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Felicia S.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

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Nutan C.

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Simon E.

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All right, so we're ready to talk about surface and girls and the idea behind the surface, integral is very similar to the idea behind the line. Integral. So recall that when we had the line integral, we had a curve in space or in the plane. And what we did is we broke this line into little pieces and each one of these pieces we called a line element called a DS. And then what we did is we took some function whether that was a vector field or a scaler field, and we added up the function values along these little pieces of the curve and then add them all up. And we called that a line in girl. But remember that the most important part of a line integral or at least the initial part, was that we needed to give a parameter ization of the curve. And so that's gonna be the same thing we're doing with the surface in space. So suppose that I have some surface in three dimensional space, and what I'm gonna do is I'm going to break it up into small pieces like this, and I can think about either a vector field or a scaler field defined over this region. This little surface and what I'm gonna do is I'm gonna break it up into these small pieces D sigma that I'm going to call surface elements. And then I'm going to add up the values of the functions on all of these service elements, and that's going to be a double integral. It's going to turn out, but what we need to do, just like with a line at a girl we needed to parameter rise the curve. We need to parameter rise the surface before we can do a surface integral. So how do we do that? Well, a parameter ization of a surface we'll call the surface Capital s looks like this. Well, it's just like a premature ization of a curve. It's a vector valued function. But this time we're gonna have two parameters because inherently a surface is a two dimensional object, whereas a curb is a one dimensional object and so we'll have still three component functions, and each one of the component functions will depend on both of the parameters S and T. And again, these component functions will assume that there have continuous partial derivatives Notice that we really need to specify partial derivatives because all the component functions are functions of more than one variable of two variables specifically, and so s and t thes coordinate pairs will live in some region inside the S T plane. And so this will be a region that will act is our area region when we actually do thean migration over the service. But this is a typical parameter ization of a curve what it would look like. And you know what we can think about is, you know, if this was a rectangle in the S T plane, what we're doing is basically taking a small little square and stretching it out as a surface in space like this. So this is what our premature ization is doing is taking a small little square and stretching it out into space. And so, fundamentally, we're going to be doing double integral over this region. But somehow we're going to be composing with our parameter ization. So we want to get review just a couple important backs about the parameter ization of a surface. So let's draw a picture we'll have here some region that will call our and this is going to be in the S T plane and I'm drawing It is a rectangle, but of course, maybe it's not actually a rectangle. And what my parameter parameter ization is going to do is it's going to take this region, are and it's going to turn it into a surface like we saw before in space. And so there's my surface in space taken draw my trace lines here so I can think about each one of these lines, you know, here for the t parameter. If t is constantly equal toe one, this is being represented over here. Same thing for s. So these lines kind of represent curves that have fixed s values and fixed T values. Okay, so probably the most important property of the prime tries curve is that if I look at the partial derivatives of are as a function of s and T, So if I look at our success and I look at our Sebti, okay, so notice that the component functions of my premature ization are functions of more than one variable. So when I go to take the derivative, I really need to specify I'm taking the partial with respect to s of all the component functions or the partial with respect to t But these vectors R S A T R C s and our city I can actually draw what they are. So recall that when we take the partial derivative with respect s, we're fixing a T. And we're looking at the change in the S direction. And so if I go up here in my surface, our sub s is going to point tangent to that trace and then the same thing for our city. Looking at the partial with respect to t. I'm just fixing an S and looking at the T direction. And so my partial derivative is tangent to the curve in that direction. But what that means is these vectors, assuming that they're not both zero, lie in the plane that's tangent to the surface. And that's really cool, because this is actually going to give us away to figure out the tangent plane at a point on the surface. And moreover, if I look at this cross product now, of course, this cross product is going to be vector valued function itself. But each point this cross product of the two partial derivatives is going to be a normal vector. So this vector in which is the cross product of those two planes are sorry of those two vectors is going to be normal to the plane. So if I wanted to normal vector, I could just take the cross product of the two partial derivatives of my premature ization. And that's super cool. But we can actually go a step further. So let's zoom in on my partial derivative