Probably the weirdest function I encountered as an engineering student

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[@edgeman1135]

Dirac rolls "worst function ever" Asked to leave mathematics

[@fightocondria]

The other neat thing about the dirac delta is that it can be presented as a smooth function. It's actually kind of mathematically brilliant

[@ALX112358]

Have you never heard of the derivate of the delta function?

[@kangmoabel]

I forgot that zach made educational videos 😅

[@louisrobitaille5810]

Don't worry, you're not alone. We all remember when he uploads a video, then forget the next day 😂.

[@Zero-ef4sc]

I pretty much only watch his comedy skits in Zach Star Himself. I have to watch countries after WW2 video once every day.

[@zachstar]

Same

[@clayel1]

@@zachstarits been awhile lol

[@ErikOlexiewicz]

Are you willing and able to watch his educational videos?

[@boium.]

As a mathematician I am obligated by law to point out that the Dirac delta function isn't actually a function, but a distribution (or measure).

[@erikhicks6184]

Function in the sense of vertical line test, agree. Function in the sense of mapping, disagree.

[@arnabbiswasalsodeep]

but isn't the distribution defined as a function? a bell curve distribution has its own function too, in the sense y = f(x)

[@boium.]

@@erikhicks6184 Sure, you can define a function to something like the extended real numbers, or the reals with some extra element called ∞, but that still doesn't immediately clear up what the integral of that function should be. Moreover, if you define multiplication for this new element called ∞ as a*∞ = ∞ for all nonzero a, (or maybe you allow for a sign), then the integral still wouldn't have the desired property for that. It really isn't a function.

[@dojelnotmyrealname4018]

As a european engineering graduate I learned it as the Dirac Impulse.

[@GaussianEntity]

The Dirac Delta function is what you get when you allow engineering students to do pure math 😂

[@77mxb1]

The Dirac Delta function can also be thought of as the Normal distribution with mean and variance both at 0

[@Tymon0000]

That's pretty cool

[@chrisdupre2862]

In fact that can be made rigorous! The nerd way would be to say that as both mean and variance tend to zero, the Gaussian distribution converges weakly to the Dirac delta distribution!

[@CalvinWiersum]

Or you could say this is the one way that a normal distribution is also a uniform distribution. Also, sick pfp! Always good to see a fellow fractal enjoyer!

[@NathanSimonGottemer]

That's actually the most rigorous definition of the delta if you're putting it as a function -- delta (t) = lim (s->0) e^-(t/s)^2

[@chrisdupre2862]

@@NathanSimonGottemer Except you are missing the 1/sqrt(s) factor such that the pointwise limit doesn't exist at 0 😂. If you pick the right topology though, yes absolutely. It's also not the most rigorous, because any rigorous definition is equally rigorous. Personally, I prefer to define it as a distribution because once you know the set-up, it's the easiest to make sense of (for me).

[@PitchWheel]

The convolution is the way in music the reverb is calculated. They make the shortest and loudest possible hit inside a church or a concert hall and record the response. This gives everything is needed to matematically simulate that exact space and allow us to imagine any instrument playing there with that exact ambient. Fascinating.

[@danielpetka446]

Or you can just record the transfer function directly with a chirp

[@FF_Fanatic]

In one of my differential equations assignments, we were given four impulse responses from the four cardinal directions around a microphone (L+R channels rather), which could be a clap, a balloon popping, anything like that. We were tasked with recording something and then manipulating it with convolutions using a few lines of MATLAB to change our sound to appear as if it were coming from different places around your head. The assignments in that course were pretty neat sometimes. Plus we got the obligatory "how to make rudimentary autotune" explanation. I looked back at my assignment and I have no idea why I did this, but my MATLAB file was named oriAndTheBlindForest.m lmao

[@Pradowpradow]

As an engineer in dynamics and vibration, when we study a system, we need to know how it responds freq by freq. To do this, we have two solutions : - make the system vibrate at specific frequencies and sweep the frequency to the max value and see at each moment how it responds, - make a "bang" test : use a hammer, smash the object, and see the response. Then with convolution and Fourier transform, we get back to the response frequency by frequency

[@morganandre8887]

The latter of the two sounds much more fun! Until your nightmares of performing convolution for your differential equations class come back to haunt you...

