Difference between revisions of "Set of all derivations at a point"

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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>John M Lee - Introduction to smooth manifolds</ref> uses <math>T_p(A)</math> - which is exactly what I (and the first reference) use for [[Tangent space|the tangent space]].
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'''NOTE:''' NOT to be confused with [[Set of all derivations of a germ]]
  
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This article will use the {{M|\mathcal{D} }} form.
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==Notational clash==
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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.
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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==
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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:
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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:
  
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{{M|\mathcal{D}_p(A)}}, and assume <math>\mathcal{D}_p=\mathcal{D}_p(\mathbb{R}^n)</math>
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{{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}}===
  
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<math>\mathcal{D}_p(\mathbb{R}^n)</math> can be defined as follows, where {{M|\omega}} is a [[Derivation|derivation]], of signature: <math>\omega:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}</math>
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<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>
  
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<math>\mathcal{D}_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a point derivation}\}</math>
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<math>D_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a derivation at a point}\}</math>
  
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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]]
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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==

Revision as of 02:24, 5 April 2015

NOTE: NOT to be confused with Set of all derivations of a germ

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]

See also

References

  1. ↑ John M. Lee - Introduction to smooth manifolds - Second edition
  2. ↑ Loring W. Tu - An introduction to manifolds - second edition
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