Okay, so I'm going to start a new topic here. Actually. So I'm going to be doing integration on curves, integration on surfaces. I'm actually going to not spend a lot of time on it because I'm going to assume you've done this in 243. How to integrate on a curve? How to integrate on ACEF is I'm going to remind you of some formulas. Alright? Because my goal is to get to the green Gaussian stokes theorem and the divergence theorem. Alright? So a lot of you are physics majors, engineering majors, math major. This is that these are very important theorems. Okay? So what I want to show you first in that, in that, when I'm my this, my goal for this, for the green, the stokes theorem, divergence theorem. First you need to know the statement. Okay? And it looks like a very complicated state with all kinds of parts there and stuff. So firstly, we have to make sure you know the statement, right. What does each part mean? That's number one. Okay, that's my first goal. My second goal is, what is the point of studying these green Gauss Stokes theorem? Why are we studying then? What is the point? Is there just a theorem just for this, or is, does it really actually have any use? Right? What is it trying to say? And the third thing is, well, Stokes theorem, if you recall, there's got a curl in it. Right? It's about surfaces and the boundary of a surface. That's the stokes theorem. There is no curl. As I said, curl is a cross product of the del operator with something I'll remind you. There is the cross-product in high dimensions. So is there a generalization of the Stokes theorem for the divergence theorem to high dimensions. There is. What form does it take? What is replacement for the curl? For Stokes theorem? Okay? What does replace the divergent? Divergence is actually pretty straightforward, you can guess. But really actually is going to be generalization to higher dimensions. And what is the replacement for the curl? Okay? What is the statement? Stokes theorem is about surface and its boundary, which is a curve. Divergence theorem is a solid and this boundary which is the surface. What are the, what are the corresponding thing in high dimensional, if you're in five dimensions? Is there Stokes theorem for an object in four dimensions? And its boundary, which is some three-dimensional object, is there such a thing that we're never going to get that far? But I'm going to give you a hint when we come to it as to what the generalization might look like. Alright, so that's, that's really my goal for this whole thing. Okay? But my first step is going to before I can even state the Divergence theorem or the Stokes theorem. These are all in terms of integrals on a curve or integrals on the surface. Let me remind you what they are before I even state the theorem. Okay? How do you integrate on a curve? How do you integrate on a surface? So let's start with that. Just a reminder. Okay, so I'm not going to do an example that I will just give you some formulas from from me to 43, okay, and in the homework maybe I'll give you one or two examples you can try. So firstly, so first thing is how do we integrate on a curve? Right? So let us talk of curves in 3D. Okay, a curve in 3D, it looks something like this. It can even cross-cut. Let's redraw this. It can even cross it, but I'm not going to read it with a crossing. Let me stick to what I have in the notes. Okay. That's a curve. Let's say you start here and UN, says there is a narrow, it's called an orientation. So that's, I'm going to call this curve gamma. Okay? I don't call it C because I'm going to use c for something else. Okay? So I give you a curve gamma. Well, what does it mean? So firstly, of course has some shape in 3D. But if you're going to integrate on it, you have to be given this parametric equation. Or if you don't have this parametric equation, you can't integrate on it. So what is the parametric equation? It's just a way to generate every point on this curve. So I gave you every point on this curve, right? So consider a curve gamma with the parametric equations. X is some function of t. I'm just going to call it that. Okay? So basically just prescribed some functions. Okay? And that's what a parametric equation is. So you can think of this as it's a particle which is moving in space. Okay? It's, it's a very good analogy to remember. So as a particle moves in space, it traces out a path. And the parametric equation is, tells you where is the particle at time t? That's what the parametric equation is. Okay? So for example, you may have done this in a 243. It's a popular example. Right? Or you can draw a spiral. Okay, so this is your gamma. If you can draw x cosine, y is sine t. So x and y are varying on the circle. But your height increases as you go up, as time increases, right? X and y are on the circle. But the height goes up with time. Okay? And you can take your time to interval to be anything you like. 028 pi. So it'll go around four times. Okay? So your, this is your curve comma. Ok, So that's what I meant by x is some function of t, y is some function of t, z is some function of t. And there is a starting and an ending point. Right? So whenever you're working with curve, just think of them as there is a particle which is moving in space. And x t, y t CTR its position at time t. So curve gamma with parametric equations. If we can regard PCP as the position of the particle at time t. Ok, it's easy to visualize this. So you give you the parametric equation of a curve. This is gamma. Just it tells you basically varies a particle at time t and as time travels, it moves around on this. Okay? So this is going to be important for us now. So if you think of this as a particle, you can assign a position vector to it. So we define the position vector. Ok, my artist look like gum, I'm sorry. Okay, let's are actually so you can define a position vector of the particle is just a way to think, okay? If you don't like particle, just think of this as moving on a curve. So this is the position vector. Okay? If you have a position vector, you have, well, what velocity vector? V of t, which, remember velocity is d r by d t position, differentiate position with respect to time. And so it's not DX by DT plus d y by d t j plus