[@PluetoeInc.]

can you point in a direction that goes deeper in the latter and fun solution ? THANK you in advanced .

[@pocarski]

Finally, a scientific answer to why percussive maintenance is so effective

[@DrDeuteron]

that 2nd is called a wide-band signal.

[@davidhowe6905]

I remember being amazed by this when I first saw a video of it being done both ways (though they didn't use a hammer - just played a very short click!). It has economic advantages in that the first method needed an anechoic chamber; but with a click, it was all over before any echo could return.

[@IBViRUS]

after seing a bunch of zach's videos on other channel its hard to dismiss ironic notes from his voice

[@andrerenault]

I’m waiting for an April Fools sarcastic math explanation

[@tetokesenye397]

Laplace transforms and Engineering Dynamics at play

[@SPY-ce8qf]

"Air resistance is accounted for" what is this blasphemy this is not a world I want to live in

[@fubuma534]

That’s until you get to the Discrete time Fourier transform and then they’re all Dirac functions

[@aegoni6176]

I'm just taking a class on signal theory, and can confirm this. Honestly, it's amazing the power of this function

[@staticnullhazard6966]

Mad mathematicians incoming: "It's not a function! It's a Distribution." 3..2..1..

[@username8644]

Damn right

[@VeteranVandal]

B-but it's!

[@barnabasigari3109]

Im a physicist but i must admit that it was my first thought aswell upon seeing this video

[@bestgrill9647]

a distribution is just a probability measure and a probability measure is just a function. no mathematician will complain

[@barnabasigari3109]

@@bestgrill9647 if youre saying that you didnt have a professor rant about how distributions are not functions for 10 minutes

[@Mark-dc1su]

I've been self-studying math for about 2 years now. Currently, I'm working through a DiffEq textbook and covered the Dirac Delta Function a couple of chapters ago, along with convolution. This video was great because it allowed me to prove to myself that I did actually learn it and was able to follow along and even preemptively guess the next topic. Thanks!

[@jaym0ney_]

Holy shit he’s back

[@Gameplayer55055]

Mathematicians: wow such a strange function Programmers: it's just if(x==0){return INT_MAX;} else {return 0;}

[@markgross9582]

Realistically, since everything programmers deal with is discrete, they would be dealing with the discrete analog of the Dirac delta function, which is the kronecker delta function. d[n]=1, n=0 d[n]=0, everything else.

[@U20E0]

@@markgross9582which is just logical not

[@markgross9582]

@@U20E0 what do you mean it’s logical not? Are you talking about how Boolean vars in most languages consider 0 false and every other number true?

[@U20E0]

@@markgross9582That combined with the fact that in most languages true and false are just 1 and 0 with a taped-on moustache.

[@gratux]

​@@markgross9582 in programming, there is a distinction between inverting a boolean value (logical not) and flipping all bits of a binary representation of a number (bitwise not)

[@brachy.0]

It's a distribution, not a function 🤓. You need somewhat complicated math to derive the delta distribution cleanly. In physics you typically use it to describe mass or charge density for an infinitly small particle. Also the step function is also known as the Heaviside or Theta function.

[@BederikStorm]

It's a generalized function. Using them you can find derivatives of common functions. For example, the derivative of |x| is sign(x). And the derivative of sign(x) is 2*delta(x). It's strange that 2*infinity is not the same as just infinity, but that's correct

[@excalibercuberdavid4681]

I love direc delta because remembering that the inverse laplace of a constant is that constant multiplied by the direc delta function gave me 20 extra points on a Circuits 2 quiz

[@luizmenezes9971]

They should call it the punch function

[@dielaughing73]

In engineering it's known as the 'unut impulse function' which is pretty much the same thing

[@charlesspringer4709]

Nifty. The Dirac Delta also allows an analytical form of sampling which leads to all the DSP stuff.

