Revision as of 02:21, 5 April 2015 (view source) Alec (Talk | contribs) (Created page with "'''Warning:''' the definitions below are very similar ==Definition== ===Derivation of <math>C^\infty_p</math>=== A derivation at a point is any Linear map|{{M|\mathbb{R}-}}...")		Latest revision as of 18:08, 14 October 2015 (view source) Alec (Talk | contribs) m
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		+	{{Refactor notice}}
		+	==Definition==
		+	If {{M|a\in\mathbb{R}^n}}, we say that a map, {{M|\alpha:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R} }} is a '''''derivation at {{M|a}}''''' if it is [[Linear map|{{M|\mathbb{R} }}-linear and satisfies the following<ref name="ITSM">Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM</ref>:
		+	* Given {{M|f,g\in C^\infty(\mathbb{R}^n)}} we have:
		+	** {{Highlight|{{M|1=\alpha(fg)=f(a)\alpha(g)+g(a)\alpha(f)}}}}
		+	===Questions to answer===
		+	# What is {{M|fg}}? Clearly we somehow have {{M|\times:C^\infty(\mathbb{R}^n)\times C^\infty(\mathbb{R}^n)\rightarrow C^\infty(\mathbb{R}^n)}} but what it is?
		+	==References==
		+	<references/>
		+	=OLD PAGE=
		+
	'''Warning:''' the definitions below are very similar		'''Warning:''' the definitions below are very similar

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Latest revision as of 18:08, 14 October 2015

Definition

If $a\in\mathbb{R}^n$, we say that a map, $\alpha:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}$ is a derivation at $a$ if it is [[Linear map|$\mathbb{R}$-linear and satisfies the following^[1]:

• Given $f,g\in C^\infty(\mathbb{R}^n)$ we have:
  • $\alpha(fg)=f(a)\alpha(g)+g(a)\alpha(f)$

Questions to answer

1. What is $fg$? Clearly we somehow have $\times:C^\infty(\mathbb{R}^n)\times C^\infty(\mathbb{R}^n)\rightarrow C^\infty(\mathbb{R}^n)$ but what it is?

References

1. Jump up ↑ Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM

OLD PAGE

Warning: the definitions below are very similar

Definition

Derivation of $$C^\infty_p$$

A derivation at a point is any $\mathbb{R}-$Linear map:

$$D:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}$$ that satisfies the Leibniz rule - that is $$D(fg)|_p=f(p)Dg|_p+g(p)Df|_p$$

Recall that

$$C^\infty_p(\mathbb{R}^n)$$ is a set of germs - specifically the set of all germs of smooth functions at a point

Derivation at a point

One doesn't need the concept of germs to define a derivation (at p), it can be done as follows:

$$D:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}^n$$ is a derivation if it is $\mathbb{R}-$Linear and satisfies the Leibniz rule, that is:

$$D(fg)=f(p)Dg + g(p)Df$$

Warnings

These notions are VERY similar (and are infact isomorphic (both isomorphic to the Tangent space)) - but one must still be careful.

See also

• Germ

References