Let $M$ be the circular cylinder given by the equation $x^2+y^2=9$ and the curve $c(t)=(3\cos(t),3\sin(t),0)$. Compute the Gauss map of $M$
The definition I am given is $N:X \to S^2$ where $N(p)=\frac{X_u \times X_v}{||X_u \times X_v||}(p)$ but I can't seem to relate that to my question?
Best Answer
It looks to me as if your text/prof whatever is being sloppy in describing surfaces. Surfaces in differential geometry are almost always parametric surfaces, i.e., we start with a function $$ X : [a, b] \times [c, d] \to \Bbb R^3 $$ and look at the image $S = X([a, b] \times [c, d])$, which (if $X$ is nice enough), we call a "surface". Often $[a, b]$ and $[c, d]$ are both $[0, 1]$, but not always. To keep things neat, I'm going to use $U$ to denote $[a, b] \times [c, d]$, the domain of $X$>
For such a surface $S$, we say that $X$ is a "parameterization" of $S$. Associated to such a parameterization is a Gauss map, with the same domain: $$ N : U \to \Bbb R^3 : (u, v) \mapsto \frac{X_u(u,v) \times X_v(u, v)}{||X_u(u, v) \times X_v(u, v)||} $$
Some books choose instead to define $N$ as a function on $S$ itself, saying that if $p = X(u, v)$ for some particular $u,v$, then
$$N(p) = \frac{X_u(u,v) \times X_v(u, v)}{||X_u(u, v) \times X_v(u, v)||} $$
In this form, we have $$ N: S \to \Bbb S^2 $$ Notice the difference in the domains.
This isn't really quite as nice, because it's possible that $X$ is not one-to-one, but nonetheless this notation is pretty common.
In your case, to "compute the Gauss map", you'll want to do the second thing, and you can do it without even writing a parameterization of the cylinder: you want to associate to each point $(x, y, z)$ of the cylinder a unit normal vector. For instance at the point $(3, 0, 8)$, it's pretty easy to see that the normal points in the $x$ direction, so you can write down that $$ N(3, 0, 8) = (1, 0, 0). $$
You could also have written $(-1, 0, 0)$ -- without an explicit parameterization, it's impossible to decide which was meant! But you should at least make a choice that's continuous as you vary $x,y,z$.