters:
00:00 Introduction
01:01 Linear differential operators
03:54 Dirac delta "function"
09:56 Principle of Green's functions
15:50 Sadly, DE is not as easy
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
Social media:
Facebook: / mathemaniacyt
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Patreon: / mathemaniac (support if you want to and can afford to!)
Ko-fi: ko-fi.com/mathemaniac
For my contact email, check my About page on a PC.
See you next time!

[@mathemaniac]

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

[@rockosocko86]

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".

[@mskellyrlv]

@@frankdimeglio8216 I made $1 billion while in a coma, using this one weird trick...

[@Ap-zq7lb]

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!!

[@walterufsc]

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]

damn. this comment made the video obsolete from my pov.

[@Evan490BC]

@@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.

[@winstonvpeloso]

@@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?

[@sabzimatic]

can I say Green's function is system's response when the Forcing term is Dirac's Delta? Forcing term as explained at kzbin.info/www/bejne/n6TQY4acj8x5gMk

[@sabzimatic]

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.

[@dontsmackdafish3771]

3Blue1Brown is having a video contest this summer. You should submit this! It's great!

[@mathemaniac]

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!

[@arkaroaksoe5593]

@@mathemaniac please do enter, more people need to know about this amazing channel!

[@thetrickster42]

@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.

[@astphaire]

Nah, I don't think he has the voice for it.

[@abdullahm4830]

10 minutes through the video. I love it. Wonderful. I came to this channel for the first time.

[@noahifiv]

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.

[@mathemaniac]

Thanks for the kind words!

[@benburdick9834]

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!

[@mathemaniac]

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!

[@howkudyou]

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!

[@henryginn7490]

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

[@mathemaniac]

Thanks!

[@sweepsweep5572]

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!

[@mathemaniac]

Thank you so so much for the kind words! This video did take a long time to put together!

[@washieman2445]

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.

[@mathemaniac]

Thank you very much!

[@georgemartin2221]

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.

[@markmajkowski9545]

VERY POWERFUL. When learning Green’s Functions (long forgotten) - after you do enough - you can basically just write down the answer.

[@PabloAvilaEstevez]

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

[@mathemaniac]

Glad to help!

[@lordkelvin8380]

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.

[@mathemaniac]

Thanks so much! Glad to help!

[@tehyonglip9203]

You have done it! You have taught what my lecturers have failed to teach for the whole semester in 23 minutes!

[@curiousaboutscience]

Definitely see these everywhere in higher level physics. Great to see the E&M examples!

[@XxS4NN4SxX]

This should've existed 7 months ago for my exams.

[@mathemaniac]

Haha sorry about that! Hope that your exams went well nonetheless.

[@flashmedia8953]

For me, 11 years ago. 🤣

[@bitvision-lg9cl]

The animation, the background music, the tone, the words are exaclty match the 3b1b style. Nice job.

[@rollingsnowball9095]

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

[@mathemaniac]

Thanks so much! Glad that the exercise is useful!

[@kurtoverley6560]

Wow - what a fabulously instructive and interesting video! I learned more about solving PDEs from it than an entire college course!

[@jengofrett]

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.

[@jamesjackson5955]

This is incredible! A fascinating look at Green's functions. Amazing job

[@mathemaniac]

Thanks!

[@TobyAsE120]

Thank you for this flash back to my theoretical electrodynamics lecture. Back when studying physics was kinda fun...

[@david203]

I hated electrodynamics so much I dropped out of my PhD program entirely. I wish I had found it fun.

[@Andrew-rc3vh]

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.

[@christophas]

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!

[@adriencances134]

A crystal clear introduction to the idea behind Green's functions!

[@jamespage6013]

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

[@mathemaniac]

Thanks for the kind words!

[@Daniel-ih4zh]

yeah, green's functions have like 8 different names, from propagators, to correlation functions, to response functions,

[@anamarijavego6688]

beautiful explanation, I had difficulties understanding the idea behind those functions, but you put it very simply together. Thank you! I immediately subbed :)

[@mathemaniac]

Glad it helped!

[@abdullahbinaamir8875]

Amazing video. The best yet on Green's functions on the internet in my opinion. Thanks a bunch man!

