Hi everyone. So today I'm going to talk about how to integrate on surfaces. Okay? And I talked earlier in the introduction on time back about why we want to integrate on surfaces. Remember their applications to how much lift is generated by the wing of an airplane or how much water is flowing across the surface. And there are other applications to integration on surfaces. So being able to integrate on surfaces is actually important. And I'm going to talk about how we integrate on a surface. Now again, this is, these are things you did in 243. So I'm only going to give a very quick review of that. Alright, so now before you can integrate on a surface, the surface has to be given to you. So what are the ways we as surfaces are represented? One is what I call the explicit form, right? For example, you've seen the parable Lloyd z equals x squared plus y squared. Okay? The second form is what I call the implicit form. You know, explicit means given explicitly here, x0 is given in terms of xn by implicit form is really the way most of the surfaces are given to us, for example, the sphere, right? You could solve for z in terms x and y, but there will be plus or minus two square roots. Okay? And sometimes you cannot even solve for one of the variables. Okay? So it's just an equation, all the points which obey this equation. These are the points which lie on that surface. Okay? So if something is given an explicit form, it certainly is an implicit form. You can bring the z2 one side, right? But not every implicit form gives you an explicit form surface, okay? Well, for integration actually what we need is this, what is called the parametric forms. Which means a way to generate all the points on the stiffness. Ok, so for example, me, what I wanted to say is for integration on surfaces. Okay, we need the parametric equation of the surface. So we need this part. The surface will be represented in this form then and only then can we do with the integration on surface. Okay? If it isn't explicit, form it actually, if you're somebody has, in fact, we'll just do an example and I don't need to say this. Okay? So for example, suppose someone gives you the surface, this parabola, right? So this looks like this. Again, this is stuff you've done into 43 MB, just doing it quickly. This is z equals x squared plus y squared. Okay? So what's the parametric form of the surface? I just want a way to generate all the points on this parabola. Lloyd. Ok, so the easiest way to generate points on this hyperboloid is pick an arbitrary value for x, pick an arbitrary value for y and z, right? So arbitrary value for x, arbitrary value for y. Then solve for Z. Okay? And what about the range where u and v? Well, if you're doing the whole infinite surface, okay? Then it has like this. Alright? So that's the parametric representation of this. So if it's an explicit form, it's pretty easy. But sometimes it's tricky. Right? Suppose your surface has this ellipsoid. Actually, I will leave it for you to read. Well, so what we did one example was this. Actually, I forget with section it was, right. We found a formula. Formula. We found a way to find the sink. So if the So this is obtained by revolving a curve in the YZ plane. About the YZ. The z-axis occurred in the YZ plane. Then we have shown how to obtain parametric equation. Okay? So see the section on change of variables, right? I did that there, if you remember just before the I did the torus example. Okay? So for example, parametric equation of a torus I showed you last time. The surface. Solid surface, if I remember it was five plus two. Cosine mu mg. Sine v and u varied between 02 pi and be varied between 02 pi. Ok? So this is, this generates all the points on a torus. So these are some examples of parametric surface said this is parametric equation of a surface of revolution. Okay? So given a surface, S parametric representation has the form. So x is some function of u and v. Y is some function of VI. And actually, let me just call them x and y and z. Okay? And UV varying, they lie in E. So this is what we mean by parametric equations. You can think like this. This is what a parametric equation means. Right here is some surface, some curve. S, Right Here is some region in the uv plane. So what you do parametric equation is it's a way to flattening this curve thing. This is a very curb thing. It's hard to integrate over. What we're going to do is produce that curve thing to a flat thing in the uv plane. So we have this map, okay? To x-y-z. So a parametric equation is a way to flatten the surface. Instead of considering things on this curve thing, I'm going to be working on the uv plane. There is a, every point in the uv plane gives me a point on the surface, right? We have this map. So anything you want to do on this curved surface, you rather do it on this flat thing, especially integration. Except now you have to figure out what is the correction you make when you move things from the curve thing to the flat thing. Okay? So that is what you had figured out. We've already done this. It's a change of variables formula. It's not quite okay. It's close but not quite right? Change of variables formula does two-dimensional things to two-dimensional things. It doesn't map two to three. Write the surface is a two-dimensional bar as a two-dimensional sitting in three. A change of variables was a region in two dimensions to a region in two dimensions, or a region in three dimensions to a region, It wasn't a two-dimensional patch going to a three dimensional surface. So change of variables is not quite right here. Okay? But we are going to do a kind of change of variables. But it's not the other change of variables in some sense, if you like, this is a generalization of that change of variables. Okay? So how do we integrate on the surface by working on UV, right? How is that happening? So before I can do that, there are 23 things I want to let me get rid of all these lines here. They confuse. Yeah. Okay. And here is the region. You okay. So the first thing before we even go