Difference between revisions of "Derivation"
(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}-}}...") |
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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 | ||
Latest revision as of 18:08, 14 October 2015
Contents
Definition
If [ilmath]a\in\mathbb{R}^n[/ilmath], we say that a map, [ilmath]\alpha:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R} [/ilmath] is a derivation at [ilmath]a[/ilmath] if it is [[Linear map|[ilmath]\mathbb{R} [/ilmath]-linear and satisfies the following[1]:
- Given [ilmath]f,g\in C^\infty(\mathbb{R}^n)[/ilmath] we have:
- [ilmath]\alpha(fg)=f(a)\alpha(g)+g(a)\alpha(f)[/ilmath]
Questions to answer
- What is [ilmath]fg[/ilmath]? Clearly we somehow have [ilmath]\times:C^\infty(\mathbb{R}^n)\times C^\infty(\mathbb{R}^n)\rightarrow C^\infty(\mathbb{R}^n)[/ilmath] but what it is?
References
- ↑ Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM
OLD PAGE
Warning: the definitions below are very similar
Definition
Derivation of [math]C^\infty_p[/math]
A derivation at a point is any [ilmath]\mathbb{R}-[/ilmath]Linear map: [math]D:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}[/math] that satisfies the Leibniz rule - that is [math]D(fg)|_p=f(p)Dg|_p+g(p)Df|_p[/math]
Recall that [math]C^\infty_p(\mathbb{R}^n)[/math] 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:
[math]D:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}^n[/math] is a derivation if it is [ilmath]\mathbb{R}-[/ilmath]Linear and satisfies the Leibniz rule, that is:
[math]D(fg)=f(p)Dg + g(p)Df[/math]
Warnings
These notions are VERY similar (and are infact isomorphic (both isomorphic to the Tangent space)) - but one must still be careful.
