Definition:Differential of Mapping/Manifolds

Definition

Let $M$ and $N$ be smooth manifolds with or without boundary.

Let $F : M \to N$ be a smooth mapping.

Let $T_p M$ be the tangent space at $p \in M$.

Then the differential of $F$ at $p$ is defined and denoted as the smooth mapping $\d F_p : T_p M \to T_{\map F p} N$ such that:

$\forall f \in \map {C^\infty} N : \forall p \in M : \forall v \in T_p M : \map {\map {\d F_p} v} f = \map v {f \circ F}$

Sources

• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): $\S 3$: Tangent Vectors. The Differential of a Smooth Map