# 外形式的外积

$f$是向量空间$V$上的$r$次外形式，令$({A}_{r}(f))$$\left({u}_{{1}},···,{u}_{{r}} \right)$$=$$\cfrac{1}{r!}$$\delta_\left( 1···r\right)^\left({i}_{1}···{i}_{r} \right)$$f$$\left({u}_{{i}_{1}},···,{u}_{{i}_{r}} \right)$（求和约定）

\begin{align*}
A_q \circ a_p(f) = A_q(f).
\end{align*}

\begin{align*}
(f \wedge g) \wedge h
&= \frac{(r+s+t)!}{(r+s)!t!} A_{r+s}((f \wedge g) \otimes h) \\
&= \frac{(r+s+t)!}{(r+s)!t!} A_{r+s+t} \left ( \frac{(r+s)!}{r!s!} A_{r+s}(f \otimes g) \otimes h \right ) \\
&= \frac{(r+s+t)!}{r!s!t!} A_{r+s+t} \circ a_{r+s} (f \otimes g \otimes h) \\
&= \frac{(r+s+t)!}{r!s!t!} A_{r+s+t}(f \otimes g \otimes h),
\end{align*}

\begin{align*}
& a_{r+s}(f \otimes g \otimes h) (u_1, \cdots, u_{r+s+t}) \\
&= \frac{1}{(r+s)!} \delta^{i_1 \cdots i_{r+s}}_{1 \cdots r+s} (f \otimes g \otimes h) (u_{i_1}, \cdots u_{i_{r+s}}, u_{r+s+1}, \cdots, u_{r+s+t})\\
&= \frac{1}{(r+s)!} \delta^{i_1 \cdots i_{r+s}}_{1 \cdots r+s} (f \otimes g)(u_{i_1}, \cdots, u_{i_{r+s}}) \cdot h(u_{r+s+1}, \cdots, u_{r+s+t}) \\
&= A_{r+s} (f \otimes g) \otimes h (u_1, \cdots, u_{r+s+t}).
\end{align*}