We discuss at the moment about the gaussian Formula (divergence theorem). There we have that the set on which we integrate must be compact and have a smooth edge. I know what compact means, but I can't imagine how a smooth edge looks like. We had the following definition
Let $A\subset \mathbb{R}^n$ be a compact subset. We say that A has a smooth edge if for eacht $a\in \partial A$ there exists an open neighbourhood $U\subset \mathbb{R}^n$ and a $C^1$-function $$\psi:U\rightarrow \mathbb{R}$$ such that
- $A\cap U=\{y\in U: \psi(x)\leq 0\}$
- $\nabla \psi(x)\neq 0 \,\,\,\,\,\,\forall x\in U$
But somehow I don't see how we can think about that. So i think that a sphere has always smooth edge. But in the lecture we also computed an integral over a cube with the divergence theorem, therefore I assume that also a cube must have smooth edge right? So are there other sets such as a thetraeder or a cylinder which have smooth edges?
Could someone explain this to me? Maybe also with some intuition?