A subset $S$ of $\mathbb R^n$ is called a hypersurface of class $C^k (1\leq k \leq \infty)$ if for every $x_0\in S$ there is an open set $V\subset \mathbb R^n$ containing $x_0$ and a real-valued function $\phi \in C^k(V)$ such that $\nabla \phi$ is nonvanishing on $S\cap V$ and $S\cap V = \{ x\in V: \phi(x)=0 \}.$
With $S, V, \phi$ as above, the vector $v(x)$ is perpendicular to $S$ at $x$ for every $x\in S\cap V.$ We shall always suppose that $S$ is oriented, that is, that we have made a choice of unit vector $v(x)$ for each $x\in S$, varying continuously with $x$ which is perpendicular to $S$ at $x.$ $v(x)$ will be called the normal to $S$ at $x.$
Note: On $S\cap V,$ we have $v(x)= \pm \frac{\nabla \phi (x)}{|\nabla \phi (x)|}.$
Fix $r, a, b\in (0, \infty)$ with $a<b.$
Let $A= \{ (x,t) \in \mathbb R^n \times (a,b): |x|<r, 0<a<t<b \}.$
Questions: (a) What is the boundary of $A$ in $\mathbb R^n \times (0, \infty)$? (The boundary of $A$ is denoted by $\partial A.$) (b) How to find the normal $v(z)$ to $\partial A$ at $z$ in $\partial A$? (Is $\partial A$ hypersurface of class $C^1$? What is $\phi $ for $\partial A$ ?)