Definition:Derivation of Germs of Smooth Functions

Definition

Let $M$ be a smooth manifold with or without boundary.

Let $p \in M$.

Let $\map {C_p^\infty} M$ be the set of all germs of smooth functions at $p$.

Let $\sqbrk{f}_p \in \map {C_p^\infty} M$ denote the germs of the smooth functions $f$ at $p$.

Then a derivation of $\map {C_p^\infty} M$ is a linear map $\delta: \map {C_p^\infty} M \to \R$ satisfying the Leibniz law:

$\delta \sqbrk{fg}_p = \map f p \delta \sqbrk{f}_p + \delta \sqbrk{g}_p \; \map g p$

Also see

• Definition:Derivation

Sources

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