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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

  1. 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?


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


Warning: the definitions below are very similar


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]


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

See also