Set of all derivations at a point
I chose to denote this (as in[1]) by [math]\mathcal{D}_p(A)[/math] however at least one other author[2] uses [math]T_p(A)[/math] - which is exactly what I (and the first reference) use for the tangent space.
This article will use the [ilmath]\mathcal{D} [/ilmath] form.
Definition
We denote the set of all derivations of smooth or [ilmath]C^\infty[/ilmath] functions from [ilmath]A[/ilmath] at a point [ilmath]p[/ilmath] (assume [ilmath]A=\mathbb{R}^n[/ilmath] if no [ilmath]A[/ilmath] is mentioned) by:
[ilmath]\mathcal{D}_p(A)[/ilmath], and assume [math]\mathcal{D}_p=\mathcal{D}_p(\mathbb{R}^n)[/math]
In [ilmath]\mathbb{R}^n[/ilmath]
[math]\mathcal{D}_p(\mathbb{R}^n)[/math] can be defined as follows, where [ilmath]\omega[/ilmath] is a derivation, of signature: [math]\omega:C^\infty_p(\mathbb{R}^n)\rightarrow\mathbb{R}[/math]
[math]\mathcal{D}_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a point derivation}\}[/math]
Recall [math]C^\infty_p=C^\infty_p(\mathbb{R}^n)[/math] and denotes The set of all germs of smooth functions at a point
