$k-$形式の計算

$$\begin{array}{rl}\omega (X_1,\cdots ,X_k)& =\frac{1}{k!}{\omega }_{{\mu }_1,\cdots ,{\mu }_k}\mathrm{d}{x}^{{\mu }_1}\wedge \cdots \wedge \mathrm{d}{x}^{m_k}{X}_1^{{\nu }_1}\cdots X_k^{{\nu }_k}(\frac{\partial }{\partial x^{{\nu }_1}},\cdots ,\frac{\partial }{\partial x^{{\nu }_k}})\\ & =\frac{1}{k!}{\omega }_{{\mu }_1,\cdots ,{\mu }_k}{X}_1^{{\nu }_1}\cdots X_k^{{\nu }_k}\sum _{\sigma \in S_k}\mathrm{s}\mathrm{g}\mathrm{n}\left(\begin{array}{ccc}{\mu }_1& \cdots & {\mu }_k\\ \sigma ({\mu }_1)& \cdots & \sigma ({\mu }_k)\end{array}\right)({\delta }_{{\nu }_1}^{\sigma ({\mu }_1)}\cdots {\delta }_{{\nu }_k}^{\sigma ({\mu }_k)})\\ & ={\omega }_{{\mu }_1,\cdots ,{\mu }_k}{X}_1^{{\mu }_1}\cdots X_k^{{\mu }_k}\end{array}$$