[@mathemaniac]

Wow, thanks!

[@fattimiv]

This is a fantastic explanation! A lot of pieces suddenly fell into place after watching this.

[@mathemaniac]

Glad it helps!

[@jannikdettmer9279]

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.

[@mathemaniac]

Hope it helps!

[@jackshi7613]

Good video about the introduction of green's functions. Thanks a lot

[@QuantumConundrum]

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.

[@AJ-et3vf]

This is a very nice video! Glad that the algorithm recommended me this!

[@johnchessant3012]

I found your channel today! Your videos are great and I hope/expect you will reach a larger audience soon

[@mathemaniac]

Thanks so much for the kind words!

[@michamiskiewicz4036]

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!

[@mathemaniac]

Thanks so much for the compliment!

[@jessstuart7495]

Very nice introduction to Green's functions, Thank you!

[@mathemaniac]

Thanks so much!

[@arsenzatikyan]

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.

[@mathemaniac]

Thanks so much for the kind words!

[@mr_zmt7152]

Excited to learn something thanks to you!

[@mathemaniac]

Glad to hear it!

[@arthsojitra53]

What a wonderful explanation! I hope every university teaches this way!

[@mathemaniac]

Thanks so much for the kind words!

[@sololy_r]

Thanks for bringing this

[@sachs6]

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.

[@david203]

I graduated in physics, yet could not follow much of this video. Too many basics not understood, I guess.

[@johnchristian5027]

Great video! looking forward to seeing more!

[@mathemaniac]

Thanks!

[@nirajangupta77]

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 🙂🙂🙂.

[@mathemaniac]

Thanks! Will consider more advanced topics in the future!

[@subhasnandy391]

Brilliant work. Would love to see videos on integration of complex functions and their applications in engineering someday.

[@mathemaniac]

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).

[@ernstuzhansky]

Many thanks for the video. Well done!

[@tariq3erwa]

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!

[@d3scripted672]

Great video, your explanations are really good, and the visuals are super pleasent. I hope your channel because more popular - you deserve it! (:

[@mathemaniac]

Thanks so much for the kind words!

[@d3scripted672]

@@mathemaniac thank you for the amazing content!

[@reinakousaka3838]

Thank you for making this video! It is very useful & friendly!

[@mathemaniac]

Thanks for the compliment!

[@algorithminc.8850]

Great channel for those wanting to quickly understand a topic ... thanks ... just subscribed.

[@mathemaniac]

Thanks so much!

[@aashsyed1277]

Cool video ! Thanks so much! This is a cool visualisation! Exited to learn something new !

[@mathemaniac]

Glad you liked it!

[@FrozenArtStudio]

wow dude, the quality of video is amazing!!

[@mathemaniac]

Thanks for the kind words!

[@soumyadipghosh6925]

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! ❤️

[@mathemaniac]

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!

[@UnforsakenXII]

@@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.

[@DrDeuteron]

@@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.

[@NoNTr1v1aL]

Absolutely amazing video!

[@mathemaniac]

Thanks!

[@robinashaheen1713]

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.

[@mathemaniac]

Thanks so much for the compliment!

[@user-vg7zv5us5r]

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.

[@mr_rede_de_stone916]

Actually very well explained, congrats!

[@mathemaniac]

Glad it was helpful!

[@student99bg]

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. kzbin.info/www/bejne/n2q6mqt-m86Nhrc

[@user-xn6jt6qu8m]

Excellent video I was struggling to understand green functions in quantum field theory

[@grounded9623]

omg- THE BEST MATH EXPLANATIONS EVER. Thanks.

[@mathemaniac]

Happy to help!

[@shubhamdawda7288]

Really nice one. Thanks!

[@joelcurtis562]

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!

[@mathemaniac]

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.

[@mortezamaleki691]

Thanks for this great lecture!

[@marcovillalobos5177]

Your visuals are incredible

[@mathemaniac]

Thanks so much for the kind words!

[@tarkesdora20]

Thanks for putting efforts and making this video

[@mathemaniac]

Thanks for the kind words!

