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Our teacher is claiming that (in $\mathbb{R}^3$) the following surfaces are "cylinders":

  • $3x+y+\frac{7}{2}=0$
  • $y=x^2$
  • $z^2 = y$
  • $\frac{x^2}{4} + \frac{y^2}{4} = 1$

Is there any definition of cylinder that can justify this statement, and if so, which definition?

The last one is the only one that fits the definition that I know (which he calls an "elliptic cylinder")

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Only the last might possibly qualify as a cylinder (in $\mathbb{R}^3$, with the $z$ coordinate unbounded, so more an infinite pipe than a cylinder) in any usage I am aware of. –  copper.hat Oct 23 '13 at 18:54
    
Take a look at math.stackexchange.com/questions/28602/… –  Arthur Oct 23 '13 at 19:00

1 Answer 1

up vote 1 down vote accepted

When most people heat the word "cylinder" they are thinking of a "right circular cylinder" -- that is -- a circle which is copied up and down along a line perpendicular to the plane that circle lies in.

The term "cylinder" can more generally refer to a curve copied along any line. So for example, $y=x^2$ is a parabola in the plane, but since there is no reference to $z$, the parabola $y=x^2$ and $z=5$ also satisfies this equation (same for $z=$any constant). So in 3-space, the graph of $y=x^2$ is a parabola copied up and down the $z$-axis. Instead of a circular (regular old) cylinder, you have a "parabolic cylinder".

If you notice, all of the equations you have listed above only list 2 out of the 3 variables (x,y,z). So the graphs of these surfaces are graphs of a curve copied along the remain variable's axis.

For example: Here is (part of) the graph of $z^2=y$ (another parabolic cylinder)...

enter image description here

Notice that the parabola $z^2=y$ (in the $yz$-plane) has been copied along the $x$-axis.

Wikipedia Page on Cylinders -- see "other types of cylinders"

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