= Pullback of a differential form
{title2=$\phi^*$}
= Pullbacks of a differential form
{synonym}
A smooth map $\phi:M\to N$ pulls a <differential form> on $N$ back to one on $M$ by applying its <differential> $d\phi$ to each <tangent vector> argument. In particular, a parametrization of a <curve> pulls a <one-form> back to an ordinary integrand on an interval.
For $\alpha=x\,dy$ and the <parabola> $\phi(t)=(t,t^2)$, $0\leq t\leq1$, substitute $x=t$ and $dy=2t\,dt$ to obtain
$$
\phi^*\alpha=2t^2\,dt,\qquad \int_\phi\alpha=\int_0^1 2t^2\,dt=\boxed{\frac23}.
$$
The same endpoints gave $0$ and $1$ along the two square-edge paths in the introduction, so this third answer is consistent with path dependence. By contrast, $\phi^*d(xy)=d(t^3)=3t^2\,dt$ gives integral $1$, the same endpoint difference as before.
The <pullback of a differential form> commutes with the <wedge product of differential forms> and the <exterior derivative>: $\phi^*(\alpha\wedge\beta)=\phi^*\alpha\wedge\phi^*\beta$ and $\phi^*(d\alpha)=d(\phi^*\alpha)$. These identities make changes of coordinates and parametrizations compatible with differentiation and integration.
It is the alternating-tensor case of <pullback (differential geometry)>.
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