In this lecture we define the derivative of a smooth map $\varphi \colon M \to N$ between two manifolds. At each point $p \in M$, the derivative is a linear map

$$D \varphi(p) \colon T_p M \to T_{ \varphi(p)}N.$$

We then give an alternative definition of tangent vectors as velocity vectors $\gamma’(t)$ of curves $\gamma(t)$. This approach is often more useful in computations than our earlier definition of tangent vectors as derivations.

Finally, we prove that the tangent space to a vector space is canonically isomorphic to the vector space, via the so-called “dash-to-dot” map[1].

1. Here by “so-called” what I really mean is “called by me”, since I made the name up... ↩︎