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
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