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 map.

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

Then the differential of $F$, denoted by $d F$, is the mapping $d F_p : T_p M \to T_{\map F p} N$ such that:


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