Derivation
Warning: the definitions below are very similar
Contents
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.
