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]] | '''NOTE:''' NOT to be confused with [[Set of all derivations of a germ]] | ||
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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== | ==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. | 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. | ||
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]
