Difference between revisions of "Set of all derivations at a point"

Revision as of 01:51, 5 April 2015

I chose to denote this (as in^[1]) by $$\mathcal{D}_p(A)$$ however at least one other author^[2] uses $$T_p(A)$$ - which is exactly what I (and the first reference) use for the tangent space.

This article will use the $\mathcal{D}$ form.

Definition

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

$\mathcal{D}_p(A)$, and assume $$\mathcal{D}_p=\mathcal{D}_p(\mathbb{R}^n)$$

In $\mathbb{R}^n$

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

$$\mathcal{D}_p(\mathbb{R}^n)=\{\omega|\omega\text{ is a point derivation}\}$$

Recall $$C^\infty_p=C^\infty_p(\mathbb{R}^n)$$ and denotes The set of all germs of smooth functions at a point

See also

• Derivation
• Tangent space
• Manifolds

References

1. Jump up ↑ Loring W. Tu - An introduction to manifolds - Second edition
2. Jump up ↑ John M Lee - Introduction to smooth manifolds