Démonstration. [01U4] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
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Démonstration.
Comme les formes de bidegré ( k , k ) (k,k) j = 1 j=1
d ′′ ( u 1 d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ α ) OPEN = d ′′ u 1 ∧ d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ α ) + u 1 d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ d ′′ α , \mathop{\mathrm{d}^{\prime\prime}}(u^{1}\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\alpha)\\
=\mathop{\mathrm{d}^{\prime\prime}}u^{1}\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\alpha)+u^{1}\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\mathop{\mathrm{d}^{\prime\prime}}\alpha,
puis
d ′ d ′′ ( u 1 d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ α ) \displaystyle\hskip-28.45274pt\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}(u^{1}\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\alpha)\hskip-170.71652pt
OPEN = d ′ d ′′ u 1 ∧ d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ α ) − d ′′ u 1 ∧ d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ d ′′ α \displaystyle=\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{1}\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\alpha)-\mathop{\mathrm{d}^{\prime\prime}}u^{1}\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\mathop{\mathrm{d}^{\prime\prime}}\alpha
+ d ′ u 1 d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ d ′′ α − u 1 d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ d ′ d ′′ α \displaystyle\quad+\mathop{\mathrm{d^{\prime}}}u^{1}\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\mathop{\mathrm{d}^{\prime\prime}}\alpha-u^{1}\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}\alpha
OPEN = d ′ d ′′ u 1 ∧ d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ α ) − u 1 d ′ d ′′ u 2 ∧ ⋯ ∧ d ′ d ′′ u p ∧ d ′ d ′′ α . \displaystyle=\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{1}\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\alpha)-u^{1}\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{2}\wedge\dots\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}u^{p}\wedge\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}\alpha.
L’assertion résulte alors de la formule de Green.
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