Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 50 / 3 / a / Solution

Treat $V$ as a derivation. For an arbitrary smooth function $f$ on $\mathcal{N}$, the definition of the differential of a smooth map and the chain rule giveComparison with $({\varphi }_{\ast }V{)}^{\alpha }{\partial }_{\alpha }f$ for all $f$ provesThe Jacobian is an $n\times m$ matrix; it need not be square or invertible.