# Tangent space at a point

From Diffgeom

This article defines a basic construct that makes sense on any differential manifold

View a complete list of basic constructs on differential manifolds

## Contents

## Definition

Let be a differential manifold and . The *tangent space* at to , denoted , is defined in a number of equivalent ways.

### In the language of derivations

Consider the algebra of infinitely differentiable functions on . Then, the tangent space is defined as the vector space of all derivations:

By being a derivation, we mean:

- is a -linear map
- Given functions we have:

### In the language of curves

The tangent space is defined, as a *set*, as the quotient of the set of all curves where , by the following equivalence relation: if, under the mapping to an open subset in via a coordinate chart, .

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 under a local coordinate chart. *Fill this in later*