[@agastyasanyal4026]

I love the explanation on convolution, never seen it explained more intuitively

[@robertomsgomide]

Probably one of the best videos for intuition about signals and systems that I have ever seen

[@Ocro555]

I'm willing to learn more from Zach's engineering channel and enjoy these videos equally much, but sadly as a highschooler I am not able to understand most of the topics and content :(

[@rivigantentertainment]

Thank you for making a video on the Dirac Delta function, I have studying it for some time and I hope this video will help me understand it better.

[@agranero6]

It works for solving no homogeneous differential equations, but strictly (formally) speaking it is not a function. Mathematicians had to create a new theory to formalize those weird things Physicists were using; it is called distribution theory, some call it generalized function: because the formal definition of function does not include it.

[@Vlakrov]

I want to add that in mathematics this dirach delta he defined does not exist, infact the condition that the integral is 1 is impossible for a function 0 everywhere except for one point. Indeed in mathematics we use the dirach delta a lot but without this condition. Still this is very useful in physics as explained in this video. This makes this function even more wonderfull.

[@robegatt]

In math everything you define clearly exists.

[@olli1886]

@@robegatt yeah, but the dirac distribution is not a "well-defined function", it's as ill-defined as "defining" f(x) as a function that's always negative, but its integral is positive.

[@username8644]

@@robegatt In math it's not a function, it's a distribution. It does not satisfy the requirements of a function.

[@robegatt]

@@username8644 technically is a limit of the definition of a function, but since it fits with differential calculus, which is based on the concept of limit, it all goes well.

[@jacoblojewski8729]

Yeah, the Dirac Delta "Function" is a misnomer. Still very useful. If you want to be a mathematician about it, there are a lot of ways to define it. A method accessible to a intro integral calculus course would be: Define a sequence of functions d_n with the property: 1) integral from -oo -> oo of each d_n = 1 2) d_n >= 0 for all x. 3) For any integrable function f, lim n-> oo of integral from -oo -> oo of f * d_n = f(0). Then we abuse notation and say any time d(x) is inside the integral, we really mean to take the limit as n -> oo of that integral, where we replace d(x) with d_n(x). Lots of sequences of functions satisfy this property, one is the one Zach gave. There are also "easier" ways to define the dirac delta, but require further math (measure theory and Lebegue integration is the most common way).

[@josgibbons6777]

While several comments have already noted it's not a function (it's a distribution as well as a measure), it's worth knowing the true functions whose distributional limit is the Dirac delta are called nascent delta functions, in case you want to look up the rigorous details.

[@Impatient_Ape]

Great job on this one Zack!

[@skyscraperfan]

In our university we had a laser that had the power of 10^14 watts for 10^-12 seconds.

[@dielaughing73]

Holy shit

[@skyscraperfan]

@@dielaughing73 The total energy is just 100 joules, but as it is released in such a tiny amount of time, it can turn the air into plasma.

[@dielaughing73]

@@skyscraperfan that's friggin awesome

[@DigitalJedi]

I remember the university I did my PhD at had a similar laser. I didn't ever get to play around with it for anything but I remember a few stories of what it did to the surfaces of different materials. Putting little craters into titanium blocks and such.

[@ariadne2746]

i started off thinking of this as just the derivative of the unit step

[@gamerpedia1535]

I want to mention that your rectangular formation of Dirac Delta function can be fudged to provide any value to the integral Eg. From -a -> a we have an area of n This means our height would need to be n/2a Take a -> ∞ and you get the same resuly, just with an area of n

[@highgroundproductions8590]

In math we call the "impulse response" a Green's function. We integrate the Green's function, and that's the convolution.

[@NathanSimonGottemer]

TBF the Laplace Transform is still useful here because it turns out that convolution gets turned into multiplication in the frequency domain and also the FT and LT of the delta function are both 1

[@ShadowSlayer1441]

Dirac functions were definitely the craziest thing my differential equations course covered.

[@ThePrimeMetric]

For me it was the Laplace Transform. I wasn't exposed to delta functions until later in my physics classes.

