设 $\omega\in A^r(M)$, $X_1,\ldots,X_{r+1}$ 是 $M$ 上任意 $r+1$ 个光滑切向量场, 则

\[
\begin{split}
d\omega(X_1,\ldots,X_{r+1})&=\sum_{i=1}^{r+1}(-1)^{i+1}\bigl(\langle X_1\wedge\cdots\wedge\hat{X}_i\wedge\cdots\wedge X_{r+1},\omega\rangle\bigr)\\
&\quad +\sum_{1\leq i < j\leq r+1}\bigl\langle [X_i,X_j]\wedge\cdots\wedge\hat{X}_i\wedge\cdots\wedge\hat{X}_j\wedge\cdots\wedge X_{r+1},\omega\bigr\rangle
\end{split}
\]