z by d t. So that's the velocity vector. Okay? So I have a curve. So a particle is moving and then I have this velocity vector. Velocity is physics. So there is, but there is geometry here, right? Every curve has a tangent vector. How do I find the tangent vector to a curve? If I gave you the parametric equation I gave you will like for example here, read this spiral. I tell you, okay, find me the tangent vector at this point, which corresponds to, let's say T equals two pi. Right? How do we find that? The velocity tangent to the curve, the velocity vector is always tangential to the curve. So that's something which you know, to 43 we talked about. Maybe it was proved or not. Why is it the time, but the velocity vector? And if I want the unit tangent vector, so I want a vector which is tangent to the curve, but of length one, it take that vector and then you find its magnitude and that's an exilic nine, right? So the velocity vector is tangential comma. So the unit tangent vector to gamma is defined as capital T. And you take the velocity vector divided by its length. And of course, the unit tangent vector changes with time. Right? So here is my curve. It's a function of time. Okay? So I've defined some, just reviewing stuff from 243 here. Okay? Then the next thing we define, which is really important, is the arc length function. Okay? Do you recall how do you find if I gave you this parametric equation like this? Sorry. If you cooperate, then I would go to the next thing. I give you this spiral. I tell you what is the length of that spiral? It goes around four times, right? So is its radius is one it looks like. So is it four times the circumference of the unit circle? What do you think? This is a setting on the unit circle and its height is going up and this is going around four times c. You think the length of this arc is four times the circumference of the unit circle are really more or less. I think it has to be a little bit more because you also going up anhydrite to stretching it. Okay. And how do you find this link? You'll remember had 243. Yeah, I don't remember if anyone. Okay, alright, here. So somebody remembers it now. So get the formula for length is or actually let me just write it down instead of copying. Right? You integrate, remember t goes from a to b, and you just integrate the velocity vector. The magnitude of the velocity vector, sorry, not the velocity vector. Right? If you're like, I mean, it's good to think like this. I'm in fact I'm not going to interstate, not right. The other thing you can write this in terms of dx, d y, d t, think. But I intentionally want you to remember is the length of the velocity vector. And how do you compute the velocity? Velocity is d r by d t. So there is another formula, but I'm not going to write it, right? You can write this in terms of square root of d x by d t squared dy by d d square, and so on. But I'm not going to write it. I want you to remember this way. It's easier to remember magnitude of the velocity, not the integral of the velocity. All right? Okay, so that's the length of a curve. But there is a very important function. Ok, which I'm going to come here and okay, so gamma has this parametric equation. Okay, that's a parametric equation. No. Notice something very strange here if you think of this as the motion of a particle, right? So let's think of it as a spacecraft. Starts at this point, a goes to this point B, takes this path and new track, or where was it at any given time? And that's what parametric equation curriculum that describes the path. If you know the parametric equation, you know the path. So can this part have more than one parametric representations? Or is a parametric representation for this path unique? And more than one? Because this is a path like anyone flies from, say, Cape Canaveral to the moon, right? Let's say, right? So assuming you know, you can take the same path, right? But one space goes very slowly. Another one goes much faster, speeds up, slows down. They're both going to the moon. We're both taking the same path. But they'll give you different ways of different x t, y t, and z t like for example, right? You're driving from here to Houston. Right? You'll probably all take the same way. You and I are 995 south and somewhere you will go to, I don't know, 70 or 80, so on, right? So you're all going to go from here to use the same path. You will trace out the same path. But some of us speed demon, some are going to go very fast. And you will take different time. Your parametric equation for that path will be different from someone else driving exactly the same way. So you can see that a curve can have more than one parametric representation. So which parametric representation do I use when I want to calculate, right? When you can use any one, you suppose you went to get the same answer. Right? But let me ask you this. Is there another, is there a natural parametric representation for this curve? You're driving from here to Houston, right? So one person be both start at the same time. Ok. Let's say I and someone else, right? So I and I'm reporting where I am. So I said Okay, I started at times 0 after one hour. I'm in Baltimore. After two hours. I mean, I don't know. Somewhere. The Somerset was someplace three hours. I'm somewhere else. Six hours, Iowa and I don't know. It's little rock. After 20 hours IN houston, someone else's driving. Right? They will certainly report something quite different. After one hour they were there after two hours, they were somewhere else. Same path. So if I just gave you that, you would not be able to tell whether they were really taking the same path or not. Right. But is there another way to report your position so that no matter who drives, you will audit report exactly the same thing. I'm driving and I keep reporting to someone. You drive, you keep reporting to someone. Clearly, if your report based on time, you're going to get very different answers. Is there another way you can report your possession and You know, not time but something so that you have the path and any two drivers would report exactly the same thing, then that's a natural representation of that path. Do you understand my point here? Right. Something which