Unit sphere as a regular level set
= Unit sphere as a regular level set
{title2=$S^n$}
The unit sphere $S^n\subseteq\mathbb R^{n+1}$ is the regular level set $\lVert x\rVert^2=1$. Its tangent space is
$$
T_xS^n=x^\perp=\{v\in\mathbb R^{n+1}:x\mathbin\cdot v=0\}.
$$