Difference between revisions of "Set of all derivations at a point"
(Created page with "I chose to denote this (as in<ref>Loring W. Tu - An introduction to manifolds - Second edition</ref>) by <math>\mathcal{D}_p(A)</math> however at least one other author<ref>Jo...") |
m (→See also) |
||
| Line 19: | Line 19: | ||
* [[Derivation]] | * [[Derivation]] | ||
* [[Tangent space]] | * [[Tangent space]] | ||
| − | * [[ | + | * [[Manifolds]] |
| + | |||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Differential Geometry|Manifolds}} | {{Definition|Differential Geometry|Manifolds}} | ||
Revision as of 01:51, 5 April 2015
I chose to denote this (as in[1]) by [math]\mathcal{D}_p(A)[/math] however at least one other author[2] uses [math]T_p(A)[/math] - which is exactly what I (and the first reference) use for the tangent space.
This article will use the [ilmath]\mathcal{D} [/ilmath] form.
Definition
We denote the set of all derivations of smooth or [ilmath]C^\infty[/ilmath] functions from [ilmath]A[/ilmath] at a point [ilmath]p[/ilmath] (assume [ilmath]A=\mathbb{R}^n[/ilmath] if no [ilmath]A[/ilmath] is mentioned) by:
[ilmath]\mathcal{D}_p(A)[/ilmath], and assume [math]\mathcal{D}_p=\mathcal{D}_p(\mathbb{R}^n)[/math]
In [ilmath]\mathbb{R}^n[/ilmath]
[math]\mathcal{D}_p(\mathbb{R}^n)[/math] can be defined as follows, where [ilmath]\omega[/ilmath] is a derivation, of signature: [math]\omega:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}[/math]
[math]\mathcal{D}_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a point derivation}\}[/math]
Recall [math]C^\infty_p=C^\infty_p(\mathbb{R}^n)[/math] and denotes The set of all germs of smooth functions at a point
