![]() T over 2 tau, and you have to evaluate thisįrom minus tau to tau. You get- I'll do this just to satiate your curiosity. You could get this is going toīe equal to- you take the antiderivative of 1 over 2 tau, We don't even have to knowĬalculus to know what this integral's going And then over that boundary, theįunction is a constant, 1 over 2 tau, so we could just Goes to positive infinity or minus infinity. Way too, right? Because we can just take theīoundaries from here to here, because we get nothing whether t Infinity, because there's no area under any of those Integral as the integral from minus tau to tau- and we don'tĬare if infinity and minus infinity or positive Say, well, first of all, it's zero everywhere else. The area under this curve, this is a pretty straightforward thing to calculate. Going to be equal to? Well, if the integral is just Minus infinity to infinity of d sub tau of t dt, what is this So why did I construct thisįunction this way? Well, let's think about it. So this point right here on theĭependent axis, this is 1 over 2 tau. Then we stay constant at that level until we So it's going to be zeroĮverywhere, and then at minus tau, we jump to this level, and Until we get to minus t, and then at minus Happen here? It's going to be zero everywhere We could call it the y-axis or the f of t-axis, or This will actually look likeĪ combination of unit step functions, and we can actuallyĭefine it as a combination of unit step functions. Less then tau and greater than minus tau. Over 2 tau, and you'll see why I'm picking these numbers ![]() We're doing in the Laplace transform world,Įverything's been a function of t. Well, let me put it as a function of t because everything Intuition for how this Dirac delta functionĬan be constructed. Is all just to satisfy this craving for maybe a better Let's say that I constructedĪnother function. Transform of this, and then we'll start manipulating Then we're going to start taking the Laplace Understanding of how someone can construct a function Under this- I'm telling you- is of area 1. Narrow base that goes infinitely high, and the area Of x is a function such that its integral is 1. Now, you might say, Sal, youĭidn't prove it to me. Number line, if I take the integral of this function, Infinity to infinity, so essentially over the entire real Were to take the integral of this function from minus I deal with that? How do I take the integral Look, when it's in infinity, it pops up to infinity Represented by the delta, and that's what we do Manipulate it? And I'm going to make one moreĭefinition of this function. This video, but we'll call it a function in this video. Under this, it becomes very- to call this a function isĪctually kind of pushing it, and this is beyond the math of So it'd be infinitely high rightĪt 0 right there, and then it continues there. To some point and then bam! It would get infinitely strong,īut maybe it has a finite area. What would happen is that nothing happens until we get Some type of function that could model this type And you'll learn this in theįuture, you can kind of view this is an impulse. Nothing happens forĪ long period of time, and then whack! Something hits you really hardĪnd then goes away, and then nothing happens for a very Nothing happening for a long period of time. Like this, but it can be approximated by the unit Happens for a long period of time and then bam! Something happens. Introduced is because a lot of physical systems kind What you've seen in maybe your Algebra courses. What you've seen in just a traditional Calculus course, ![]() It's more exotic and a little unusual relative to Unit step function, I said, you know, this type of function, In the above example I gave, and also in the video, the velocity could be modeled as a step function. ![]() The Dirac delta function usually occurs as the derivative of the step function in physics. It may also help to think of the Dirac delta function as the derivative of the step function. So they use the Dirac delta function to make these "instantaneous" models. Physicists and engineers make the assumption that some things happen instantly, because they are so fast that trying to model them using actually equations would over complicate the problem, with no gain. Sounds like the Dirac delta function, huh? In this scenario, the force applied to the object could be modeled as 2*m*delta(x), where m is the mass of the accelerated object. To model this, we would need a function the represents an infinite acceleration (to accelerate the object in an infinitely small time) but has a finite area (the area under the acceleration function is velocity). We know that this force could not have accelerated the object instantly, but for our purposes, let's assume it did. Consider a physical system, in which a object is at rest, and an external force is quickly applied, accelerating it to a velocity of 2 m/s.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |