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“I introduced you to the nunit step function. I said you know this type of of function nit s more exotic and a little unusual relative to nwhat you ve in just a traditional calculus course nwhat you ve seen in maybe your algebra courses. But the reason why this was nintroduced is because a lot of physical systems kind nof behave this way that all of a sudden nothing nhappens for a long period of time. And then bam.
Something happens and you go like that and it doesn t happen exactly nlike this. But it can be approximated by the unit. Nstep function similarly. Sometimes you have nnothing happening for a long period of time nothing happens for na long period of time.
And then whack something hits you really hard nand then goes away. And then nothing happens for a very nlong period of time and you ll learn this in the nfuture you can kind of view. This is an impulse and we ll talk about nunit impulse functions and all of that so wouldn t it be neat if we had nsome type of function that could model this type nof behavior and in our ideal function nwhat would happen is that nothing happens until we get nto some point and then bam. It would get infinitely strong nbut maybe it has a finite area and then it would go back to nzero and then go like that so it d be infinitely high right nat.
0. Right there and then it continues. There and let s say that the area nunder this it becomes very to call this. A function is nactually kind of pushing it and this is beyond the math of nthis video.
But we ll call it a function in this video. But you say well what good nis this function for how can you even nmanipulate. It and i m going to make one more ndefinition of this function. Let s say we call this function nrepresented by the delta and that s what we do nrepresent this function by it s called the dirac ndelta function and we ll just informally say nlook when it s in infinity.
It pops up to infinity nwhen x. Equal to 0. And it s zero everywhere. Else nwhen x.
Is not equal to 0. And you say how do ni deal with that how do i take the integral nof that and to help you with that i m ngoing to make a definition. I m going to tell you what nthe integral of this is this is part of the definition nof. The function.
I m going to tell you that if i nwere to take the integral of this function from minus ninfinity to infinity. So essentially over the entire real nnumber line. If i take the integral of this function ni. m defining it to be equal to 1.
I m defining this now you might say sal you ndidn t prove it to me. No. I m defining. It.
I m telling you that this delta nof. X. Is a function. Such that its integral is 1.
So it has this infinitely nnarrow base that goes infinitely high and the area nunder this. I m telling you is of area 1. And you re like hey sal nthat s a crazy function. I want a little bit better.
Nunderstanding of how someone can construct a function nlike this so let s see if we can satisfy nthat a little bit more. But then once that s satisfied nthen. We re going to start taking the laplace ntransform of this and then we ll start manipulating nit and whatnot. Let s see let me complete nthis delta right here.
Let s say that i constructed nanother function. Let s call it d sub. Tau and this nis all just to satisfy this craving for maybe a better nintuition for how this dirac delta function ncan be constructed and let s say. My d sub.
Tau of nwell let me put it as a function of t. Because everything nwe. re doing in the laplace transform world neverything s been a function of t. So.
Let s say that it equals. 1 nover. 2 tau and you ll see why i m picking these numbers nthe way. I am 1.
Over 2 tau when t is nless. Then tau and greater than minus..
Tau and let s say it s 0 neverywhere else so this type of equation nthis is more reasonable this will actually look like na combination of unit step functions and we can actually ndefine. It as a combination of unit step functions. So if i draw. That s nmy x.
Axis. And then if i put my ny. Axis. Right here.
That s my y axis. Sorry. This is a t axis. I have to get out nof that habit.
This is the t axis and i mean nwe could call it the y axis or the f of t. Axis or nwhatever. We want to call it that s the dependent variable. So what s going to nhappen here.
It s going to be zero everywhere nuntil. We get to minus t. And then at minus nt. We re going to jump up to some level just let me put that npoint here.
So this is minus tau and nthis is plus tau. So it s going to be zero neverywhere and then at minus tau. We jump to this level and nthen. We stay constant at that level until.
We nget to plus tau and that level. I m saying nis. 1. Over 2 tau.
So this point right here on the ndependent axis. This is 1 over 2 tau. So why did i construct this nfunction this way well let s think about it what happens. If i take nthe integral let me write a nicer nintegral sign if i took the integral from nminus infinity to infinity of d sub tau of t dt.
What is this ngoing to be equal to well if the integral is just nthe area under this curve. This is a pretty straightforward thing to calculate you just look at this and you nsay well first of all it s zero everywhere else. It s zero everywhere. Else and nit s only the area right here.
