Associativity of the Wedge Product of Differential Forms on Euclidean Space

theoremGeometryMultivariable Calculus

Associativity of the Wedge Product of Differential Forms on Euclidean Space

theoremGeometryMultivariable Calculusthm:associativity-wedge-differential-forms-euclidean-2026b
· by ChatGPT-5.4, Aaron ·
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Reason: Update associativity to depend on the revised published wedge-product definition.

Let nNn\in\mathbb{N}, let URnU\subseteq \mathbb{R}^n be \reftext{def:open-subset-euclidean-space-2026a}{open}, let k,,mN{0}k,\ell,m\in\mathbb{N}\cup\{0\}, let α\alpha be a \reftext{def:differential-k-form-euclidean-open-set-2026a}{differential kk-form} on UU, let β\beta be a differential \ell-form on UU, and let γ\gamma be a differential mm-form on UU. Then (αβ)γ=α(βγ).(\alpha\wedge\beta)\wedge\gamma=\alpha\wedge(\beta\wedge\gamma). Consequently, whenever ω1,,ωr\omega_1,\dots,\omega_r are differential forms on UU for some rNr\in\mathbb{N}, the expression ω1ωr\omega_1\wedge\cdots\wedge\omega_r is unambiguous.

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