Pullback of a covector
= Pullback of a covector
{title2=$\phi^*\eta=\eta\circ d\phi$}
The <pullback of a covector> is the dual linear map to the <differential of a smooth map>. In coordinates its components are $(\phi^*\eta)_i=(\partial_i y^\alpha)\eta_\alpha$. The same rectangular Jacobian defines vector pushforward, with its other index contracted. No invertibility is needed.