# Derivation

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## 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

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?

## References

1. 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 [ilmath]\mathbb{R}-[/ilmath]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 [ilmath]\mathbb{R}-[/ilmath]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.