Preliminaries/Notation: Let $U \subset \mathbb{R}^2$ be open, and let $(f^1, f^2, f^3)=f: U \to \mathbb{R}^3$ be a regular parametrization/surface (i.e. immersion). Let $p \in U$ be given. Then there exists a neighborhood $V \subset U$ of $p$ such that $f|_V$ is an embedding and thus $f(V)$ is diffeomorphic to $V$. Let $(x_1, x_2)$ be the coordinate system for $U$. Finally, let $(N^1, N^2, N^3)=N: U \to \mathbb{S}^2 \subset \mathbb{R}^3$ be the unit normal vector field for $f$, $N(p)=\frac{f_{x_1} \times f_{x_2}}{|| f_{x_1} \times f_{x_2}||}$.
Definition of Weingarten Operator: The definition of Weingarten operator I have is as follows: for any $(p, (v_1, v_2)) = v \in T_p V$, the Weingarten operator is the function $A: T_p V \to T_p V$ such that for $(p, (w_1,w_2))=w:=AV \in T_p V$, one has $$Df(p)(w) = \nabla_v N(p)$$ where $Df(p)(w)=(f(p), ( f^1_{x_1}w_1 + f^1_{x_2}w_2, f^2_{x_1}w_1 + f^2_{x_2}w_2, f^3_{x_1}w_1 + f^3_{x_2}w_2) )$ and $\nabla_v N(p) = (f(p),(N^1_{x_1}v_1 + N^1_{x_2}v_2, N^2_{x_1}v_1 + N^2_{x_2}v_2, N^3_{x_1}v_1 + N^3_{x_2}v_2)) $. I am aware that this definition differs by a minus sign from the typical one, and also that $\nabla_v N(p) \perp N(p)$ is what guarantees us the necessary but not sufficient condition for $A$ to be well-defined that $\nabla_v N(p) \in T_{f(p)}f(V)$.
Obviously what we really need for $A$ to be well-defined is that $w$ exists and is unique. The above definition leads to following system of linear equations: $$\begin{bmatrix} N^1_{x_1} & N^1_{x_2} \\ N^2_{x_1} & N^2_{x_2} \\ N^3_{x_1} & N^3_{x_2} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} f^1_{x_1} & f^1_{x_2} \\ f^2_{x_1} & f^2_{x_2} \\ f^3_{x_1} & f^3_{x_2} \end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} $$ Therefore the question of the well-definedness of $A$ reduces to multivariable calculus.
My question: How can we use the implicit function theorem to show that a solution for this system of equations always exists, and that it must be unique?
That $f$ is an immersion and therefore $Df(p)$ has rank $2$ must surely be relevant somehow.
a. How does the implicit function theorem allow us to conclude that there exists a change of coordinates such that $DN(p)$ can be written as a $2 \times 2$ matrix? (Since $\mathbb{S}^2$ is a 2-manifold.)
b. Does the exact same reasoning as for 1. allow us to write $Df(p)$ as a $2 \times 2$ matrix? (Since $f$ also maps $U$ to a 2-manifold?)
c. Since $f$ and $N$ are different (albeit related) functions, how can we ensure that there is a coordinate system which satisfies both 1. and 2.? (I think the answer might have something to do with "normal coordinates", but I don't understand Riemannian geometry, so I can't tell.)
Note: A very similar question has been asked here before, but the answer given seems incorrect. Matrix representation of shape operator. The reason why is that, at least with the definition of unit normal vector field which I have, $DN$ should be a $3 \times 2$ matrix, because it maps a subset of $\mathbb{R}^2$ into the unit sphere $\mathbb{S}^2$, which is a subset of $\mathbb{R}^3$.
Also, even if the given answer is correct, I am still unsatisfied with it, because it must be using somewhere the assumption that the given parametrization of the surface is an immersion, i.e. a regular parametrization/regular surface, but I don't see where.
I am aware that the proof of the Weingarten equations most likely answers my question; however, I want to use the answer to my question to prove the Weingarten equations. Specifically, I want to be able to use the implicit function theorem to get from questions about $3 \times 2$ matrices to questions about invertible $2 \times 2$ matrices, rather than hide the implicit function theorem "under the rug".
This is a follow-up to three of my previous questions:
What is the dimension of a tangent space?
Motivation behind notation for tangent space
Why is the image of the implicit function in the implicit function theorem not open?
