Tangent space
= Tangent space
{title2=$T_pM$}
{wiki}
The tangent space $T_pM$ is the <vector space> of tangent vectors to a <smooth manifold> $M$ at $p$. For an embedded manifold $M\subseteq\mathbb R^N$, it is
$$
T_pM=\{\gamma'(0):\gamma:(-\varepsilon,\varepsilon)\to M
\text{ is smooth and }\gamma(0)=p\}.
$$
For a regular parametrized surface $X(u,v)$, it is spanned by $X_u$ and $X_v$.