# Tangent space at a point

This article defines a basic construct that makes sense on any differential manifold
View a complete list of basic constructs on differential manifolds

## Definition

Let $M$ be a differential manifold and $p \in M$. The tangent space at $p$ to $M$, denoted $T_p(M)$, is defined in a number of equivalent ways.

### In the language of derivations

Consider the algebra $C^\infty(M)$ of infinitely differentiable functions on $M$. Then, the tangent space $T_p(M)$ is defined as the vector space of all derivations: $d: C^\infty(M) \to \R$

By $d$ being a derivation, we mean:

• $d$ is a $\R$-linear map
• Given functions $f,g \in C^\infty(M)$ we have: $d(fg) = (df)(g(p)) + f(p)(dg)$

### In the language of curves

The tangent space $T_p(M)$ is defined, as a set, as the quotient of the set of all curves $\gamma: (-1,1) \to M$ where $\gamma(0) = p$, by the following equivalence relation: $\gamma_1 = \gamma_2$ if, under the mapping to an open subset in $\R^n$ via a coordinate chart, $\gamma_1'(0) = \gamma_2'(0)$.

With this definition, addition is not very clear.

### In the language of local coordinate charts

In this language, the tangent space is defined as the tangent space to its image in $\R^n$ under a local coordinate chart. Fill this in later