Definition of uniformly $C^1$-regular boundary: Why last coordinate of diffeomorphism $\mathbb{e_d} \cdot \Phi(y) >0 \iff y \in \Omega $

Question

In my partial differential equations class, we introduce Sobolev spaces. We define that for an open, connected set $\Omega$ to have a uniformly $C^1-$regular boundary, it means that there are positive constants $\rho, C$ such that $\forall x \in \partial \Omega,$ there is:

a $C^1$-diffeomorphism $\Phi: \mathbb{B^{d}_\rho(x)} \rightarrow \mathbb{B}$ with

• $\sup_{y} ||\Phi'(y)||<C$,
• $\sup_{y} ||\Phi'(y)^{-1}||<C$,
• $\mathbb{e_{d}}\cdot \Phi(y) >0 \iff y \in \Omega \cap \mathbb{B^{d}_\rho(x)}.$

I have little experience with manifolds and submanifolds. I would like to understand:

1. By the second condition, do we mean that the supremum over the preimage of the Jacobian of $\Phi$ is bounded? Do I understand the expression correctly?

2. What is the importance, the implications of the fact that the last coordinate of that diffeomorphism is only positive inside $\Omega?$ Any intuition on this? How should I imagine this condition visually?

$\mathbb{B^{d}_\rho(x)}, \mathbb{B}$ are the ball around point $x$ with radius $\rho$ and the unit ball, both in dimension $d.$