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.