Tangent space at a point

Let $M$ be a smooth manifold and let $p\in M$.

Definition (Derivations)

The tangent space $T_pM$ is the real vector space of all derivations at $p$, i.e. all $\mathbb{R}$-linear maps

such that $v(fg)=f(p)\,v(g)+g(p)\,v(f)$ for all $f,g\in C^\infty(M)$ (the Leibniz rule).

Elements $v\in T_pM$ are called tangent vectors at $p$.

Coordinate description

If $(U,\varphi)$ is a smooth chart with $p\in U$ and $\varphi=(x^1,\dots,x^n)$, then there are canonical basis derivations $\left.\frac{\partial}{\partial x^i}\right|_p\in T_pM$ defined by

Every $v\in T_pM$ is uniquely expressible as $v=\sum_{i=1}^n v^i \left.\frac{\partial}{\partial x^i}\right|_p$, so a chart identifies $T_pM$ (non-canonically) with $\mathbb{R}^n$.

Functoriality (pushforward)

Given a smooth map $f:M\to N$ between smooth manifolds, there is an induced linear map on tangent spaces

called the differential (pushforward) of $f$ at $p$. As $p$ varies, these tangent spaces assemble into the tangent bundle .

Examples

1. Euclidean space. For $M=\mathbb{R}^n$ and any $p\in \mathbb{R}^n$, there is a canonical identification $T_p\mathbb{R}^n\cong \mathbb{R}^n$, with basis $\left.\frac{\partial}{\partial x^1}\right|_p,\dots,\left.\frac{\partial}{\partial x^n}\right|_p$ in the standard coordinates.

2. The sphere as a constraint. For $S^2=\{x\in \mathbb{R}^3:\|x\|=1\}$ and $p\in S^2$, the tangent space is

   $$T_pS^2=\{v\in \mathbb{R}^3 : v\cdot p = 0\},$$

   i.e. the plane through the origin orthogonal to $p$.

3. Lie groups. For a Lie group $G$ with identity element $e$, the tangent space $T_eG$ is the underlying vector space of the Lie algebra of $G$.