The goal of the next few lectures is to associate to an $n$-dimensional smooth manifold $M$ an $n$-dimensional vector space $T_x M$ to every point $x \in M$. We call $T_xM$ the tangent space to $M$ at $x$. This will allow us to define the derivative of a smooth map $\varphi \colon M \to N$ between manifolds—for each $x \in M$ we will obtain a linear map $D \varphi(x) \colon T_x M \to T_{ \varphi(x)} N$. In the special case where $M$ is an open subset of $\mathbb{R}^n$, this notion will coincide with the usual definition of the derivative.

We define the tangent space as the set of linear derivations of the space of germs at a given point $x \in M$. This definition will at first seem somewhat abstract… 🤯

But don't worry: we will gradually simplify the definition over the next couple of lectures (next lecture we will get rid of the germs), and by Lecture 5 the definition should agree with what you might naively guess the “tangent space” should be.