Proving the Equivalence of Two Definitions for a $C^1$ Boundary of an Open Set $\Omega \subset \mathbb{R}^n$

I want to prove the equivalence of the two definitions below:

• Definition 1: An open set $\Omega \subset \mathbb{R}^d$ has a $C^1$ boundary if, for each point $x \in \partial \Omega$, there exists a neighborhood $V_x$ of $x$ and a $C^1$-diffeomorphism $\Phi_x : ]-1,1[^d \longrightarrow V_x$ such that $\Phi_x(0) = x$ and $V_x \cap \Omega = \Phi_x(\{ y \in ]-1,1[^d \ | \ y_d < 0 \})$.

• Definition 2: An open set $\Omega \subset \mathbb{R}^n$ has a $C^1$ boundary if, for each point $a \in \partial \Omega$, there exists an open neighborhood $O_a$ of $a$ and a function $f^a \in C^1(O_a, \mathbb{R})$ such that:

  1. $\overline{\Omega} \cap O_a = \{x \in O_a : f^a(x) \leq 0\}$,
  2. $\nabla f^a(x) \neq 0$ for each $x \in O_a$.

Here is an attempt:

Direction 1: Definition 1 $\Rightarrow$ Definition 2

Assume that $\Omega$ satisfies Definition 1. Let $x \in \partial \Omega$ and consider the $C^1$-diffeomorphism $\Phi_x : ]-1,1[^d \longrightarrow V_x$ such that $\Phi_x(0) = x$ and $V_x \cap \Omega = \Phi_x(\{ y \in ]-1,1[^d \ | \ y_d < 0 \})$.

Define a function $f^a : V_x \to \mathbb{R}$ by $f^a(z) = (\Phi_x^{-1}(z))_d$, where $(\cdot)_d$ denotes the $d$-th component. This function is $C^1$ because $\Phi_x$ is a $C^1$-diffeomorphism.

Let's verify that this function satisfies the conditions of Definition 2:

1. $\overline{\Omega} \cap V_x = \{ z \in V_x : f^a(z) \leq 0 \}$ because $V_x \cap \Omega = \Phi_x(\{ y \in ]-1,1[^d \ | \ y_d < 0 \})$.
2. The gradient of $f^a$ is non-zero because $\Phi_x$ is a $C^1$-diffeomorphism, which implies that the differential of $\Phi_x^{-1}$ is invertible.

Thus, $\Omega$ also satisfies Definition 2.

Direction 2: Definition 2 $\Rightarrow$ Definition 1

Assume that $\Omega$ satisfies Definition 2. Let $a \in \partial \Omega$ and consider the function $f^a \in C^1(O_a, \mathbb{R})$ such that:

• $\overline{\Omega} \cap O_a = \{ x \in O_a : f^a(x) \leq 0 \}$,
• $\nabla f^a(x) \neq 0$ for each $x \in O_a$.

By the implicit function theorem, there exists a neighborhood $V_x \subset O_a$ of $a$ and a $C^1$-diffeomorphism $\Psi : ]-1,1[^{d-1} \times ]-1,1[ \longrightarrow V_x$ such that $\Psi(0) = a$ and$$ \Psi(y', y_d) = x \quad \text{with} \quad f^a(\Psi(y', y_d)) = y_d. $$

We then define a diffeomorphism $\Phi_x : ]-1,1[^d \longrightarrow V_x$ by $\Phi_x(y) = \Psi(y', y_d)$. By construction, $\Phi_x$ is a $C^1$-diffeomorphism satisfying $V_x \cap \Omega = \Phi_x(\{ y \in ]-1,1[^d \ | \ y_d < 0 \})$.

Thus, $\Omega$ satisfies Definition 1.