[@ShadowSlayer1441]

@@ThePrimeMetric Laplace transforms are so cool. Higher level mathematics are truly a marvel.

[@ThePrimeMetric]

@@ShadowSlayer1441 In my opinion, Fourier Transforms are even cooler. To be honest I haven't really used Laplace transforms since my first ODE class. I don't know what their applications are outside of solving differential equations but Fourier transforms seem to do the trick just as well. Their pretty similar, Laplace transforms are just the real-valued analog I guess, but I haven't seen them used for anything besides solving differential equations. I've used Fourier transforms in many classes though and even used it for some physics research. My favorite applications for them is Fraunhofer diffraction from Optics and using them to parametrize any curve or surface.

[@ThePrimeMetric]

I was actually wrong here. I probably knew this at some point and forgot but the frequencies of the Laplace transform can take on complex values. So the Fourier Transform is actually a special case of the Laplace transform. A Fourier transform decomposes a function into sinusoids and the Laplace transform decomposes functions into exponentials and sinusoids. So they each have their own strengths and weakness. Laplace transforms are in general probably better for solving differential equations because they are more stable with exponential growth or decay.

[@vbmalek]

I was once a TA for this subject in college. One helpful analogy that students loved was the Taco Shop or the Furniture Store. At either, ingredients or raw materials (alluding to the input curves) go into the Shop or Store (System to Convolute with) and each produced nachos, tacos, or burritos or a chair, table, or shelf (alluding to the output). The output would “take the shape/presentation” of the directive at the Shop/Store at that moment. The analogy may not be 100% accurate, but oh how fondly I remember teaching Convolution and seeing how students began to understand what it all meant.

[@whermanntx]

Dirac and Jalad at tinagra. When the walls fell.

[@Phroggster]

Temba, his arms wide.

[@fightocondria]

Convolution of impulse, then the walls feel

[@blueskull7898]

Just came across the dirac delta recently in Quantum Mechanics. It used to be a pretty strange function to me but the application in Quantum beautiful. Truly a function by mathematicians, for mathematicians

[@zorroaster8895]

Elite timing

[@slavetotheseo4127]

Love the educational content ❤..you have a gift of explaining

[@guitarhero3812]

As a computer engineering major, the delta function is something that still amazes me. The concept of an impulse response blew my mind when I first learned it; seeing its applications in things like filter design, digital signal processing, and even control systems. Also the fact that convolution in the time domain maps to multiplication in the frequency domain is something that still captivates me to this day.

[@mgm6723]

Zach being able to teach so seriously, and yet willing to teach unseriously, is such a blessing

[@melm4251]

i've been doing a project on fourier transforms and i only realised last week that the frequency spike FT of a single sine wave is modelled with a dirac delta function as well. I first read about them in a QM book where a 3D delta function describes a point particle. Very versatile piece of maths!

[@hadisyed4666]

Bruh where was this video when i needed it. I just finished my linear systems and signals class today 😭. Good intuition

[@danielberry777]

I hate that I saw the thumbnail and went “Dirac delta function!” **Diff EQ flashbacks set in**

[@imbored1253]

You just casually gave the best intuitive definition of convolution

[@jessicaraven9546]

Woah this was really cool. CUrrently studying for my differential equations final and was cool to see how the dirac delta and step function are related

[@randycasty1226]

Currently learning this as an ee student and it definitely confused me at first

[@HenrikMyrhaug]

I'd say that it isn't built into the dirac delta that it can predict the response of a system to any impulse, but instead that that's what convolution does. Convolution multiplies all points of time of an impulse with the response the system has after the time delay since that impulse, and sums/ integrates over the time the impulse has acted. In reality, the concept of convolution is not so complex, although it becomes a complex calculation requiring computers in order to be done in a reasonable ammount of time. I also love how it intersects with the topic of Fourier transforms, in that we can use Fourier transforms to compute a convolution.

