Smooth and Lipschitz domains

Question

We know that an open ball $B_{r}\subseteq R^{n}$ is a smooth domain. It follows that this is a Lipschitz domain. How can I show explicitly the function $\varphi_{x}\in C^{0,1}(R^{n-1})$ that is related with each $x\in\partial B_{r}$?

Answer 1

We recall the definition of Lipschitz domain: let $\Omega\subseteq R^{n}$ be an open set. The set $\Omega$ is said to be $\text{C}^{0,1}$ if for every $x\in\partial\Omega$, there exist a neighbourhood $U_{x}$ of $x$ and $\varphi_{x}\in\text{C}^{0,1}(R^{n-1})$ such that, up to rotation,