Proof of Permutation Rule for Wedge Products of Coordinate 1-Forms

By Every Permutation is a Product of Adjacent Transpositions, there exist an integer $N\ge 0$ and indices $r_1,\dots,r_N\in\{1,\dots,k-1\}$ such that

where for each $m\in\{1,\dots,k-1\}$ the adjacent transposition $\tau_m$ swaps $m$ and $m+1$ and fixes all other indices.

For $j\in\{0,\dots,N\}$, define permutations

for $j\ge 1$. Then $\sigma_N=\sigma$. Also define

We will compare $\Omega_j$ and $\Omega_{j-1}$.

Fix $j\in\{1,\dots,N\}$ and write $r=r_j$. Since $\sigma_j=\tau_r\circ\sigma_{j-1}$, the tuple

is obtained from

by swapping the entries in positions $r$ and $r+1$ and leaving all other positions unchanged. Thus there are wedge products $A$ and $B$ of coordinate $1$-forms such that

for suitable coordinate indices $p\neq q$.

By the definition Wedge Product of Differential Forms on Euclidean Space of the wedge product of two $1$-forms,

for every point $x\in U$ and all vectors $v_1,v_2\in\mathbb{R}^n$. Interchanging $p$ and $q$ gives

Hence

Using associativity from Associativity of the Wedge Product of Differential Forms on Euclidean Space, it follows that

Since this holds for every $j\in\{1,\dots,N\}$, repeated substitution yields

That is,

Finally, by Sign of a Product of Adjacent Transpositions, the same decomposition of $\sigma$ satisfies

Substituting this into the previous display gives

This proves the theorem.