> [!NOTE] Definition (Surface) > Given [[Real-Valued Function on Real n-Space (Multivariable Function)]] $f:U\subseteq \mathbb{R}^{n}\to \mathbb{R}$, the set of points $\{ (x_{1},\dots,x_{n+1}) \mid \forall x_{1},\dots,x_{n}\in U, x_{n+1}\in \mathbb{R} \text{ such that } x_{n+1} = f(x_{1},\dots,x_{n})\}$is a surface in $\mathbb{R}^{n+1}.$ # Properties - See [[Level Sets of Real-Valued Function of Several Real Variables]]. - See [[Fréchet Differentiation]] & [[Normal Vector of Surface]]. - See [[Critical Point of Real-Valued Function on Real 2-Space]]. - See [[Integration in Cartesian coordinates|Volume under surface]]. # Examples - A [[Graph of functions on real n-space|graph]] is a surface.