User:Austrodata/Definition:Derivation/Smooth Mapping

Definition

Let $n$ be a natural number.

Let $x$ be a point in $\R^n$.

Let $\phi: \map {C^\infty} {\R^n} \to \R$.

Then, $\phi$ is called a derivation at $x$ if and only if:

• $\phi$ is linear over $\R$
• $\phi$ satisfy the following Leibniz's product rule:

$\map \phi {fg} = f \, \map \phi g + \map \phi f \, g$

Sources

• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): Chapter $3$: Tangent Vectors : $\S$ Tangent Vectors.