Difference between revisions of "Set of all derivations at a point"
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| − | + | '''NOTE:''' NOT to be confused with [[Set of all derivations of a germ]] | |
| − | This | + | |
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| + | ==This page might be total crap== | ||
| + | I was confused about the concept at the time! DO NOT USE THIS PAGE | ||
| + | ==Notational clash== | ||
| + | Some authors use <math>T_p(\mathbb{R}^n)</math> to denote this set (the set of derivations of the form <math>\omega:C^\infty\rightarrow\mathbb{R}</math>)<ref>John M. Lee - Introduction to smooth manifolds - Second edition</ref> however other authors use <math>T_p(\mathbb{R}^n)</math><ref>Loring W. Tu - An introduction to manifolds - second edition</ref> to denote the [[Tangent space]] - while isomorphic these are distinct. | ||
| + | |||
| + | I use the custom notation <math>D_p(\mathbb{R}^n)</math> to resolve this, care must be taken as <math>D</math> and <math>\mathcal{D}</math> look similar! | ||
==Definition== | ==Definition== | ||
| − | We denote the set of all [[Derivation|derivations]] of [[Smooth|smooth or {{M|C^\infty}}]] functions from {{M|A}} at a point {{M|p}} (assume {{M|1=A=\mathbb{R}^n}} if no {{M|A}} is mentioned) by: | + | We denote the set of all [[Derivation#Derivation at a point|derivations (at a point)]] of [[Smooth|smooth or {{M|C^\infty}}]] functions from {{M|A}} at a point {{M|p}} (assume {{M|1=A=\mathbb{R}^n}} if no {{M|A}} is mentioned) by: |
| − | {{M| | + | {{M|D_p(A)}}, and assume <math>D_p=D_p(\mathbb{R}^n)</math> |
===In {{M|\mathbb{R}^n}}=== | ===In {{M|\mathbb{R}^n}}=== | ||
| − | <math> | + | <math>D_p(\mathbb{R}^n)</math> can be defined as follows, where {{M|\omega}} is a [[Derivation|derivation]], of signature: <math>\omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}</math> |
| − | <math> | + | <math>D_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a derivation at a point}\}</math> |
| − | Recall <math>C^\ | + | Recall <math>C^\infty=C^\infty(\mathbb{R}^n)</math> and denotes the set of all smooth functions on {{M|\mathbb{R}^n}} |
==See also== | ==See also== | ||
* [[Derivation]] | * [[Derivation]] | ||
* [[Tangent space]] | * [[Tangent space]] | ||
| − | * [[ | + | * [[Manifolds]] |
| + | |||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Differential Geometry|Manifolds}} | {{Definition|Differential Geometry|Manifolds}} | ||
Latest revision as of 21:51, 13 April 2015
NOTE: NOT to be confused with Set of all derivations of a germ
Contents
This page might be total crap
I was confused about the concept at the time! DO NOT USE THIS PAGE
Notational clash
Some authors use [math]T_p(\mathbb{R}^n)[/math] to denote this set (the set of derivations of the form [math]\omega:C^\infty\rightarrow\mathbb{R}[/math])[1] however other authors use [math]T_p(\mathbb{R}^n)[/math][2] to denote the Tangent space - while isomorphic these are distinct.
I use the custom notation [math]D_p(\mathbb{R}^n)[/math] to resolve this, care must be taken as [math]D[/math] and [math]\mathcal{D}[/math] look similar!
Definition
We denote the set of all derivations (at a point) 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]D_p(A)[/ilmath], and assume [math]D_p=D_p(\mathbb{R}^n)[/math]
In [ilmath]\mathbb{R}^n[/ilmath]
[math]D_p(\mathbb{R}^n)[/math] can be defined as follows, where [ilmath]\omega[/ilmath] is a derivation, of signature: [math]\omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}[/math]
[math]D_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a derivation at a point}\}[/math]
Recall [math]C^\infty=C^\infty(\mathbb{R}^n)[/math] and denotes the set of all smooth functions on [ilmath]\mathbb{R}^n[/ilmath]
