次の $k-$形式
\begin{align}
\omega &= \frac{1}{k!} \omega_{\mu_1, \cdots, \mu_k} {\rm d} x^{\mu_1} \wedge \cdots \wedge {\rm d} x^{m_k}
\end{align}
(ここに $\omega_{\mu_1, \cdots, \mu_k}$ は添字に対して完全反対称であるとする。)
に対して
\begin{align}
\omega(X_1, \cdots, X_k) &= \omega_{\mu_1, \cdots, \mu_k} X_1^{\mu_1} \cdots X_k^{\mu_k}
\end{align}
を示せ。
\begin{align}
\omega(X_1, \cdots, X_k) &= \frac{1}{k!} \omega_{\mu_1, \cdots, \mu_k} {\rm d} x^{\mu_1} \wedge \cdots \wedge {\rm d} x^{m_k} X_1^{\nu_1} \cdots X_k^{\nu_k} \left(\frac{\partial}{\partial x^{\nu_1}}, \cdots, \frac{\partial}{\partial x^{\nu_k}}\right) \\
&= \frac{1}{k!} \omega_{\mu_1, \cdots, \mu_k} X_1^{\nu_1} \cdots X_k^{\nu_k}
\sum_{\sigma \in S_k} {\rm sgn}
\begin{pmatrix}
\mu_1 & \cdots & \mu_k \\
\sigma(\mu_1) & \cdots & \sigma(\mu_k) \\
\end{pmatrix}
\left(\delta_{\nu_1}^{\sigma(\mu_1)} \cdots \delta_{\nu_k}^{\sigma(\mu_k)}\right) \\
&= \omega_{\mu_1, \cdots, \mu_k} X_1^{\mu_1} \cdots X_k^{\mu_k}
\end{align}
