the pullback of a form
Suppose you have a smooth map f from one manifold M to another N, and a differential form omega living on the target N. The pullback f^*omega is the way you carry that form backwards to M — you measure things on N using omega but read off the answer over on M. The name is apt: the form flows opposite to the map. This is the operation that makes forms the natural language for substitution and change of variables.
Concretely, for a smooth map f: M -> N and a k-form omega on N, the pullback (f^*omega)_p acts on tangent vectors v_1, ..., v_k at a point p in M by pushing them forward with the differential df_p and then evaluating omega: (f^*omega)_p(v_1, ..., v_k) = omega_{f(p)}(df_p v_1, ..., df_p v_k). For a function (0-form) g, f^*g = g composed with f. The pullback is linear, commutes with the wedge (f^*(alpha^beta) = f^*alpha ^ f^*beta), and crucially commutes with the exterior derivative: f^*(d omega) = d(f^*omega). In coordinates you literally substitute: if y = f(x), replace each y^j by its formula and each dy^j by d(f^j) = sum (partial f^j / partial x^i) dx^i.
Pullback is everywhere. It is exactly the change-of-variables rule for integrals (the Jacobian appears automatically through the wedge), it is how you restrict a form to a submanifold, and because f^* commutes with d it descends to a map f^*: H^k_dR(N) -> H^k_dR(M) on cohomology, the basis for homotopy invariance and functoriality. A key asymmetry to remember: forms pull back along ANY smooth map, but vector fields generally do NOT push forward unless f is a diffeomorphism. This contravariance is why forms, not fields, are so flexible.
Polar coordinates: f(r, theta) = (r cos theta, r sin theta) maps the (r, theta)-plane to the (x, y)-plane. Pull back the area form dx ^ dy: substitute dx = cos theta dr - r sin theta d theta and dy = sin theta dr + r cos theta d theta, wedge, and the cross terms in dr ^ d theta give f^*(dx^dy) = r dr ^ d theta. The factor r is exactly the Jacobian — the change-of-variables formula falls out of the wedge automatically.
Pulling back the area form through polar coordinates produces the r in 'r dr d theta' for free; pullback IS the change of variables.
Because f^* commutes with d, it sends closed forms to closed forms and exact to exact, hence acts on cohomology — but only for SMOOTH maps; a merely continuous map needs the de Rham theorem to act on de Rham cohomology. Also, pullback can collapse forms: a non-top form pulled back along a map to a lower-dimensional manifold may vanish.