frontier of class $C^{1}$. Studying the Divergence Theorem (Gauss theorem), found the definition of frontier of class $C^{1}$. Which means? That is, the one which is a set with boundary of class $C^{1}$? 
Can give reference books.
grateful
 A: This is the notion of submanifold. A $C^1$ submanifold of dimension $p$ is a subset of $\mathbb{R}^n$ that "locally looks like $\mathbb{R}^p$", the straightening being of regularity at least $C^1$. More precisely :
A set $M$ is a $C^1$ submanifold if for every $x \in M$ there is an open set $U$ containing $x$ and a $C^1$ diffeomorphism $\phi : U \rightarrow V$ ($V$ being an open set of $\mathbb{R}^n$) which sends $U \cap M$ to $(\mathbb{R}^p \times \{0\}^{n-p}) \cap V$ and $x$ to 0.
Examples: the sphere in the 3-dimensional space, a torus, a cylinder...
A: An explanation in more elementary terms: an open set $A$ has boundary of class $C^1$ if  for every point $p$  on the boundary of $A$ there exist: 


*

*a function $f$ of $n-1$ variables, with continuous first-order derivatives 

*a number $r>0$

*an invertible linear transformation $T:\mathbb R^n\to\mathbb R^n$


such that $A\cap \{x:|x-p|<r\} = \{Tu:u_n>f(u_1,\dots,u_{n-1})\}\cap \{x:|x-p|<r\}$. 
Unfortunately I don't have an accessible reference for this, apart from Wikipedia article on the aforementioned manifolds.
