Definition of $C^k$ boundaries I am reading the book "Partial Differential Equations" of Lawrance c. Evans by myself and started with Appendix part.
At the very beginning of Appendix C, there exists a definition
"We say $\partial U$ is $C^k$ if for each point $x^0\in\partial U$, there exists $r>0$ and a $C^k$ function $\gamma:\mathbb{R}^{n-1}\longrightarrow\mathbb{R}$ such that we have
$ U\cap B(x^0,r)=\{x\in B(x^0,r)\lvert x_n>\gamma(x_1,...,x_{n-1})\}$
I do not understand the intuition behind this definition.To my understanding it does not correspond to derivatives. Can anyone help me with that? Why we call such boundary sets as $C^k$?

The condition $$x_n > \gamma(x_1, \ldots, x_n)$$ says that, locally, the boundary can be written as the graph of a function -- it means that the boundary itself is locally the set were $$x_n = \gamma(x_1, \ldots, x_n)$$ 
This is equivalent to saying it is an $n-1$ dimensional submanifold or a hypersurface -- maybe you have heard these words.
The smoothness assumption on $\gamma$ translates to a smoothness statement about this hypersurface. If you would, e.g, only require $\gamma$ to be Lipshitz, this would allow for corners in the boundary. Asking it to be of class $C^k$ with $k\ge 1$ makes it differentiable, or (if e.g. $k\ge 2$ already implies that it curves continuously.
