In Evans's PDE book, he proves that $C^{\infty}(\bar U)$ is dense in $W^{k,p}(U)$, provided that $\partial U$ is of class $C^1$. The characterization of $C^1$ boundary he uses is the following:
The domain $U\subset \Bbb R^n$ has $C^1$ boundary if for each $x_0\in\partial U$ there is an $r>0$ such that (relabeling the coordinate if necessary) $$ B_r(x_0)\cap U = \{ x\in B_r(x_0): x_n>\gamma(x_1,\dots,x_{n-1}) \} $$ for some $\gamma\in C^1(\Bbb R^{n-1})$.
The proof in the book uses this characterization, I don't have problem with the proof itself. However, there's another characterization that I usually see.
The domain $U\subset \Bbb R^n$ has $C^1$ boundary if for each $x_0\in\partial U$ there is an $r,r'>0$ and a $C^1$ diffeomorphism $\Phi$, sending $x_0$ to $0$, such that $$ \Phi(B_r(x_0)\cap U) \cap B_{r'}(0) = \{ y\in B_{r'}(0): y_n>0 \} $$
Needless to say, this half-ball is easier to work with. However, I don't know any result concerning the behaviour of $W^{k,p}(U)$ functions under composition with $\Phi^{-1}$. I don't know what would happen if I obtain an approximation $f\in C^{\infty}$ of $u\circ\Phi^{-1}$ in the $y$-coordinate. Would $f\circ\Phi$ approximate our original $u$ (at least locally in $B_r(x_0)$)?
I'd really appreciate if anyone could tell me the result in this direction, or give me some good references. Here is another approach with link to Evans' proof I referred to.
Edit: Perhaps I was not clear enough. I am aware that the 2 characterizations are equivalent. What I want to know is if there's a proof of this result using the second definition directly.