Difference between revisions of "Tangent space"
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Revision as of 00:24, 5 April 2015
I prefer to denote the tangent space (of a set [ilmath]A[/ilmath] at a point [ilmath]p[/ilmath]) by [ilmath]T_p(A)[/ilmath] - as this involves the letter T for tangent however one author[1] uses [ilmath]T_p(A)[/ilmath] as Set of all derivations at a point - the two are indeed isomorphic but as readers will know - I do not see this as an excuse.
| Name | Preferred form | Alternate form | Definition |
|---|---|---|---|
| example | |||
| Tangent space | [math]T_p(A)[/math]
|
[math]A_p[/math]
|
[math]=\left\{(p,v)|v\in A\right\}[/math] |
| Set of all derivations at a point | [math]\mathcal{D}_p(A)[/math] | [math]T_p(A)[/math] | (see page) |
Definition
It is the set of arrows at a point, the set of all directions essentially. As the reader knows, a vector is usually just a direction, we keep track of tangent vectors and know them to be "tangent vectors at t" or something similar. A tangent vector is actually a point with an associated direction.
Euclidean (motivating) definition
We define [math]T_p(\mathbb{R}^n)=\left\{(p,v)|v\in\mathbb{R}^n\right\}[/math]
References
- ↑ John M. Lee - Introduction to Smooth Manifolds - second edition
TODO:
