# Definition(s) > [!NOTE] Definition 1 (Sphere in $\mathbb{R}^n$) > Let $n$ be a natural number. The $n$-dimensional sphere of radius $r\geq 0$ is given by $S^{n}_{r} = \{ \mathbf{x} \in \mathbb{R}^{n+1}: \Vert \mathbf{x} \Vert = r \}$where $\Vert \cdot \Vert$ denotes the [[Euclidean Norm|standard Euclidean norm]]. **Notation**: The unit sphere is denoted $S^n:=S_{1}^n.$ > [!Example] Example ($S^n$ for $n=0,1,2$) > $S^{0}= \{ -1,1 \}$ > $S^{1}$ is the unit circle. > $S^2$ is the unit sphere. # Properties(s) # Application(s) **More examples**: # Bibliography