should not depend on who's driving. So what kind of reporting can you do so that any two people driving on that path will report exactly the same thing. Parameterize it based on arc length. So which means what do you mean? So when I'm driving from here to Houston, what you report your report. The distance that you travel along the arc is that at the time that you spent traveling? So you say I've traveled a 100 miles. I'm not Washington DC. I've traveled 200 miles. I'm now in Somerset. I've traveled 300 miles. I don't I'm in Charlottesville. I will traveled 800 miles. I'm a Little Rock and it doesn't matter who's driving on that road. They will report exactly the same thing. Correct. So there is a natural parametrization for a path which will tell me where you are based on the distance you've traveled from the starting point. Okay? And that's, so that's a very important way of representing the parameterizing a curve. Okay? Now you see my motivation for why want to define the arc length function? That is the natural parametrization. It doesn't depend on who's going on that road. So when you have a curve, there are all kinds of parameterizations for the curve, but there is one natural parameterization. Okay, so let me write this down now. So Kurt can have more than one parameterization. Okay? I'd give quote an example in the notes, C example in notes. Okay, it's pretty easy to do, it's not. Okay. However, every curve has a unique, unique natural parametrization with respect to the distance travelled. Okay? So you define what is called the arc length function. So I have my curve gamma. I am starting here at t equals a, ending at T equals B, going like this. So S of t. Distance covered in time interval a2, t. So you started here at time t equals a and you are here at times t equals T. And S P is this length. So it gives you a connection between time and distance. Right? And remember the arc length formula for you. You have to integrate the velocity vector vocabulary to use sigma here instead of T, because T is right. If you had the full curve, you go a to B, but now you're going to a to t. That's your arc length functions. Right? This is called the arc length function. So you integrate the speed case, but this is actually just forgot to write this. Okay? So a magnitude of velocity is speed. So anyway, so S of t is that. Okay? So what is the derivative of the arc length function with respect to time? So use the fundamental theorem of calculus. You'll get, you'll get speed. Right? So how does the rate of change of distance travelled with respect, with time? That's exactly what speed is. Correct. So i want to write these things down. So it's a little proposition. For any curve gamma. The unit tangent vector to gamma is, which is the velocity vector divided by the magnitude of V. Secondly, okay, these are going to be very important for us when we integrate. So that's one. Okay? The second thing was natural parameterization. Or any curve comma is a parametrization. With respect to distance travelled. So you tell me where you are based on distance traveled. Okay? Where L is equal to the length. Ok. So you give me the total distance from here to Houston, 50-100 miles. For every S between 01500, you report your position. Ok. That's the path you've taken. Hold, right? Okay, so I haven't yet come to integration. All right, so now here is the first step. Integration on curbs. Okay? So consider a curve gamma with parametric equation. Okay? So there are going to be two natural integrals. Can natural means, I'm going to define integrity. Integration is addition. So you're going to add contribution from every point on a curve. Okay? And it's going to be natural in the sense it will not depend on the coordinate system. Just depend on the curve. Okay? You can, I mean, you have a, you have a curve in space, a path from here to the moon. I mean, you put it in any coordinate system that's not going to change the path, right? You can put your origin somewhere, rotate your axis whichever way you like. That doesn't change the path. I'm going to define an integral which doesn't depend on the axis. Ok? So the first one is this. So suppose f x y, z is a function on gamma. Then we can define this as the first integral ds. Okay? Am I claim is a natural integral ds has integration on the length, so it doesn't depend on the coordinate access. But this f x, y, z seems to depend on the coordinate axes and I'm calling it a natural integral. Right? But why is it natural? Right, is a function on a curve. So here is one way to see why doesn't depend the coordinate axis, right? Think of curve as a wire is some wire which has bent in some shape. Okay? And f of x, y, z is the density at each point on the wire, you give me a point x, y, z on the wired. If XYZ is its density, density here I'm going to zoom. Cross section is the same, so it's mass per unit length. So this is the mass of a small piece, fx times ds, ds with the length of the small piece, fx times that is the length of that small piece. Think of density is varying from point to point. So that's the mass of the small piece of that wire. And then you add it upon gamma, which is integrate on gamma, you get the mass of the wire, right? So you prescribe the density at each point on the wire, that's your function f. Then you integrate addict up on the curve. That's that integral. That's the motivation for that. Does it depend on the coordinate system? Does the mass of that wire depending on which coordinate system you have. It doesn't. Your function f will change in the sense that if you do different x, y, z. But the density at that point, if you think of a function as on the wired, you give me a location of the point on the wire is density will not change. So this is a natural integral. Okay, I've said this in the book, in the notes, I've stated this, so I'm just going to be very quick here. Okay, so does natural integral. Okay, time's up, so I'm going to stop here. And we will continue with this next time. Alright. Well, let's do a poll for the second part. Okay, well, let me stop the recording and then.
Arc length and the first integral on a curve
From Rakesh Rakesh September 17, 2020
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