I mean i could rewrite this nintegral as the integral from minus tau to tau and we don t ncare. If infinity and minus infinity or positive ninfinity. Because there s no area under any of those npoints of 1. Over.
2 tau. D. Tau. Sorry.
1. Over. 2 tau dt. So we could write it this nway.
Too right because we can just take the nboundaries from here to here. Because we get nothing whether t ngoes to positive infinity or minus infinity. And then over that boundary the nfunction is a constant 1. Over 2 tau.
So we could just ntake this integral and either way we evaluate it we don t even have to know ncalculus to know what this integral s going nto evaluate to this is just the area under nthis. Which is just the base. What s the base. The base is 2 tau.
You have one tau here and nthen another tau there so it s equal to 2 tau ntimes your height and your height. I just nsaid is 1 over 2 tau so your area for this function. Nor for this integral is going to be 1. You could evaluate this you could get this is going to nbe equal to you take the antiderivative of 1 over 2 tau nyou get i ll do this just to satiate your curiosity.
Nt over 2 tau. And you have to evaluate this nfrom minus tau to tau and when you would put tau in nthere you get tau over 2 tau and then minus minus tau over n2 tau and then you get tau..
Plus. Tau over 2 tau that s n2 tau over 2 tau. Which is equal to 1 maybe. I m beating na dead horse.
I think you re satisfied that nthe area under this is going to be 1 regardless nof. What tau was i kept this abstract now if i take smaller and nsmaller values of tau. What s going to happen. If my new tau is going to be nhere.
Let s say my new tau is going to be there i m just ngoing to pick up my new tau. There then my 1 over n2 tau. The tau is now a smaller number so when it s in the denominator nmy. 1.
Over 2 tau is going to be something nlike this right i mean i m just saying if i npick. Smaller and smaller taus. So then if i pick an even nsmaller tau than that then my height is going to be nhave to be higher my 1 over. 2 tau is going nto have to even be higher than that and so i think you see where ni m going with this what happens as the limit nas tau approaches zero.
So what is the limit as tau napproaches zero of my little d sub. Tau function. What s the limit of this well these things are going nto go infinitely close to zero. But this is the limit they re never going to nbe.
Quite at zero and your height here is going to ngo infinitely high. But the whole time. I said no matter nwhat my tau is because it was defined very arbitrarily nwas my area is always going to be 1. So you re going to end up with nyour dirac delta function let me write.
It now. I was going to write nan x. Again. Your dirac.
Delta. Function is a nfunction of t. And because of this. If you ask what s the nlimit as tau approaches zero of the integral from minus ninfinity to infinity of d sub.
Tau of t. Dt. Well this should nstill be 1 right because this thing right. Here.
Nthis evaluates to 1. So as you take the limit as tau napproaches zero. And i m being very generous with nmy definitions of limits and whatnot. I m not being very rigorous.
But i think you can kind of nunderstand. The intuition of where i m going this is going to nbe equal to 1. And so by the same intuitive nargument you could say that the limit from minus infinity to ninfinity of our dirac delta function of t. Dt is also ngoing to be 1.
And likewise. The dirac delta nfunction. I mean this thing pops up to infinity at nt is equal to 0. This thing.
If i were to draw my nx axis like that and then right at t equals. 0. My ndirac delta function pops up like that and you normally draw nit like that and you normally draw it so nit goes up to 1 to kind of depict its area. But you actually put an arrow nthere and so this is your dirac delta function.
But what happens if you nwant to shift it how would i represent my nlet s say i want to do t minus. 3. What would the graph nof this be well this would just be nshifting. It to the right by 3 for example.
When t. Equals. 3. Nthis will become the dirac delta of 0.
So this graph will just nlook like this this will be my x axis. And let s say that this nis..
My y axis. Let me just make that 1. And let me just draw some points nhere. So it s 1 2.
3. That s t is equal to 3. Did i say that was the x axis that s my t axis. This is t equal to 3.
And what i m going to do here is nthe dirac delta function is going to be zero everywhere. But then right at 3. It ngoes infinitely high and obviously we don t have nenough paper to draw an infinitely high spike nright. There so what we do is we ndraw an arrow.
We draw an arrow there and the arrow. We usually draw nthe magnitude of the area under that spike. So we do it like this and let me be clear. This is not saying that the nfunction just goes to 1.
