Cylinder or cylindrical surface?

Question

My professor says they are different from one another.But I don't understand it and it is not explained in the book. So can someone explain to me the difference between a cylinder and a cylindrical surface? And the difference of their equations? I would really appreciate it.Thank you!

Answer 1

A cylinder is a solid block - a 3d body - $D\times \mathbb R$ or $D\times I$ where $D$ is a disk and $I$ is a segment.

A cylindrical surface is a (side) surface of a cylinder - a 2d sheet - $C\times \mathbb R$ or or $D \times I$ where $C=\partial D$ (boundary of $D$) is a circle.

More generally, replace disk/circle with a planar domain/its boundary.

Answer 2

According to James Stewart's Calculus book (Chapter 12), a $\textbf{cylinder}$ is a surface that consists of all lines (or rulings) that are parallel to a given line and pass through a given plane curve.

$\textbf{Example 1}$. Consider the graph of the surface $y=x^2$ in the $3$-dimensional (real) space. Restricted to the $xy$-plane, $y=x^2$ looks like a parabola. But since the variable $z$ doesn't appear in the equation $y=x^2$, you should imagine picking up this parabola and sweeping it in the positive and negative $z$-direction to obtain what I call a $\textit{generalized cylinder}$ or a $\textit{cylindrical surface}$. But the book refers this surface as a cylinder as well.

$\textbf{Example 2}$. Consider the graph of the surface $y^2+z^2=1$. The cross-section along the $yz$-plane is a circle of radius $1$ centered at the origin. Since $x$ is a free variable (it doesn't appear in the equation $y^2+z^2=1$), you should imagine sliding this circle in the positive and negative $x$-direction to obtain an honest (classical) $\textit{cylinder}$ of radius $1$ centered at the origin of infinite length.

Earlier in this book in Chapter 6, Stewart defines a $\textit{cylinder}$ in a different way. Let $B_1$ and $B_2$ be two planar (bounded) regions that are parallel to each other. The $\textbf{cylinder}$ is the set of all points on line segments connecting $B_1$ and $B_2$.

After thinking about the several definitions you are seeing on this post, I suggest that you have a discussion with your professor on which definition you are using.