Let $S \subset \mathbb{R}^n$ and $0 \leq k \leq n-1 . S$ is called a $k$-dimensional **submanifold** of $\mathbb{R}^n$ if for each $x \in S$ there exists $U \subset \mathbb{R}^n$ an open neighbourhood of $x$ and $f: U \rightarrow \mathbb{R}^{n-k}$ such 0 is a regular value of $f$ and $S \cap U=f^{-1}(0)$. By the [[Implicit function theorem|implicit function theorem]], this is equivalent to asking that for each $x ∈ S$ there exists an open neighbourhood $U ⊂ \mathbb{R}^n$ of $x$ such that $S ∩U$ can be written as the [[Graph of functions on real n-space|graph]] of a function from an open set in $\mathbb{R}^k$ to $\mathbb{R}^ {n−k}$ . We say that $S$ is a **smooth hypersurface** if it is an $(n-1)$-dimensional submanifold of $\mathbb{R}^n$.