What is Pullback? — Mathematics

Definition 21.10 University Mathematics — Year 3 · Chapter 21 — Differential Forms and Stokes’ Theorem

Let $\varphi\colon U \to V$ be smooth ( $U \subseteq \R^m$ , $V \subseteq \R^n$ open). The pullback $\varphi^*\colon \Omega^k(V) \to \Omega^k(U)$ is defined pointwise by the linear pullback along the differential: $(\varphi^*\omega)_x = (D\varphi(x))^*\,\omega_{\varphi(x)}$ . Concretely, $\varphi^*$ substitutes: $\varphi^*f = f \circ \varphi$ on functions, $\varphi^*(\dd y_i) = \dd\varphi_i = \sum_j\frac{\partial\varphi_i}{\partial x_j}\dd x_j$ , and $\varphi^*(a\,\dd y_{i_1}\wedge\dots\wedge \dd y_{i_k}) = (a\circ\varphi)\,\dd\varphi_{i_1}\wedge\dots\wedge \dd\varphi_{i_k}$ .

Examples

Example 21.12 (Polar coordinates)

For $\varphi(r, \theta) = (r\cos\theta, r\sin\theta)$ : $\varphi^*\dd x = \cos\theta\,\dd r - r\sin\theta\,\dd\theta$ , $\varphi^*\dd y = \sin\theta\,\dd r + r\cos\theta\,\dd\theta$ , so

\varphi^*(\dd x \wedge \dd y) = (\cos\theta\,\dd r - r\sin\theta\,\dd\theta) \wedge (\sin\theta\,\dd r + r\cos\theta\,\dd\theta) = r\,\dd r \wedge \dd\theta,

the Jacobian of Example 11.12 appearing by pure algebra — no measure theory. Exercise 21.9 turns this remark into a statement: for oriented integrals, the change-of-variables formula is the pullback formula.

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