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# Dirac delta function

## Video transcript

when I introduced you to the unit step function I said you know this this type of a function it's more exotic and a little unusual relative to what you've seen in just a traditional calculus course what you've seen in maybe your algebra courses but the reason why this was introduced because a lot of physical systems kind of behave this way that all of a sudden nothing happens nothing happens for a long period of time then BAM something happens and you go like that and it doesn't happen exactly like this but it can be approximated by the unit step function similarly sometimes you have nothing happening you have nothing happening for a long period of time nothing happens for a long period of time and then whack something hits you really hard and then goes away and then you nothing happens for a very long period of time and you'll learn this in the future you can kind of view this as an impulse and we'll talk about unit impulse functions and all of that so wouldn't it be neat if we had some type of function that could model this type of behavior and and our ideal function what would happen is is that nothing happens nothing happens until we get to some point and then BAM it would get infinitely strong but maybe it has a finite area and then it would go back to zero and then go like that so it'd be infinitely so high right it let's say zero right there and then it go can't in use right and let's say that the area under this I mean it becomes very to call this a function is actually kind of pushing it and well this is beyond the math of this video but we'll call it a function in this video but what we can you know you say well how do you even you know what what good is this function for how can you even manipulate it and I'm going to make one more definition of this function so what I just do here let's say we call this function we represented by the Delta and that's what we do represent this function by its called the Dirac Delta function and we'll just informally say look when it's an infinity it pops up to infinity when X is equal to zero and it's zero everywhere else when X is not equal to zero and say how do I deal with that how do I take the integral of that and to help you with that I'm going to make a definition I'm going to tell you what the integral of this is this is part of the definition of the function I'm going to tell you that if I were to take the integral of this function from minus infinity to infinity so essentially over the entire a real number line if I take the integral of this function I'm defining it I'm defining it to be equal to one I'm defining this now you might say Sal you didn't prove it to me no I'm defining it I'm telling you the Delta this Delta of X is a function such that it's integral as one so it has this infinitely narrow base that goes infinitely high and and the area under this I'm telling you is of area one and you're like hey Sal that's a crazy that's a crazy function I want a little bit a little bit better understanding of how someone can construct a function like this so I don't let's see if we can if we can satisfy that a little bit more but then once that satisfy then we're going to start taking the Laplace transform of this and and then we'll start manipulating it and whatnot let's see maybe we'll complete this Delta right here let's say that I constructed another function let's call it D D Sub tau and this is all just to satisfy this craving for maybe a better intuition for how this Dirac Delta function can be constructed and let's say my D Sub tau of let me just put in it's a function of T because I want to want to everything we're doing in Laplace transform world everything's been a function of T so let's say that it equals let's say that it equals 1 over 2 tau and you'll see why I'm picking these numbers where I am 1 over 2 tau when T is less than tau and greater than minus tau and let's say it's 0 everywhere else every where else so this type of this equation this is more reasonable this will actually look like a combination of unit step functions and we can actually define it as a combination of unit step functions so if I draw that's my x-axis that's my x-axis and then if I put my y-axis right here that's my y-axis Y and this is sorry this is the T taxes to get out of that habit this is a t axis and I mean we could call what we call it the y-axis or the f of t axis or whatever we want to call it that's the dependent variable so what's going to happen here it's going to be zero everywhere until we get to minus T and then at minus T we're going to jump up to some level so let me put that point here so this is minus tau and this is plus tau right minus tau and plus tau so it's going to be zero everywhere and then at minus tau we jump to this level and then we stay constant at that level until we get to plus tau and that level I'm saying is 1 over 2 tau so this point right here the 1 the dependent axis this is 1 over 2 tau so why did I construct this function this way well let's think about it what happens if I take the integral let me write a nicer integral sign if I took the integral from minus infinity to infinity of D sub tau of T DT what is this going to be equal to well if you just I mean if the integral is just the area under this curve this is a pretty straightforward thing to calculate right you just look at this and you say well this is first of all