外形式的外积

青青子衿

\(f\)是向量空间\(V\)上的\(r\)次外形式，令
\[({A}_{r}(f))\left({u}_{{1}},···,{u}_{{r}} \right)=\cfrac{1}{r!}\delta_\left( 1···r\right)^\left({i}_{1}···{i}_{r} \right)\left({u}_{{i}_{1}},···,{u}_{{i}_{r}} \right)\]（求和约定）
定义外积
\[f\wedge g=\cfrac{(r+s)!}{r!s!}({A}_{r+s}(f\otimes g))
\]
(\(f\),\(g\)是向量空间\(V\)上的\(r\),\(s\)次外形式)
若\(f\),\(g\),\(h\)是向量空间\(V\)上的\(r\),\(s\),\(t\)次外形式
证明:
\[(f\wedge g)\wedge h=\cfrac{(r+s+t)!}{r!s!t!}({A}_{r+s+t}(f\otimes g \otimes h))\]

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How to prove that the wedge product is the determinant (Spivak's Claim)