# What does the notation “$\Omega \subset \mathbb{R}^n$ is $C^1$” mean?

In my calculus 2 lecture notes, we have the following definition:

A region $\Omega \subset \mathbb{R}^n$ is $C^1$ (or $C_{pw}^1$ or $C^k$ respectively), if for each point $x_0 \in \partial \Omega$ there exist coordinates $(x',x^n) \in \mathbb{R}^{n-1} \times \mathbb{R}$ around $x_0 = (0,x_0^n)$, a number $d>0$, an open cuboid $Q' \subset \mathbb{R}^{n-1}$ around $x_0' = 0$ and a function $\psi \in C^1(\overline{Q'})$ (or $\psi \in C_{pw}^1(\overline{Q'})$ or $\psi \in C^k(\overline{Q'})$ respectively), where $0 \leq \psi \leq 2d$ and $\psi(0) = d = x_0^n$ such that $$\Omega \cap (Q' \times [0,2d]) = \{(x',x^n) \in \mathbb{R}^n; x' \in Q', 0 \leq x^n < \psi(x')\} = \Omega_\psi.$$

As this notation is being used quite often later on and I have no idea at all what it tells me about the region $\Omega$, I ask for help. Can anyone simplify this definition or tell me how I have to imagine such a region? As examples for such regions, I am given:

• $B_1(0) \subset \mathbb{R}^2$ is $C^k$ for all $k \geq 0$.
• A $n$-cuboid $Q$ is $C_{pw}^1$.

I tried to just "apply" the definition to the first example to show that the unit circle is $C^k$ for all $k$ but I am not quite sure about this. The definition tells me that for all $x_0 \in \partial \Omega$, I should be able to find such coordinates, $d$, $Q'$ and $\psi$, however it also states that $d = x_0^n > 0$. Let's pick $x_0 = (0,1)$, then $d=1$ and $Q' = (-\varepsilon, \varepsilon)$ for some $\varepsilon >0$. Now I want

$$B_1(0) \cap ((-\varepsilon,\varepsilon) \times [0,2]) = \{(x',x^n) \in \mathbb{R}^2: x' \in Q', 0 \leq x^n < \psi(x')\}$$

for some $\psi \in C^k([-\varepsilon,\varepsilon])$ with $\psi(0)=1$. Can I just choose $\psi(x) = \sqrt{1-x^2}$?

If my attempt to apply the definition to the unit circle went horribly wrong, please tell me because I really have no idea what this definition actually tells me and what properties I can conclude from a region being $C^k$. In case what I did was right, it seems to me that a region is $C^k$, iff it is "locally" simple with respect to the last coordinate (if this makes sense, we only had simple regions in $\mathbb{R}^2$) where the lower border is $0$.

To sum up my questions: What does this notation mean? Is there a simpler definition, maybe a less abstract definition? How can I show that $B_1(0)$ is $C^k$ or is my attempt correct? Is there anything else I need to know about this notation?

Thanks for any help in advance.

An equivalent way of saying the same thing: If $x_0$ is any point on the boundary of $\Omega$, then you can rotate $\Omega$ in such a way that the portion of the boundary of $\Omega$ in some ball centered at the rotated $x_0$ can be written as the graph of a $C^1$ function.
So if $\Omega$ is the unit sphere and $x_0$ is on the boundary of $\Omega$, you can rotate your $x_0$ to make it $(0,0,...,1)$, and then near $(0,0,...,1)$ the boundary of $\Omega$ has equation $x_n = \sqrt{1 - x_1^2- x_2^2-...-x_{n-1}^2}$, which is a $C^1$ function.
It turns out you never actually have to rotate the domain; even in the original coordinates there's always some collection of $n-1$ variables where the portion of the graph near $x_0$ can be written as a graph as a function of those $n-1$ variables; i.e. there's always some $i$ such that near $x_0$ it's the graph of some $x_i = \phi(x_1,...,x_{i-1},x_{i+1},...,x_n)$. That's why the definition you have there is equivalent.
I'm not familiar with the $pw$ notation, but it probably just means the boundary can be written as the union of finitely many pieces such that the above condition holds except at the boundaries of the pieces.
The fact that it is stated locally is a technical trick to avoid the fact that the whole domain might not be an epigraph (like a sphere in $\mathbb{R}^3$). You can consider it as a magnifying lens: any small portion of the boundary of the domain must be described as a (sufficiently regular) hypersurface.