And then spikes back down this tells me that the narea under the function is equal to 1. This spike would have to be ninfinitely high to have any area considering. It has an ninfinitely small base. So the area under this impulse function.
Nor under this dirac delta function now this one right here is t nminus. 3. But your area under this is still going to be 1. And that s why i made nthe arrow go to 1 let s say i wanted to graph nlet me do it in another color let s say i wanted to graph n2 times.
The dirac delta of t minus. 2. How would i graph this well. I would go to t minus 2.
When t is equal to 2 you get nthe dirac delta of zero. So that s where you would nhave your spike and we re multiplying. It by 2 nso. You would do a spike twice as high like this now both of these go to ninfinity.
But this goes twice as high to infinity. And i know this is all being na little ridiculous now. But the idea here is that the narea under this curve should be twice the area under nthis curve and that s why we make the arrow ngo to 2 to say that the area under this arrow is 2. The spike would have to ngo infinitely high.
So this is all a little nabstract. But this is a useful way to model things that are nkind of very jarring. Obviously nothing actually nbehaves like this. But there are a lot of phenomena in nphysics or the real world that have this spiky behavior instead of trying to say noh.
What does that spike exactly look like we say hey that s a dirac ndelta function and we ll dictate its impulse nby something like this and just to give you a little nbit of motivation behind this and i was going to go here in nthe last video. But then i kind of decided not to but i m just going to show it nbecause. I ve been doing a lot of differential equations and ni ve been giving you no motivation for how this applies nin. The real world.
But you can imagine if i have njust a wall and then i have a spring attached to some mass nright there and let s say that this is a natural state nof. The spring. So that the spring would want to be here nso. It s been stretched a distance y from its kind of nnatural.
Where it wants to go and let s say i have some nexternal force right here. Let s say. I have some external nforce right here on the spring and of course. Let s nsay.
It s ice on ice. There s no friction nin all of this and i just want to show you nthat. I can represent the behavior of this system with nthe differential equation and actually things like the nunit step functions the dirac delta function actually start nto become useful in this type of environment. So we know that f is equal to nmass times acceleration that s basic physics nright.
There now what are all nof. The forces on this mass right here well you have this nforce right here. And i ll say. This is a positive nrightward direction.
So it s that force and then nyou have a minus force from the spring. The force from the spring nis hooke s law..
It s proportional to how far nit s been stretched from its kind of natural point. So its nforce in that direction is going to be ky or you could ncall. It minus ky. Because it s going in the opposite direction nof.
What we ve already said is a positive ndirection. So the net forces on this is f nminus. Ky. And that s equal to the mass of our object.
Times. Nits acceleration. Now what s its acceleration. If its position is y.
So if y nis equal to position. If we take the derivative of y. With nrespect to t. Y.
Prime. Which we could also say dy dt. This nis going to be its velocity and then if we take the nderivative of that y prime prime. Which is equal to d nsquared y.
With respect to dt squared. This is equal nto acceleration. So instead of writing a we ncould right y prime prime and so if we just put this non. The other side of the equation.
What do we get we get the force. This force nnot just this force. This was just f equals ma. But this nforce is equal to the mass of our object.
Times. The nacceleration of the object. Plus whatever the spring nconstant is for the spring plus k. Times.
Our position ntimes y. So if you had no outside force nif. This was zero you d have a homogeneous differential nequation and in that case. The spring nwould just start moving on its own.
But now this f all of a nsudden. It s kind of a non homogeneous term. It s what nthe outside force you re applying to this mass. So if this outside force was nsome type of dirac delta function.
So let s say it s t nminus. 2. Is equal to our mass times. Y.
Prime prime plus. Our nspring constant times. Y. This is saying that at time is equal nto 2 seconds.
We re just going to jar. This thing nto the right and it s going to have an and ni ll talk more about it it s going to have nan impulse of 2. It s force times time is going nto be or its impulse is going to have 1. And i don t want to get too much ninto.
The physics here. But its impulse or its change in nmomentum is going to be of magnitude 1 depending on nwhat our units are but anyway i just wanted to ntake a slight diversion because you might think sal is nintroducing me to these weird exotic functions. What are they ever going nto be good for but this is good for the idea nthat sometimes you just jar. This thing by some magnitude nand.
Then let go and you do it kind of infinitely nfast. But you do it enough to change the momentum of nthis in a well defined way anyway in the next video. We ll ncontinue with the dirac delta function. We ll figure out its laplace ntransform and see what it does ” .
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