Difference between revisions of "Tangent space"
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| + | What is defined here may also be called the '''Geometric tangent space''' | ||
==Definition== | ==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. | 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. | ||
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We define <math>T_p(\mathbb{R}^n)=\left\{(p,v)|v\in\mathbb{R}^n\right\}</math> | We define <math>T_p(\mathbb{R}^n)=\left\{(p,v)|v\in\mathbb{R}^n\right\}</math> | ||
| + | Generally then we may say: <math>T_p(A)=\left\{(p,v)|v\in A\right\}</math> | ||
| + | |||
| + | ==Notation== | ||
| + | A tangent vector (often {{M|v}} is used) shall be left as just {{M|v}} if the point to which it is a tangent to is implicit (ie "{{M|v}} is a tangent at {{M|p}}") | ||
| + | |||
| + | Rather than writing {{M|(p,v)}} we may write: | ||
| + | * {{M|v}} (if it is implicitly understood that this is a tangent to the point {{M|p}}) | ||
| + | * {{M|v_a}} | ||
| + | * <math>v|_a</math> | ||
| + | |||
| + | ==Why ordered pairs== | ||
| + | Ordered pairs are used because now the tangent space at two distinct points are disjoint sets, that is <math>\alpha\ne\beta\implies T_\alpha(A)\cap T_\beta(A)=\emptyset</math> | ||
| + | |||
| + | ==Vector space== | ||
| + | <math>T_p(A)</math> is a vector space when equipped with the following definitions: | ||
| + | * {{M|1=v_a+w_a=(v+w)_a}} | ||
| + | * {{M|1=c(v_a)=(cv)_a}} | ||
| + | |||
| + | It is easily seen that the basis for this is the standard basis <math>\{e_1|_p,\cdots, e_n|_p\}</math> and that the tangent space {{M|T_p(A)}} is basically just a copy of {{M|A}} | ||
| + | |||
| + | ==See also== | ||
| + | *[[Set of all derivations at a point]] | ||
| + | *[[The tangent space and derivations at a point are isomorphic]] | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| − | + | ||
{{Definition|Differential Geometry|Manifolds}} | {{Definition|Differential Geometry|Manifolds}} | ||
Revision as of 00:48, 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) |
What is defined here may also be called the Geometric tangent space
Contents
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]
Generally then we may say: [math]T_p(A)=\left\{(p,v)|v\in A\right\}[/math]
Notation
A tangent vector (often [ilmath]v[/ilmath] is used) shall be left as just [ilmath]v[/ilmath] if the point to which it is a tangent to is implicit (ie "[ilmath]v[/ilmath] is a tangent at [ilmath]p[/ilmath]")
Rather than writing [ilmath](p,v)[/ilmath] we may write:
- [ilmath]v[/ilmath] (if it is implicitly understood that this is a tangent to the point [ilmath]p[/ilmath])
- [ilmath]v_a[/ilmath]
- [math]v|_a[/math]
Why ordered pairs
Ordered pairs are used because now the tangent space at two distinct points are disjoint sets, that is [math]\alpha\ne\beta\implies T_\alpha(A)\cap T_\beta(A)=\emptyset[/math]
Vector space
[math]T_p(A)[/math] is a vector space when equipped with the following definitions:
- [ilmath]v_a+w_a=(v+w)_a[/ilmath]
- [ilmath]c(v_a)=(cv)_a[/ilmath]
It is easily seen that the basis for this is the standard basis [math]\{e_1|_p,\cdots, e_n|_p\}[/math] and that the tangent space [ilmath]T_p(A)[/ilmath] is basically just a copy of [ilmath]A[/ilmath]
See also
References
- ↑ John M. Lee - Introduction to Smooth Manifolds - second edition
