Tangent space at a point

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

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

v: C^\infty(M)\to \mathbb{R}

where $C^\infty(M)$ is the space of smooth real-valued functions on $M$ , and the maps satisfy the Leibniz rule $v(fg)=f(p)v(g)+g(p)v(f)$ for all $f,g\in C^\infty(M)$ . Its elements are the 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

\left.\frac{\partial}{\partial x^i}\right|_p (f) = \frac{\partial (f\circ \varphi^{-1})}{\partial x^i}\bigl(\varphi(p)\bigr).

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

d f_p : T_pM \to T_{f(p)}N,

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

Examples

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.

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 $p$ .

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 $G$ .