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(a) What does the equation $ y = x^2 $ represent as a curve in $ \mathbb{R}^2 $?(b) What does it represent as a surface $ \mathbb{R}^3 $?(c) What does the equation $ z = y^2 $ represent?

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a) parabolab) parabolic cylinderc) parabolic cylinder

01:09

Wen Zheng

01:43

Anthony Ramos

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 6

Cylinders and Quadric Surfaces

Vectors

Johns Hopkins University

Campbell University

Oregon State University

Harvey Mudd College

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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(a) What does the equation…

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Describe the surface whose…

party. We're gonna talk about. What does the equation why? Well Sex square represent a secure in our two. The plane in purple. What does it represent? The surface in our three. And pursue with that. We discuss what does the equations the equals Y square represent. So uh Y equals was X square. We all know that this is a parabola into the dimensions. That is if we join points X with this square we get this type of curve which Increased without any bound to the right, into the left. At zero equals 0 At one. It is one At at negative, one is 1 At two is 4 like that. So is a parable. What happens if we consider this same equation in the space? In three dimensions? Well the equation is the same. It means that the 3rd variable Z appears with his cereal coefficients. So C can be any value. And the equations who feel it means that drawing three dimensions we have here X axis, Y axis and Z axis. Then in the X Y axis. Let's say we have the negative part of X here and the negative part of why I hear negative policy down here. And on the Y X. Y axis we have the same curve. Let's say this parable here drawn on the plane. Xy But then a C. Is the same. Is any values or is any value then we has that that problem can be drawn at any value of C. And that will form the surface. That is the surfaces what we call a cylinder. So we will have this. That is if we look from certain inclination here, we can see this that's a figure with infinity here and here, that is. This surface doesn't came on above and below. He goes upward all the time and downward and at any hide. That is if we put a plane product to xy at anybody of C, we get the same problem. So as we that's what we call parabolic seen cylinder. That is. We have the shape of the parable on the plain X. Y. And that is extruded up and down to form a surface. Which then is this shape. Here, if we look different angles, it is clear that the cylinder in this case goes up. That is if we make some traces, we see that we see. The problem is replicated at any height. So that's the surface that is formed and when we talk about Z equals y square, it happens the same. That is C equals y. Square will be a parabola on are too. But in this case it is drawn in plain to see why. That is if this is a plan C. Y. Then the problem is this just the same as this. So that the only difference that here we have the independent variable is the original eggs and the very core of accesses the dependent variable. Y. So here the independent variable is why and the dependent variable dizzy, brought the shape is exactly the same. And we have a sexually the same type of curve. There is a parabola. But in the space now if you look X. Y and Z. And we have the parable at see why there is. Now this is let's see and we drove to rob a lot at the plane, see what there is something like this. And now we got to put this problem at any value of X. Because now X is a variable doesn't appear so X can be any value and we'll have this. Mhm. This shape here that is we put this problem everywhere along the X. Axis and we get this cylinder again it's a parabolic cylinder. But the shape goes along the X. Axis. Yeah that is. We have this without this. Sorry. It is shape here. Yeah. And in this case goes along to see access.

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