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  • Identify a cylinder as a type of three-dimensional surface.
  • Recognize the main features of ellipsoids, paraboloids, and hyperboloids.
  • Use traces to draw the intersections of quadric surfaces with the coordinate planes.

We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces , to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.

Identifying cylinders

The first surface we’ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word cylinder , here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don’t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.

In the two-dimensional coordinate plane, the equation x 2 + y 2 = 9 describes a circle centered at the origin with radius 3 . In three-dimensional space, this same equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the z -axis ( [link] ), forming a hollow tube. We can then construct a cylinder from the set of lines parallel to the z -axis passing through circle x 2 + y 2 = 9 in the xy -plane, as shown in the figure. In this way, any curve in one of the coordinate planes can be extended to become a surface.

This figure a 3-dimensional coordinate system. It has a right circular center with the z-axis through the center. The cylinder also has points labeled on the x and y axis at (3, 0, 0) and (0, 3, 0).
In three-dimensional space, the graph of equation x 2 + y 2 = 9 is a cylinder with radius 3 centered on the z -axis. It continues indefinitely in the positive and negative directions.

Definition

A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or cylinder    . The parallel lines are called rulings    .

From this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line ( [link] ).

This figure has a 3-dimensional surface that begins on the y-axis and curves upward. There is also the x and z axes labeled.
In three-dimensional space, the graph of equation z = x 3 is a cylinder, or a cylindrical surface with rulings parallel to the y -axis.

Graphing cylindrical surfaces

Sketch the graphs of the following cylindrical surfaces.

  1. x 2 + z 2 = 25
  2. z = 2 x 2 y
  3. y = sin x
  1. The variable y can take on any value without limit. Therefore, the lines ruling this surface are parallel to the y -axis. The intersection of this surface with the xz -plane forms a circle centered at the origin with radius 5 (see the following figure).
    This figure is the 3-dimensional coordinate system. It has a right circular cylinder on its side with the y-axis in the center. The cylinder intersects the x-axis at (5, 0, 0). It also has two points of intersection labeled on the z-axis at (0, 0, 5) and (0, 0, -5).
    The graph of equation x 2 + z 2 = 25 is a cylinder with radius 5 centered on the y -axis.
  2. In this case, the equation contains all three variables x , y , and z so none of the variables can vary arbitrarily. The easiest way to visualize this surface is to use a computer graphing utility (see the following figure).
    This figure has a surface in the first octant. The cross section of the solid is a parabola.
  3. In this equation, the variable z can take on any value without limit. Therefore, the lines composing this surface are parallel to the z -axis. The intersection of this surface with the yz -plane outlines curve y = sin x (see the following figure).
    This figure is a three dimensional surface. A cross section of the surface parallel to the x y plane would be the sine curve.
    The graph of equation y = sin x is formed by a set of lines parallel to the z -axis passing through curve y = sin x in the xy -plane.
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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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Do somebody tell me a best nano engineering book for beginners?
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there is no specific books for beginners but there is book called principle of nanotechnology
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
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CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Do you know which machine is used to that process?
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for screen printed electrodes ?
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of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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