[@xiaoyu5181]

The great explanation I have ever seen!

[@mathemaniac]

Thanks!

[@farbodrassouli6939]

Love this, keep going

[@rushikeshambekar2185]

Very clear explanation!

[@mathemaniac]

Glad you liked it!

[@ig5r140]

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!

[@mathemaniac]

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.

[@Caspar__]

Good luck 🤞

[@alex_zetsu]

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.

[@brendawilliams8062]

I would like the phase for 4 as 4295

[@weltkaiserendzeit2417]

This is absolutly great ! A truly stunning and interesting way of introducing Green's function. Congrats ^^ By the way, great animation, I assume you use manim, did you have any experience with it before ?

[@mathemaniac]

Thanks so much for the kind words! I don't use Manim, see more in the description.

[@bikramdas9994]

Good explanation. Thanks

[@John_Shanks]

Very instructive video.

[@angelisaacroaarias9278]

Perfect, keep up the good work!

[@mathemaniac]

Thanks!

[@tanvirfarhan5585]

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 KZbin. (and I watch nothing but science and physics docs on KZbin 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.

[@mathemaniac]

That's so kind! Thank you!

[@victorscarpes]

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!

[@odeia18]

i had these exact problems in my electrodynamics exam last week! wish i had this sooner

[@dan1204hc]

You are inspiring! Thank you for the great video!

[@mathemaniac]

Thanks for the appreciation!

[@MessedUpSystem]

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

[@MathPhysicsEngineering]

What an amazing job you did! How many hours of work did you put into it? What software did you use to animate?

[@mathemaniac]

Thanks so much for the kind words! I usually tell people that for 10 minutes of video, I need about 30 to 40 hours of work (without any distraction). As for the software, see the description!

[@ackinito]

Fantastic content!

[@mathemaniac]

Thanks!

[@ikarienator]

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.

[@mathemaniac]

That's an interesting perspective!

[@Ricocossa1]

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]

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.

[@david203]

@@Ricocossa1 I could not follow this. I was lost with the very first sentence, sorry.

[@Ricocossa1]

@@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).

[@jamesmosher6912]

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!

[@mathemaniac]

Yes, your first point has already been covered by another commenter as well, but I can't edit the video on KZbin.

[@PearlKhurana]

Amazing! Thank you 😍😍😍

[@KyleDouglass]

This is a great video and has been really useful as I brush up on electrodyanmics. Thanks so much for sharing. I'm having trouble understanding the justification in chapter 2 on the Dirac delta function for using the indicator function instead of integrating over the volume D surrounding the charge and taking the limit as V goes to 0. You say it's "not ideal because the region is going to change in the limiting process." Could you elaborate more on why this is a problem that warrants solving by using an indicator function and changing the integration bounds?

[@jonathanengwall2777]

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.

[@mimimi3440]

you are so good at this thank you

[@mathemaniac]

Happy to help!

[@adixo1851]

Great Video!!, very informative 💯💯

[@mathemaniac]

Thanks so much for the kind words!

[@Giganesh_exe]

Beautiful and just the right timing as well! I'm about to start MATH2100 which covers PDEs.

[@mathemaniac]

Hope that it will help!

[@alejrandom6592]

I always love to watch ur vids😊

[@DeGuerre]

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.

[@brinsino]

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!

[@user-sm6fv6kw7h]

Beautiful!

[@morbidmanatee5550]

I love Green's functions! Delta function rock!

[@mathemaniac]

Indeed!

[@jojo_jo2212]

Delta what you said I think you spelled "distribution" wrong

[@mathemaniac]

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.

[@morbidmanatee5550]

@@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.

[@muhammedatef1193]

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

[@mathemaniac]

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.

[@igorb4650]

very good video on subject!

[@leon_noel1687]

Thank you, I just learn Electrodynamics, perfect video, greetings from Berlin

[@michaelupdike-bz6rg]

This is hella good; another sub!

[@mathemaniac]

Thanks!

[@elimeril2377]

More things of Green's functions !!

[@cycleSCUBA]

Getting the hang of Green's now, thanks.

[@mathemaniac]

Glad to help!

[@cristhiangalindo4800]

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.