it's 0 everywhere else it's 0 everywhere else and it's only the area right here I mean I could write this I could rewrite this integral as the integral from minus tau 2 tau we don't care infinity and minus infinity and positivity because the there's no area under any of those points of 1 over 2 tau D tau we could write it we could DT sorry 1 or 2 tau DT so we could write it this way too right because if we can just take the boundaries from here to here because you we get nothing the whole other whether T goes to positive infinity or minus infinity and then over that boundary the function is a constant 1 over 2 tau so we could just take this integral and either way we evaluate it we don't even have to know calculus to know what this integral is going to evaluate to this is just the area under this the area under this which is just the base what's the base the base is to Tao right the base is to Tao you have one Tao here and then another tile there so it's equal to 2 tau times your height and your height I just said is 1 over 2 tau 1 over 2 tau so your area for this function over for this integral is going to be 1 you could evaluate this you could get this is going to be equal to you take the antiderivative of 1 over 2 tau you get I'll do this just to satiate your curiosity T over 2 tau and you have to evaluate this from from minus tau 2 tau you put Talon there you get tau over 2 tau and then minus minus tau over 2 tau and then you get tau plus tau over 2 tau that's 2 tau over 2 tau which is equal to 1 maybe I'm I'm beating a dead horse I think you're satisfied that the area under this is going to be 1 regardless of what tau was I kept this abstract now if I take smaller and smaller values of tau what's going to happen if my new tau is my new tau is going to be let's say here let's say my new tau is going to be there I'm just going to pick up my new tau there then my 1 over 2 tau tau is now a smaller number so when it's in the denominator my 1 over 2 tau is going to be something like this it's going to be something like this right I mean I'm just saying if I pick smaller and smaller tau so if I pick me an even smaller tile than that then my height is going to be have to be higher right my 1 over 2 tau is going to have to be going to be higher than that and so I think you see where I'm going this what happens is the limit as tau approaches 0 so what is the limit the limit as tau approaches 0 of my little D sub tau function what's the limit of this well you're going to these things are going to go infinitely close to 0 but this is the limit they're never going to be quite at 0 and your height here is going to go infinitely high but the whole time I said no matter what my Tao is because it was defined very arbitrament my area is always going to be one so your end up you're going to end up you're going to end up with your Dirac Delta function let me write it it's going to write it X again your Dirac Delta function is a function of T and because of this if you say you know if you ask well you know what's the limit as tau approaches zero of the integral from minus infinity to infinity of D Sub tau of T DT well this should still be one right because this thing right here this evaluates to 1 so as you take the limit as tau approaches zero and I'm being very generous with my definitions of limits and whatnot I'm not being very rigorous but I think you can kind of understand the intuition where I'm going this is going to be equal to 1 and so by the same I guess in intuitive argument you could say that the limit as from minus infinity to infinity of our Dirac Delta function of T DT is also going to be 1 and likewise Dirac Delta function I mean this thing this thing pops up to infinity at at T is equal to 0 right this thing if I were to draw my x-axis my x-axis like that and then right at T equals 0 my Dirac Delta function pops up like that and you normally draw it like that and you normally draw it so it goes up to 1/2 kind of depict its area but you put actually put an arrow there and so this is your Dirac Delta function but what happens if you want to shift it what happens if you want to shift it what would how would I represent how would I represent my let's say I want to do t minus 3 what would the graph of this be well this would just be shifting it to the right by 3 for example when T equals 3 this will become the Dirac Delta of 0 so this graph this graph will just look like this this will be my x-axis let's say that this is my y-axis let me just make that one and let me just draw some points here so it's one two three that's T is equal to three so I say that was x-axis that's my t axis so this is T equal to three and what I'm going to do here it's the Dirac Delta function it's going to be zero everywhere everywhere zero everywhere but then right at three goes infinitely high and obviously we don't have enough paper to draw an infinitely high spike right there so what we do is we draw an arrow we draw an arrow there and the arrow we usually draw the magnitude of the area under that spike so we'd do it like this and let me be clear this is not telling me that the function just goes to 1 and then spikes back down this tells me that the area under the function is equal to 1 this spike would have to be infinitely high to have any area considering it has no an infinitely small base so the the area the area under this impulse function or under this Dirac Delta function now this one right here is t minus 3 but your