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

Evaluate the pullback of a covector on the coordinate basis vector ${\partial }_i$. Its pushforward is $({\partial }_i{y}^{\alpha }){\partial }_{\alpha }$, soEquivalently, pulling back the coordinate one-forms gives ${\varphi }^{\ast }(d{y}^{\alpha })=d(y^{\alpha }\circ \varphi )=({\partial }_i{y}^{\alpha })d{x}^i$. The same Jacobian enters the vector pushforward and the covector pullback, with the different index contractions reflecting their duality.