什么是 tangent space？

Let $M$ be a $d$-submanifold and $a \in M$. The tangent space $T_aM$ is the set of velocity vectors $\gamma'(0)$ of $\mathcal C^1$ curves $\gamma \colon \intoo{-\varepsilon}\varepsilon \to M$ with $\gamma(0) = a$.