ahead, thing is remember when we did this for a curve, we had a parametric equation of the tangent line. Same thing here. I need to figure out the unit normal. Making the unit normal vertical. It doesn't have to be right? It's the normal. Normal means perpendicular to the tangent plane. Okay? I'm not going to draw the tangent plane, but that's a unit, Dartmouth. So first question is, how do we find the unit normal to a surface? And then the second question is, how to integrate on the surface. Okay? Now this one, I can give a reasonable answer. This one I will just tell you a formula because you will see why. Alright. So again, this is stuff you've done in 243, but I'll try to do this quickly. Okay? So UV and here is your stiffness. Ok? So you want to find the normal at this point P, which comes from this point, UV. Okay? So what is a normal? Ok. So the normal is, again, I would if I was to 43, I would ask you and probe you, but I'm going to just tell you quickly, right? What's the normal means is at this point there is a tangent planes. And the normal is the normal to that plane. But then the question becomes, what is a tangent planes? Right? Now the tangent plane is basically, you look at all the curves which lie on that surface S. And you look at the tangent lines. He has another one. Here is another one. So all the lines that are on this are the curves that lie on as the tangent lines actually lie in one plane and that is the tangent plane. That's the definition of a tangent plane. And the normal to that is called the normal to the plate. Alright? So I'm just saying it very quickly. As I said, it's from 243. You look at all the curves on the surface s. We've talked of a tangent vector to occur. We've defined that already. So look at all the tangent vectors, two curves which lie on S. They form a plane called the tangent planes. The normal to the tangent plane is called the normal to the surface, right? So I'm just saying this, I am not writing it. Okay, so how do we find the normal to the surface? Okay? The trick is this right? Points here. And you go to points one s. So if I looked at this line parallel to the u axis, it will go to some curve. It will go to some curve here. I don't. Let me raise the normal for the moment. Okay? This horizontal curve parallel to u some, goes to some curve here. Okay? So it has a tangent line. Okay? If you take this curve parallel to the v axis, it goes to some other curve. So it has a tangent lines. So now you have two lines in the tangent plane that determines the tangent plane. And how do you find a vector normal to two vectors? That will be a normal. You find the normal to two vectors by taking the cross-product, right? Okay. So how do we find these two vectors? Again, I'm showing you a derivation for calculation. You don't need it. So what we do is we define this vector r uv coming from the parametric equations. Alright? Remember there was a parametric given equation for S. We define this vector r. So let me erase this and come back to it again. Okay? So if you change only you, okay? So that means you're changing. And then you look at your tracing out this curve here. And then I ask you, what is the tangent line to it? So you're asking as you change it, literally you changes, right? What does it so the tangent line in this direction will be delta r by delta u. U is changing, which is another way that what you call Ru, sorry. Okay, so this is the derivative of the tangent vector in this direction. Okay? Now if you change your just your v, that will trace out some other curve. Let me draw it somewhere. Let's say it's this one. Okay? And then you change v, So that gives you a tangent vector in this direction. Are we? Alright? So just by changing the, you are just changing the V. I get two curves on this surface S. So I get two tangent vectors. And then the cross product will give me something perpendicular. Not that many. Ok, I mean, if a vector is perpendicular to as many other multiple, the latter also perpendicular. So that's the answer to the first question. How do you find the normal to a surface, right? I've given you a proof. Okay? Second thing is how do you integrate on the self? And so I'm going to summarize all this, including the second piece in this proposition. If S is an oriented surface, I'm not explaining oriented. Oriented means you choose the direction of the normal. Meaning. The normally should be going this way or going opposite to it. Orientation means choosing a direction. Don't worry about it for the moment. If S is an oriented surface with parametric equation, x equals x u v y u v u v in you. Then number one, how do you cross RV is normal 2S. And second demo to write something which the ds, okay, which we'll come back to later on, how you cross r v d u, d v. So what I'm saying here is this. Here is my region you hear is my surface s. So remember I said I was going to transfer integration on the surface has two integration on this region. So integration on S means you are integrating on a small patch. That's the ds. Okay? And here is it coming from some patch du dv? So you want to convert integration on that, the integration on this, this is your correction factor. This is the replacement for the Jacobians. Alright? So that's what this is telling you. It's, I'm just stating I'm not explaining where that comes from. I'm just giving you a result because I haven't really even explain what ds means. So you know, it's if you don't know what things mean, you can't prove anything about them. So, but this is just very heuristic statement that the way you convert integration on a surface to an integration of flat region is through this corrections. And of course, the second thing is that I'm giving you the direction of the normal. Ok? So this proposition is crucial for integration on surfaces, particularly this formula. Alright. So let me stop this video and then I will restart with how to integrate on surfaces.
The normal to a surface and dS
From Rakesh Rakesh September 23, 2020
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