Let \(\omega\) be a differential \(k\)-form and \(F\colon M\to N\) smooth. Then the pullback in local coordinates is given by
\begin{equation*} F^*(\sum_{I}' \omega_Idy^I)= \sum_{I}' (\omega\circ F)d(y^{i_1}\circ F)\wedge \cdots \wedge d(y^{i_k}\circ F). \end{equation*}[@lee2013smooth_manifolds, Lemma 14.16] This follows from the local coordinate representation of \(\omega\), definition and properties of the pullback.
Remarks
- If \(M\) and \(N\) are \(n\)-dimensional we obtain the more specific formula. Let \((x^i)\) and \((y^i)\) are smooth coordinates on open subset \(U\subset M\) and \(V\subset N\), respectively, and \(u\) is a continuous real-valued function on \(V\), then the following equality holds on \(U\cap F^{-1}(V)\). \begin{equation*} F^*(u dy^1\wedge \cdots \wedge dy^n)=(u\circ F) (\det DF) dx^1\wedge \cdots dx^n, \end{equation*}