[@LeoDaLionEdits]

Thank you! I'm reading a book and was confused by this today and of course you read my mind from the future and upload this

[@JFCBFilms]

God damn, I am absolutely blessed by the timing of this video. I have an exam in "Systems and transforms" math course in 4 weeks.

[@kerrynewman1221]

Taking me back to my college electrical engineering undergraduate days in the early eighties at UC Berkeley. Thanks.

[@sirandrewthethird]

huh... So that's what my controls professor was on about.

[@TrevorDiMezza-ly6tt]

Thank you for explaining this better than any of my professors!

[@no-bk4zx]

I learned this for both electromagnetism and laplace transformations. Beautiful function, I really like how it behaves so neatly despite such an unorthodox definition . I am curious though, what is the co-domain of the function? As far as I know, infinity isnt a number and is not an element in the set of real numbers but the approximated functions leading up to the dirac delta do have a co-domain of R. Great video btw, these really help me understand what I am doing in my physics class to a deeper level.

[@alperakyuz9702]

To be mathematically precise, dirac delta function does not make sense as a function, but as a distribution. In fancy math language qe say its defined as a continious linear functional from the space of smooth, compactly supported functions topologized with an inductive limit topology, but in human language you can think of it as something that only makes sense under the integral sign multipled by a continious function.

[@Azakadune]

I just did dynamics, system modelling, and control systems and I never noticed this lol

[@MH-sf6jz]

I like to think that Dirac delta function is the laplacian of the fundamental solution to the laplace equation.

[@jonetyson]

The delta function also has a derivative. A good place to learn how to make sense of such "functions" (which are distributions, not functions) is Lieb & Loss's analysis book.

[@alitheeternity4230]

1:20 that was unexpected but glad to see Zach star himself on Zach star channel.

[@ckq]

Zach's in his signal processing era right as I finish 2 courses on it.

[@ncolyer]

your timing is impeccable seeing I got an exam on this in 3 hours ❤

[@GwenBennett]

literally took my signals and systems finals yesterday lmao

[@looper6394]

nice theory you got there 😮

[@LambOfDemyelination]

you could construe a function with an arbitrary area k by saying y=k/2a when x=0, then take the limit to say the Dirac delta function has area k under the graph. point is, the area is undefined, and the function being "at infinity" is meaningless for real valued functions.

[@foxghost1814]

I'm a matematicians so I have to say that. Delta dicara is a distribution of funtion with has 1 if x=a and 0 x=/=a. Area over this funcition is 0. However measure od any set with have element a is 1.

[@pixelguitarman]

Now that's something I'm proud to know too much about

[@ImLucky6]

i just learned convolution like 2 days ago in my differential equations class, nice to see it might actually come up in my engineering degree again

[@dielaughing73]

It will if you study mechanical or electrical engineering at least

[@johanngambolputty5351]

I kinda skimmed through, but wanted to mention, the reason the dirac delta might not make sense as a function, is because it is sometimes used in place of a density, but a point mass has no sensible density arguably. But if we are integrating against distributions, it totally can still make sense as a measure, with respects to a Lebesgue integral. Measures just give you the amount of stuff in a set, so for a continuous distribution you have the riemann integral over a region as usual, but for a dirac, you just get all the mass if the measured set contains the point mass and none of the mass if the measured set does not contain the point mass.

[@AstraGroupDF]

Dirac delta function only makes sense as limit of sequence of functions. It could also be the limit of a gaussian with the standard deviation going to zero

[@massipiero2974]

It also makes sense as probability measure supported on a point, as the "evaluation" distribution or as hyperfunction with representative 1/z, and i guess it has many more equivalent definitions

[@AstraGroupDF]

@@massipiero2974 the interpretation of the probability measure is that it's the distribution of a variable that can only be one value

[@pyropulseIXXI]

I was a physics student, and the function seemed totally obvious and not weird at all; you can litearlly integrate and pick out values since you times any function by 0 every except at x = a, where the integral is = f(a). int_a_b; f (x) d(x-a) dx = f(a)

[@pogan1983]

Space vs time perspective. No space = eternity, no time = infinite plane, a film slide.

