The opening exposition of an older differential geometry text made the following statement about the paramaterization of a surface, using fairly archaic language to describe the implicit, explicit, and parameteric forms of the surface.
Although a surface may be represented by $f(x_1,x_2,x_3)=0$, or by the Monge form $x_3=\phi(x_1,x_2)$, it is more convenient for most purposes to describe the surface by the equations in the Gauss form
$$x_i=x_i(u_1,u_2),~~~(i=1,2,3)$$
where the parameters $u_1, u_2$ are linearly independent.
My question is this: assuming a surface $S$ is arbitrarily well-behaved, for what classes of surfaces are each of these three forms guaranteed to exist? More specifically:
Is it always possible for a (well-behaved) surface embedded in $\mathbb{R}^3$ to be written as the level set of some function of three variables? For all closed surfaces?
For some surfaces, a coordinate system can be found in which the surface can be described explicitly in the so-called Monge form. For instance, many closed "star-shaped" surfaces can be written explicitly in spherical coordinates, and presumably other coordinate transformations could do the same for other surfaces. For what surfaces can a coordinate system be found in which they can be written explicitly in form (b)?
Can a two-variable parameterization always be found for any surface? For all closed surfaces?
I just finished a Calc III/IV course, and have no formal education in differential geometry (please don't feel obliged to limit the sophistication of your answers though). These seem like obvious and perhaps significant questions, but I haven't found an answer to them. Is there some underlying principle here that I'm missing?
First of all, one needs to define what a "parameterization" of a surface $S$ means. As I said in the comments, the modern way to do so is to define an atlas of local parameterizations (charts) which are diffeomorphisms $h_i$ of open subsets $U_i$ of $R^2$ to open subsets $V_i$ of $S$: $$ \{(U_i, h_i): h_i: U_i\subset R^2\to V_i\subset S\}, $$ so that the union of $V_i$'s is the entire $S$. Such an atlas is what one always works with. One can also ask for a parameterization via a single chart. This is not always possible. For instance, if $S$ is a sphere or any compact surface for this matter. One can settle for something else instead and require a parameterization to be a single map $h: U\subset R^2 \to S$ which is surjective and is a local diffeomorphism, i.e. for every $x\in U$ there exists a neighborhood $U_x$ of $x$ in $U$ such that the restriction of $h$ to $U_x$ is a diffeomorphism to its image in $S$. In one dimension lower, as an example of such, think of the map $$ h(t)= (\sin(t), \cos(t)), t\in {\mathbb R}. $$ While such parameterizations are not as useful for surfaces, nevertheless, they exist for every surface. One can prove this via elementary methods but it is quite tedious. The most interesting and important result in this direction is the Uniformization Theorem.To state this, one should allow the domain of the map $h$ to be $S^2$ and not just an open subset of $R^2$. Then the theorem says the following (I am restricting to the case of surfaces in $R^3$, but the result is more general):
Given any smooth connected surface $S\subset R^3$, there exists a conformal parameterization $h: U\to S$, where $U$ is either $S^2$ or $R^2$ or the open unit disk in $R^2$. Here conformality of $h$ means that it is a local diffeomorphism which preserves angles between tangent vectors: Given any point $x\in U$ and two vectors $u, v$ tangent to $U$ at $x$, we have $$ \angle(dh_x(u), dh_x(v)) = \angle(u,v). $$ Moreover, the mapping $h$ in this theorem is a universal cover of $S$.