Let $S^{n}:=\{x\in\mathbb{R}^{n+1}\;:|x|=1\}$, then $S^{n}$ is a smooth manifold. (Spivak ''Calculus on manifolds'' page 111)
Spivak's remark is ''note that $S^{n}=g^{-1}(0)$ and $g:\mathbb{R}^{n+1}\to \mathbb{R}$ is defined by $g(x)=|x|^2-1$. I guess, again I have to use the implicit function theorem and the fact that if (Spivak also p.111) $g:A\to \mathbb{R}^{p}$ is differentiable such that $g^{\prime}(x)$ has rank $p$ whenever $g(x)=0$ then $g^{-1}(0)$ is a smooth manifold of dim $n-p$.
Here is the similar problem, using the very same theorem: Is the following set a manifold?
So, in particular $M:=S^{n}=g^{-1}(0)$, and then should I find the differential of $g(x)$ and determine for which points $x\in S^{n}$ it's value is zero?
Any help appreciated!
Note that the stereographic projections give an atlas for $\mathbb{S}_n$; you can verify by hand that the transition maps are smooth.