[@haushofer100]

The title is like " This cow is the weirdest human I've ever met".

[@jorgedaniel1653]

Zach always makes me want to study a certain topic on my own 😂. By the way Zach, I highly recommend you get the textbook called “The Physics of Energy”. If an apocalypse occurs, that’s the book we need to restart civilization.

[@ruffifuffler8711]

Thinking of the impulse function as the reflexivity after re-settlement of a pole flip. The pole flips, decides to correct the flip which is incomplete since the flip is trying to confiscate or conquer the instability of a new territory or experience, ..., so, it does correct itself, but now has to account for and carry the instability which it did not conquer, now taking its' uninterrupted time to present itself. The presence of this "abberation" has to be presented clearly and cleanly, and so its' reflexivity accumulates after settlement in a virtual capacitor, which suddenly materializes at a fixed point on the time line, and so is such realized. In other words, time and space displacement chacterized by returning back to zero, leaves the impulse function as a memory of the previous now unknown event, other then, it happened. The sliding thining box does slide in virtual space, and does have a fixed consistency time to present itself, before expiring in validity, ...however, the measure is not infinite, it is just long enough to be recognized by the system it is now part of., and absorbed by, and cannot be measured without disturbance, and risk of determination.

[@ruffifuffler8711]

An after thought: It's the displacement spectra of a forgotten event that you know happened, but in itself does not carry enough info to explain what exactly it was. Not that sure whether it needs to be shown on the upper y-axis, as half could be below, and one being chosen just for the convenience of not disturbing any further measures.

[@Jakub1989YTb]

Aww, those times at university when I studied cybnernetics, control and regulators.

[@AlexanderTheMiddle]

I literally have an exam on this tomorrow. I was watching youtube as a way of avoiding revision, but you fooled me into preparing for the exam!!!😤😤😤!! you!!!!!!! Thanks❤

[@badmath9099]

Playing around with: ((-8)(2abs(x) - abs(x-0.5) - abs(x+0.5) - (-8)(2abs(x) - abs(x+0.25) - abs(x-0.25))) / 4 You could set the height to an infinite amount, then Subtract basically all of it. This would leave a platform with slope of infinity, width of 1, and height of whatever you want. I'm too lazy to explain further, or to simplify, so... just trust me bro.

[@VeteranVandal]

It's a distribution, not a function per se. You can define a function in the intermediate limits.

[@mathematicsreadinggroup7288]

IT'S A DISTRIBUTION.

[@ozzymandius666]

Heh. I only knew of its use in quantum mechanics. Thanks, man!

[@kerryfracasso]

Dirac delta is not a function, it is a linear functional on the space of test functions.

[@konradkania4963]

A little correction: technicaly the Dirac's delta is not a function but (Schwartz) distribution

[@tomkerruish2982]

What if Washington and Lincoln encountered a Dirac delta function?

[@kezzyhko]

I remember having a Digital Signal Processing course at university. We recorded a clap, and used convolution between clap recording and any other sound. This was effectively applying a filter to the sound and making it sound like it was recorded in the room where clap was recorded. I wonder if these things have a connection

[@toshiboyig]

They do, that technique is called Impulse Response Reverb, and uses exactly this principle!

[@markgross9582]

Well yes. The equations modeling sound are linear time invariant, so the you essentially convoluted a general input with the impulse response.

[@dielaughing73]

​@@markgross9582 convolved

[@MusicEngineeer]

Well done. Would it be correct to say that a Green's function is a sort of generalization of the idea of an impulse response when applied to partial differential equations? I found Green's functions always kinda scary until I looked at them from this angle. If so, it would be cool to see a follow up video on Green's functions.

