Let U be an open set in \mathbb R^n and let \phi : U \to \mathbb R^m be a smooth map. The differential of \phi is the smooth map
f\phi : U \times \mathbb R^n \to \mathbb R^m \times \mathbb R^m
where \mathbf v \in U \times \mathbb R^n is a vector \mathbf v = (p, v). Let alpha: I \to U be such that \dot \alpha(t_0) = \mathbf v. We define d\phi(\mathbf v) as the vector at \phi(p) given by
d\phi(\mathbf v) = \dot{(\phi \circ \alpha)}(t_0)