area under this is still going to be 1 and that's why I made the arrow go to 1 if I had if I were to if I let's say I wanted to graph let me do it in another color let's say I wanted to graph 2 times 2 times the Dirac Delta of t minus 2 how would i graph this well I would go to t minus 2 it went when T is equal to 2 you get the Dirac Delta of 0 so that's where you'd have your spike and we're multiplying it by 2 so you would do it a spike twice as high like this now both of these go to infinity but this goes twice as high to infinity well I know this is all being a little ridiculous now but the idea here is that the area under this curve should be twice the area under this curve and that's why we make the arrow go to two to say the area under this arrow is to the spike would have to go infinitely high so this is all a little abstract but this is a useful way to model things that are kind of very jarring that you know all of a sudden obviously nothing actually behaves like this but there a lot of phenomena in in in in I guess physics or the real world that kind of you know have the spiky behavior that you know and it's instead of trying to say well what does that spike exactly look like we say hey that's a Dirac Delta function and will dictate its impulse by something like this and just to give you a little bit of motivation behind this and I was going to go this I was going to go here in the last video but then I kind of decided not to but I'm just going to show it because I've been doing a lot of differential equations I've been giving you know motivation for how this applies in the real world but you can imagine if I have just a table wall and then I have a spring attached to some mass some mass right there and let's say that this is the natural state of the spring so the spring would want to be here so it's been stretched a distance Y from it's kind of natural where it wants to go and let's say I have some external force external force right here let's say I have some external force right here on the spring and and of course let's say it's ice on ice there's no friction in all of this I just want to show you that I can represent this behavior of this system with a differential equation and actually things like the unit step function and the Dirac Delta function actually start to become useful in this type of an environment so we know that we know that F is equal to mass times acceleration that's the basic physics right there now what are all of the forces on this on this on this on this mass right here well you have this you have this force right here you would and we'll say this in the positive rightward direction so it's that force then you have a minus force from the spring right the force from the spring is Hookes law it's it's proportional to how far it's been just stretched from it's kind of natural point so it's force in that direction is going to be KY or you could call it minus KY because it's going in the opposite direction of what we've already said is a positive direction so the net force is on this is F minus K Y and that's equal to the mass of our it's equal to the mass of our of our object times its acceleration now what's its acceleration if its position is Y so if Y is equal to position if we take the derivative of Y with respect to T Y prime which we could also say dy DT this is it's going to be its velocity and then if we take the derivative of that Y prime prime which is equal to e squared Y respect to DT squared this is equal to acceleration acceleration so we can write instead of writing a we could write Y prime prime Y prime prime and so if we just put this on the other side of the equation what do we get we get the force this force not just this force this is just F equals MA but this force is equal to the mass of our object times the acceleration of the object plus whatever the spring constant is for this spring plus K times our position times y so if you had no outside force if this was zero you'd be happy you'd have a homogeneous differential equation and in that case you know the spring would just start moving on its own but now this F all of a sudden are the kind of a non-homogeneous term it's it's what the outside force you're applying to this to this mass so if this outside force was some type of Dirac Delta function so let's say it's let's say it's t minus two is equal to you know our mass times Y prime prime plus our spring our spring constant times y this is saying that at time is equal to two seconds or I'll we're just going to jar this thing to the right and it's going to have and I'll talk more about it it's going to have an impulse of two its force times time is going to be or not its impulse is going to have a 1 so it's it's and I don't want to get too much into the physics here but its impulse or its change momentum is going to be of magnitude one depending on what our units are but I just wanted to take that slight diversion because you know you might be saying Sal is introducing me to these weird exotic functions what are they ever going to be good for but this is good for the idea of it sometime you just you just jar this thing by some magnitude and then let go and you do a kind of infinitely fast but you do it with some enough to change the momentum of this in a in a well-defined way anyway in the next video we'll continue with the Dirac Delta function we'll figure out its Laplace transform and see what it does to the Laplace transforms of other functions