Im reading Evans PDE second edition.
In his appendix page 710 he gives the definition of a boundary $dU$ of an open set $U\subset \mathbb{R}^{n}$ as
A $dU$ is $C^{k}$ if for all $x^{0}$ $\in dU$ there exists $r>0$ and a $C^{k}$ function $\gamma:\mathbb{R}^{n-1}\to\mathbb{R}$ such that (upon relabelling and reorienting the coordinate axes if necessary) we have $$U\bigcap B(x^{0},r)=\{x\in B(x^{0},r) : x_{n}>\gamma(x_{1},...,x_{n-1}) \} $$
I have no intuition as to what these different boundaries could be like? What is special/useful about a boundary being $C^{1}$ , or $C^{k}$? Im really confused/lost with this definition. (Hopefully someone will include the relabelling and reorienting the coordinate axes in there answer? ) .
Thanks in advance.