pullback
When you change variables in an integral — switch from x to a new variable, or describe a surface by parameters u and v — you substitute and a Jacobian factor appears. The pullback is the clean, automatic version of that substitution for differential forms. Given a smooth map F that sends one space into another, the pullback (written with a star, F-star) takes a form living on the target space and produces a form on the source space, by pure substitution.
The recipe is exactly what you already do by hand. To pull back omega under F, replace every coordinate by its formula in terms of the new variables and replace every dx by its differential. If F sends (u, v) to (x(u,v), y(u,v)), then F-star of dx is x_u du + x_v dv, and F-star of dy is y_u du + y_v dv; you substitute these into omega and simplify with the wedge rules. The Jacobian determinant of the change of variables falls out by itself when you wedge: F-star of (dx^dy) becomes (the Jacobian) du^dv. So the pullback is the change-of-variables formula, but bookkept by the algebra of forms rather than remembered as a separate rule.
Pullback is the property that makes forms the right objects to integrate. The central fact is that pullback commutes with the exterior derivative, F-star(d omega) = d(F-star omega), and with the wedge product, so the whole calculus survives any change of coordinates or parametrization unchanged. This is exactly why the integral of a form does not depend on how you parametrize the region — reparametrize, and the pullback adjusts the integrand by precisely the Jacobian needed to keep the answer the same. It is the engine behind integrating over parametrized surfaces and the reason Stokes' theorem is coordinate-free.
Pull the area form dx^dy back through polar coordinates x = r cos t, y = r sin t. Then F-star dx = cos t dr - r sin t dt and F-star dy = sin t dr + r cos t dt. Wedging, the cross terms give F-star(dx^dy) = r (cos^2 t + sin^2 t) dr^dt = r dr^dt — the familiar r dr d(theta) area element, produced automatically.
The pullback of the area form through polar coordinates hands you the Jacobian factor r without any extra rule.
Pullback runs forms backward (from target to source) even when the map F runs forward. Vector fields do not pull back this cleanly — you can only push them forward, and only when F is invertible. This asymmetry is a real advantage of forms over vector fields.