What does it mean by piecewise smooth boundary? I will be highly obliged if anyone can give me any reference where i can get the definition of domain (in $\mathbb{R^n}$) with piecewise smooth boundary.
My question is when a domain in $\mathbb{R^n}$ is said to have a piecewise smooth boundary?
I tried a lot to find in Google but i didn't get.Please help me out.  
 A: I do not think there is a commonly agreed on definition. Here is one, based on the discussion in
William Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997.
A  domain $\Omega$ in $R^n$ has piecewise smooth boundary if for each $x\in cl(\Omega)$ there exists a neighborhood $U$ of $x$ in $R^n$ and a diffeomorphism $f: U\to U'\subset R^n$, such that $f(cl(\Omega))\cap U$ equals $\Omega'\cap U'$, where $\Omega'\subset R^n$ is a convex polyhedron.  
A: I got stuck on the definition of piecewise smooth boundary while reviewing material for a calculus course I am taking. I can't intuitively see the what the definition means but it happens to solve your question.
The definition is: if the boundary of a set is a finite union of piecewise, simple closed curves, then the set has a piecewise smooth boundary and moreover, the set itself called regular region.
My own interpretation is that if every collection of curves which forms the boundary of such a set does not self-intersect in any way, then the boundary formed from those curves is said to be piecewise smooth boundary.