vectors here. So here's our tea Partial with respect to t of my premature ization. Here's the partial with respect at S so recalled that the cross product or the magnitude of the cross product so magnitude of R S Cross R T is actually the area of this parallelogram spanned by Rs and rt. So this little area is equal to that. But that's super cool, because if I think about this little parallelogram now, this parallelogram is really going to be in the tangent plane. But if I consider a really small parallelogram, actually this parallelogram is going to approximate a little piece of the surface really well and so I can actually just define what I mean by a surface element d sigma. So what I'm gonna mean is I'm gonna mean Well, I'm gonna take the area of the parallelogram along the in the tangent plane and then I'm just gonna multiply it by a small change in s and a small change in t So I'm going to call that d a And here D a. I really mean ds times DT So a small change It s times a small change in t And now what that's going to give me is it's going to give me a parallelogram, the area of a parallelogram lying in the tangent plane. It's just a really, really small piece of the tangent plane, so it's a really good approximation to a small piece of the surface. So it really is just a small surface element in what we see right off the bat is we have a really nice way now if S and T live in some regions are that's acting is the domain of the premature ization of the surface. If I just integrate overall s and T this, uh, surface element here, what I'm going to get is the surface area of the region or of the surface which we're going to call s. So the surface area of a parameter rise surface, which we're calling s is just okay. What's symbolically I'm just adding up over s all of the small surface elements of s. But what that really is is it say double in a girl over this region in the plane. So just a typical double integral over this region. Now we're calling it S and T instead of x and y of the magnitude of the cross product and then with respect to this little area element here, which is just d S N d t. So I just need to describe the domain region of my parameter ization and I can compute the surface area of a region. We're sorry of a surface described by the region. So just like when we talked about art length of a curve that allowed us to pretty much immediately define what we meant by the line and a girl of a scaler value function, we could just go ahead. And now that we have a way to talk about surface area of a parameter eyes surface, we can just go ahead and talk about a surface integral of a scaler valued function. So the surface integral of f Now this will be a scaler valued function over a surface which we'll call s. But we'll think about s is being parameter rised by the specter value function with S and T living in some region in the SD plane. So this is sort of like our interval for our parameter rised curve. Now we have It's a little bit more complicated to just describe the domain of a parameter rise surface. It's not just a interval like it is for a parameter ice curve. Yes. So what is the surface integral? Well, it's just a noted as a double integral because it really is a generalization of a double. Integral. You're doing an area integral, but over not just a plane over, actually a surface that's kind of curved in space. So over that surface s of F. And then this notation really tells you what you're doing. You're just adding up the function values over those small surface elements. And since we have the premature ization, we can write it as a double integral that we actually know how to evaluate. So it's just going to be f of now I'm going to plug in. Of course, the parameter ization. So again, that means for X, y and Z and my function. I'm gonna plug in X of S t y s t z of S t. And then just multiply by d sigma which, since we have parameter ization, is just r C s cross our city that magnitude and then d a which would be de s DT. Okay, so it's again. It's just a nice generalization of both a line integral. And then sort of also the double integral just merging those things together. And so, you know, one thing that you can do right off the bat is talk about the mass of a region described by parameter eyes service. So if this is a density function, then what you're doing is okay. Instead of just Matt adding up all of the little pieces of surface to get the surface area. If each little piece of surface has a mass per unit area, we can actually find the mass just like we did for parameter rised curves. Okay, so recall that we had two types of line into girls. We had the line integral over a scaler value function, which we've already generalized and talked about the surface integral of a scaler value function. But we would also like to talk about the second type and talk about that with respect to surface intervals, and that's the surface integral of a vector field. But before we do that, we need to talk about something that's a little subtle, and we're not gonna get into a lot of detail. But we did at least just want to introduce the idea. So in order to do a surface integral over a vector field the surface integral of a vector field, we need to be dealing with what's called oriented surfaces and loosely what this means are surfaces with two sides. Okay, And this seems a little silly because, well, maybe you're thinking all surfaces have two sides. Or maybe you're thinking, What does it mean for a surface to have one side or three sides? And so certainly most surfaces that you deal with on a regular basis have two sides. If you take a basketball the surface of a basketball, it has an inside and an outside. If I take a piece of paper, it has a front and back. Okay, so maybe you're starting to believe me that most surfaces do you have two sides? But what about an example of the surface that