[@ThePrimeMetric]

I just started learning about Green's functions so I'm not an expert but I would say Green's functions and impulse responses are one in the same, the Green's function is just mathematically exact. The Green's function is the exact output you get if the input is some shifted delta function. If you can model your system as a linear differential equation of the form Ly(x)=f(x), where L is a linear differential operator, you can define LG(x,x')=delta(x-x') and solve for G using the form of L and the boundary conditions. The hammer banging method, or whatever you want to call it, I believe is just a more empirical way of getting the approximate impulse response. After all you can't actually apply a delta function of force on something. You can get close though by hitting something very hard over a small area and contact time. An engineer probably has less of a reason for finding the exact impulse response (or Green's function) because: 1.) they are using a idealized model (simplified differential equation) to model a more complicated system and there are greater sources of error involved or 2.) The system their dealing with is so complicated they don't have a differential equation that models it. If you can get an approximate impulse response you don't really need to know what your differential equation is, all you need to know is the input or driving function. Then you can take the convolution of these two functions to get the response to the driving function.

[@TheStillWalkin]

The convolution looks a lot like a crosscorrelation with time shifted function?

[@scar6073]

That's my fav function

[@peterg76yt]

Today I learned that the Flash is able and willing to lift Thor's hammer.

[@whong09]

Please cover laplace transform and fourier transforms too. This is where my eyes glazed over in my EE classes and I said fck it I'm switching to CS.

[@ef-tee]

I have actually had a class on signal theory as part of my CS major 😅 Although I guess it wasn't _too_ in-depth on Laplace and Fourier

[@whong09]

@@ef-tee I just didn't get convolution integrals. The class I was in the prof was just throwing around integrals as if this just justifies how the transforms work, no derivation or trying to solve the integrals. If I were to do EE again I would focus more on application than theory.

[@drslyone]

The flash can't yield Thor's hammer. They are in different universes!

[@JamesD2957]

@3:55, how did you compute that h(t)?

[@themissingsamurai]

the guy who found out what 1 was made this video... wow

[@MrSN99]

damn the algorithm is rough these days

[@latt.qcd9221]

When the dirac delta is combined with Green's functions, magic happens.

[@PluetoeInc.]

what is a green's function ? what does it do i mean whats the big idea behind green's functions

[@ThePrimeMetric]

@@PluetoeInc. I haven't worked with them much yet but my understanding is their used to solve entire families of differential equations with specific boundary conditions without needing to know the driving or forcing function. They can be used to solve both ODE's and PDE's but I think as an introduction I think it's best to start with solving ODE's, even if it might be a bit overkill. If you think of a differential equation as a differential operator acting on some function to get the driving function, Green's functions are sort of like the inverse of that operator. From the fundamental theorem of calculus we know the opposite of a derivative is basically an integral, so intuitively the inverse operator is a integral. More precisely, the green function is part of the integrand for the integral. If you know the green function of any family of ODE's, with specific boundary conditions, you can solve a specific ODE, with the same boundary conditions and from the same family, by multiplying the green function by the driving function and integrating it. Another way of thinking about greens functions is they are the partial solution to a differential equation or the impulse response for an LTI system with the delta function as the input. The formal definition would be L*G*(x,x')=delta(x-x') where L is a linear operator, G(x,x') is the Green's function and delta(x-x') is a shifted delta function. I'll try to give the motivation for this definition. If you want to know the electric potential of a static charge distribution the green function G(x,x'), tells you the electric potential at x due to a point charge at position x'. There may not actually be a point charge at x', so it may just be zero. Only at the exact position of the point charges is the potential non-zero, so the potential due to each of the charges can be thought of as the impulse response due to the charge. Therefore, as we learned in this video, the solution is the convolution of the impulse function G with the input or driving function f. (Intuitively this can be thought of as a weighted average of f(x) where G(x,x') are the weights. Except, in this case, the weights are just the contributions to the potential at every point, so this is actually the exact solution.) That's where the delta function comes in. Let's say Ly=f(x) where y is the solution to the differential equation or output of the linear system and f(x) is the driving function or input of the linear system. Let's say we have some function G such that, LG(x,x')=delta(x-x'). Multiplying both sides by f and integrating you get L*integral of [G(x,x')*f(x)] dx' = integral of [f(x')*delta(x-x')]dx' =f(x). This is the same form as the differential equation in operator form, therefore, y = integral of [G(x,x')*f(x)] dx'. For a translationally invariant system, which would be the case here, we can express the Green's function in the more familiar form G(x,x')=G(x-x'), so that it is in agreement with the definition of a convolution. Greens functions for families of differential equations can be difficult to find, depending on the differential equation and boundary conditions, but once you find them solving a specific differential equation in that family is very easy. Lots of greens functions for families of differential equations, with common boundary conditions, have already been found. So if you don't care to know the details for how those green functions are found, you can solve lots of otherwise hard ODE's and PDE's pretty easily. As long as the integral you get out the green function and driving function is elementary you will get a nice closed form solution. When the integral isn't so nice you can use it to get a series solution or you may find some numerical method with the integral is faster and more efficient than numerically solving the differential equation directly. For more details there are some pretty good KZbin videos on the topic. I watched Mathemanic's, Faculty of Khan's and Andrew Dotson's videos on it and I watched Andrew Dotson's video solving an example problem. After that I felt like I came away with a decent understanding.

