# 2. Tangent Spaces

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

The Zoom lecture recording is available on the DMATH forum.

Comments and questions?