Set of all derivations at a point

NOTE: NOT to be confused with Set of all derivations of a germ

Notational clash

Some authors use $$T_p(\mathbb{R}^n)$$ to denote this set (the set of derivations of the form $$\omega:C^\infty\rightarrow\mathbb{R}$$ ) however other authors use $$T_p(\mathbb{R}^n)$$ to denote the Tangent space - while isomorphic these are distinct.

I use the custom notation $$D_p(\mathbb{R}^n)$$ to resolve this, care must be taken as $$D$$ and $$\mathcal{D}$$ look similar!

Definition

We denote the set of all derivations (at a point) of smooth or $C^\infty$ functions from $A$ at a point $p$ (assume $A=\mathbb{R}^n$ if no $A$ is mentioned) by:

$D_p(A)$, and assume $$D_p=D_p(\mathbb{R}^n)$$

In $\mathbb{R}^n$

$$D_p(\mathbb{R}^n)$$ can be defined as follows, where $\omega$ is a derivation, of signature: $$\omega:C^\infty(\mathbb{R}^n)\rightarrow\mathbb{R}$$

$$D_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a derivation at a point}\}$$

Recall $$C^\infty=C^\infty(\mathbb{R}^n)$$ and denotes the set of all smooth functions on $\mathbb{R}^n$

See also

• Derivation
• Tangent space
• Manifolds