[@ethanbottomley-mason8447]

​@PluetoeInc. You can imagine you want to solve a PDE like laplacian of u is f. To do this, you first solve laplacian of u is the Dirac delta. You can solve for u by taking the Fourier transform. It will be a tempered distribution and will in fact be a function, called the Green's function. Then to solve laplacian u is f, you can take the Green's function and convolve with f to get your solution.

[@PluetoeInc.]

@@ethanbottomley-mason8447 that flew right above 😭

[@ThePrimeMetric]

@@PluetoeInc. I'm not sure if you saw my long comment from earlier because I don't see it anymore. But the basic idea is the Green's function is the same as the impulse response in the video, it's just found directly from a system of linear differential equations and their boundary conditions. And then, just like in the video, you can take the convolution of that with the input function to get the response function to any arbitrary input. "Hitting your system with a hammer" is an approximate way of finding the Green's function basically. It's a powerful indirect technique for Engineers because the systems their dealing with are often so complicated they don't know how to model it with differential equations. Or the differential equations they have are overly idealized models of the system, so using the Green's function would be less accurate anyway.

[@ahmednagh9398]

Dang

[@pronaykhan9734]

Sir you told before you will make more videos on how to take physics classes from computer like you couldn't find relativity there but didn't make till now. Please make a video on it😢😢😢

[@kkgt6591]

Why is there a need to flip the impulse response?

[@username8644]

Hearing function and not distribution as a mathematician.

[@LeoHoulston]

So the Dirac function is just a specific impulse applied to a system?

[@hydropage2855]

Am I misunderstanding, or is the “flip, slide along, and integrate the product” a convolution? Edit: oh my god I commented this literally the second before “this is known…as convolution”

[@GhostyOcean]

As a mathematician, it annoys me that you say the derivative of the unit step function is the delta function, but this is true for *any* step function with a single change. But i see how that's useful for physics and calculation.

[@ethanbottomley-mason8447]

To be fair, it is the distributional derivative of the unit step function.

[@GhostyOcean]

@@ethanbottomley-mason8447 ah yes. That can exist when the function's true second derivative doesn't.

[@mattpeacock5208]

Forget Flash and Thor's hammer! There are no analogies for something that occurs for zero seconds. The "infinite" hight is just an idea, this is exactly why you cannot divide by zero!

[@DoesMahBlockLookBig]

That was sort of the point of the analogy………. there is no true dirac delta function in nature, instead it is approximated by high amplitude and short impact signals (like striking a hammer). the purpose was to show that for any arbitrary input to a linear time-invariant system, the output will be the convolution of the input signal with the impulse response. Say an engineer wanted to characterize an electrical circuit and determine any output Vout(t) based on input voltage Vin(t). No, the engineer wouldn’t send infinite voltage over an infinitesimal period of time. That would be impossible. To get the impulse response, the engineer would send a high voltage over a short period of time. (Or just send a step response and differentiate the output, but that’s besides the point)

[@DrDeuteron]

1:34. I know you meant "currents".