Parametrisation of a surface. I'm currently going through my course notes, currently teaching the theory leading up to surface integrals. In particular, I am reading the section on the parametrisation of surfaces.
However, there are some elements that I am not completely understanding. I guess I could try to memorise formulas such as the Jacobian of a parametrisation but I really do dislike moving on without a complete understanding of what's going on! I would really appreciate it if some of the conceptual roadblocks were cleared.
I will type out what is provided in the course notes below. Please let me know if there was something I forgot to define when re-typing this chunk of theory. 

We only want to work with smooth (or at least piecewise smooth) surfaces. To define exactly what we mean by a smooth surface, let us assume that $G:D\rightarrow \mathbb{R}^3$ is the parametrisation of (part of a surface. Suppose that $A=(a_1,a_2) \in D$, where $D \subset\mathbb{R}^2$ in space, so that $G(A)$ is the corresponding point on $S$. 
Then the map $x_1 \rightarrow G(x_1,a_2)$ is the parametrisation of a curve on $S$ through $G(A)$. Similarly, the map $x_2 \rightarrow G(a_1,x_2)$ is the parametrisation of another curve on $S$ through $G(A)$. In order for $S$ to be smooth, we assume that these curves are smooth. More precisely, this means that the above are regular parametrisations, so in particular, we require that $v_i=\dfrac{\partial}{\partial x_i} G(A) \neq 0$ for $i=1,2$.
For $S$ to be smooth, we require more. If we are very unlucky, the image of $D$ is only a curve, and not a surface. In that case, the vectors $v_1$ and $v_2$ are parallel.

Below are my questions, and what I've tried to make of it so far.


*

*Regarding the map $x_1 \rightarrow G(x_1,a_2)$. My understanding is that $x_1$ is something that moves freely around whilst $a_2$ is fixed, hence forming a curve. If both components were able to move freely, then it would form a surface. Is my understanding correct?

*What do they mean by the term 'regular parametrisations'? It appears that the condition is that the partial $\dfrac{\partial}{\partial x_i} G(A)$ is non-zero, but I do not understand conceptually what a 'regular parametrisation' is. 

*Why is it 'unlucky' if the image of $D$ is only a curve, not a surface? Surely if the image of $D$ has one less dimension, then it simplifies the problem? Or at least, this is what my intuition says.

*Geometrically, what do the vectors $v_1$ and $v_2$ mean? I understand that it is the partials of the parametrisation, but I'm having trouble understanding what this means 'in the bigger picture'.

My sincere apologies if this is a lot! I've been understanding the theory quite smoothly so far, but this is my first huge roadblock in understanding! Also, I have not formally studied any differential geometry, though I guess this is a bit of an intro to it.
 A: *

*You have to look at G as a function of 2 variables, no $a_2$ fixed. As you figured out earlier, this would result in a (special) curve (a coordinate line).

*You need to totally differentiate G, not only consider partial derivatives. That is, you have to look at the linear map $DG: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ If that map has maximal rank ($2$ in your case) in a neighbourhood of a given point $A$, the map is a called regular in $A$.

*If you want to consider surfaces, the objects you are looking at should be something $2$-dimensional. If you are talking about (abstract or embedded) manifolds in general (and surfaces are $2$-manifolds) then you don't want to have singular points or other kinds of degenerations, since several concepts work out nicely only if the object is 'nice' in that sense.

*$v_1, v_2$ represent the tangent vectors in that point pointing into the direction of the coordinate lines.

A: When you see a computer-drawn picture of a surface $S = G(D)$, there's often a "curvilinear grid" on the surface. These curves are precisely the "coordinate curves" obtained by holding one variable in the parametrization $G$ constant and varying the other. Two coordinate curves pass through each point $G(a_{1}, a_{2})$ of $S$, each obtained by holding one parameter constant:
$$
c_{1}(t) = G(t, a_{2})\quad\text{and}\quad c_{2}(t) = G(a_{1}, t).
$$
Intuitively, $S$ is "surface-like" at $G(A)$ if these curves form a smooth net, i.e., each curve $c_{i}$ has a tangent line at $G(A)$, and the two tangent lines at $G(A)$ are non-parallel (so are contained in a unique plane).
To address your questions:


*

*Exactly right; the coordinate curves of $G$ are the restrictions of $G$ to horizontal or vertical lines in the domain $D$.

*Every calculus student learns that the graph of a differentiable function has a tangent line at each point. It therefore comes as a small shock to discover that the image of a smooth (continuously-differentiable) curve need not have a tangent line at each point; "real" examples include the astroid and the cycloid,
$$
c(t) = (\cos^{3} t, \sin^{3} t),\qquad
c(t) = (t - \sin t, 1 - \cos t),
$$
each of which has infinitely-differentiable component functions, yet traces a curve with cusps. The potential "problem" occurs if the velocity $c'$ vanishes at a point; intuitively, the image curve can "change direction suddenly" at $c(t_{0})$. By definition, a "regular curve" has non-vanishing velocity. The image of a regular curve has a tangent line at each point; the velocity $c'(t_{0})$ is a direction vector for the tangent line.

*The surface $S$ is regarded as given, and we're trying to pick a parametrization $G$ of $S$. If the image of $G$ is a curve, we've missed most of $S$ (which is "unlucky").

*If the coordinate curves of $G$ at $A$ have proportional velocities, the parametrization $G$ is still "mildly defective", in that $G$ does not specify the tangent plane to $S$ at $G(A)$. (See the dark line in the surface below, where the surface has a tangent plane but the parametrization is not regular.) By definition, $G$ is regular if at each point of $D$, its "partial velocities" $v_{1}$ and $v_{2}$ are linearly independent (not proportional, hence non-zero), i.e., the $v_{i}$ span the tangent plane of $S$ at $G(A)$.

*Asking that the gradient (of a smooth function $f$ of three variables) be non-vanishing on a level set $S = \{f = c\}$ allows one to invoke the implicit function theorem, which guarantees that $S$ is a "smooth surface", i.e., for each point $p$ of $S$, there exists a neighborhood $U$ of $p$ and a regular parametrization $G$ whose image is $U \cap S$, the portion of $S$ inside $U$.

