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

Revision as of 01:18, 5 April 2015 (view source) Alec (Talk | contribs) (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...")		Latest revision as of 21:51, 13 April 2015 (view source) Alec (Talk | contribs) m
(2 intermediate revisions by the same user not shown)
Line 1:		Line 1:
−	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]].	+	'''NOTE:''' NOT to be confused with [[Set of all derivations of a germ]]
−	This article will use the {{M|\mathcal{D} }} form.	+
		+
		+	==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|\mathcal{D}_p(A)}}, and assume <math>\mathcal{D}_p=\mathcal{D}_p(\mathbb{R}^n)</math>	+	{{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>\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>	+	<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>\mathcal{D}_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a point derivation}\}</math>	+	<math>D_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a derivation at a point}\}</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]]	+	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]]
−	* [[Manifold]]	+	* [[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

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

$$T_p(\mathbb{R}^n)$$ to denote this set (the set of derivations of the form $$\omega:C^\infty\rightarrow\mathbb{R}$$ )^[1] however other authors use $$T_p(\mathbb{R}^n)$$ ^[2] to denote the Tangent space - while isomorphic these are distinct.

I use the custom notation

$$D_p(\mathbb{R}^n)$$ to resolve this, care must be taken as $$D$$ and $$\mathcal{D}$$ look similar!

Definition

We denote the set of all derivations (at a point) of smooth or $C^\infty$ functions from $A$ at a point $p$ (assume $A=\mathbb{R}^n$ if no $A$ is mentioned) by:

$D_p(A)$, and assume

$$D_p=D_p(\mathbb{R}^n)$$

In $\mathbb{R}^n$

$$D_p(\mathbb{R}^n)$$ can be defined as follows, where $\omega$ is a derivation, of signature: $$\omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}$$

$$D_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a derivation at a point}\}$$

Recall

$$C^\infty=C^\infty(\mathbb{R}^n)$$ and denotes the set of all smooth functions on $\mathbb{R}^n$

See also

• Derivation
• Tangent space
• Manifolds

References

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