does not have two sides? Well, there are examples of non oriented surfaces, and there's a famous example, and we're not going to say much about this now, but the example is called a Mobius strip. So the moment you strip is not oriented, since it only has one side. And this is really interesting. Especially if you haven't seen this before. So how do you make a movie strip? Well, what you dio is, you take a rectangular strip of paper and you turn it once and connect the edges. Okay, so let me attempt to draw a picture. So imagine that this is your piece of paper like this. And, uh, what I'm going to do kind of signify that there's a turn in it like this. Okay, so here's what the Mobius Strip would look like if you're looking at it. But let me show you how it came about. So I took a rectangular piece of paper, and I twisted it once. So turned it into Yes, so I'm sort of drawing this out of order, but we take the strip of paper and we turn it. So this is step one, and then we connect the two edges. So this is like at one, this is edge to we just gloom together like that. Okay, so you can see. Just imagine, a little ant was walking around. It would come around here, it would be on the back side, but then it would sneak around here and be on the front side and then follow around. And now I would be on the back side over here having to be behind the strip where you couldn't see the ant, and then I would come around and I'd be on the front over here, and then I'd go through the loop and I would be on the back. And if you're having trouble visualizing this, you know, I encourage you toe, just get a piece of paper and do it. But, I mean, we're not like I said, this is all I'm going to say about oriented services. Just say that First of all, way Need to deal with oriented surfaces to do surface in your girl's a vector fields And we want to rule out things like this Mobius strip that is not oriented because it only has one side. Okay, so now let's say that we have a vector field and this will be a vector field in space. So you have X y Z, And what else do we have? We have Well, just like we had a oriented curve. We need an oriented surface. We talked about what that means, and so we'll call the service s. But we want Thio assume that we have a premature ization. So the premature ization is going to look like are of S t and S and t live in some region in the S t plan. So maybe it's just a square. Or maybe it's a little bit more complicated. That's the idea. So a vector field oriented surface. Here we go, the line integral. Sorry, not the line integral. The surface integral of the vector field over S is okay. So it's gonna be a double integral because it's just naturally a two dimensional object. Then it really is generalizing the idea of the double integral. Okay, so we'll have our vector field. And just like with the line integral, we want to take the component of the vector field that lies in a certain direction. And so let me draw a sample oriented surface like this. We'll put in some surface elements just so you can see and so remember that this surfaces oriented. It has a front and a back so you can think about this, has a top in the bottom. And so we're going to have to pick an orientation, just like with the curve. With the line integral, we have to pick a direction that the particles moving. So here I'm going to pick well, a normal direction. And we'll explain what this means physically in a second. But what I'm gonna do is that each one of thes surface elements I'm gonna find a normal direction, and I'm going to project F onto a unit vector in the normal direction. And then I'm gonna add up all of those contributions over all of the various surface elements. Okay, so this is exactly analogous to the flux integral in the plant, because remember, a flux integral we were measuring the amount of the field that was passing out of the curve. Well, what we're doing here is where actually adding up the amount of the field that's passing out of the surface. So this really is a flux. Integral. But here, by flux mean a two dimensional flux of flux passing out of a surface of the amount of water passing through, you know, particular area or particular surface or an electric field passing, you know, over the surface of the cylinder. Something like that. How much of the field is passing perpendicular across the surface? So that's the idea. And again, this is just notation. But since the surfaces parameter rised, we can actually be a little bit more specific about what we mean. What we're really doing is we're integrating over a region in the S T plane that we call our Yeah. Now something similar is gonna happen when we plug in the normal direction. So the normal direction What is that? What we can do is we can take this vector that we know is in the normal direction divide by its length, and there we go. So there's a normal vector but notice that D Sigma actually has a factor of the magnitude of this cross product as well. So similarly tow line into girls the complicated magnitude factor is going to cancel. So what's gonna be left is we're going to be looking at the vector field. Of course we want to evaluate at the parameter ization, so plug in X s t y s d c f s t etcetera And then we dot with, well, the normal vector And now not the unit Normal vector, just the vector that we know is normal. So the cross product of the two partial derivatives of our parameters ation And then we just put in d A and this just becomes the in a grand is just a scaler value function of S and T, and we're doing a double integral, so it really is just a double integral. But we're just accounting for the fact that we're living on a surface in space, just like a line Integral was just a single integral. But we're accounting for the fact that we're living on a curve and space or in the plane

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