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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?

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    Since the Gauss map of $M$ has nothing to do with a curve, I think that perhaps something is missing from this question. Perhaps "Compute the Gauss map of $M$ along the image of $c$" or "Compute the composition $N \circ c$ of the Gauss map of $M$ with the curve $c$"? – 2017-01-11
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    @JohnHughes There is multiple questionss after computing the Gauss map where the curve may come into it. I am just not understanding how M relates to the equation N(p). How would you apply it to this example in particular? – 2017-01-11

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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 $x,y,z$.

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    I think I understand the theory behind it. I can also see how the point you have chosen has the normal point in the x direction. However, as you have said you need to associate each point (x,y,z) a unit normal vector, I am assuming I end up with 3 vectors? How would this be done? Especially as from what I understood in your theory, a Gauss map is literally just a way of mapping? – 2017-01-11
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    A cylinder has infinitely many points; at each point $P$, you find a normal vector $n_P$. For $P = (3,0, 8)$, we have $n_P = (1, 0, 0)$. The association $P \mapsto n_P$ defines a map from the surface to the unit sphere; that map is called the Gauss map. – 2017-01-11