ScalingStacks

Proposition 2.10 (Green’s formula) [0351]

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Proposition 2.10 (Green’s formula)

We consider a nn-dimensional integral ℝ{\mathbb{R}}-affine polyhedron σ\sigma contained in the open subset UU of NℝN_{\mathbb{R}}. Assume that α∈Ap,p​(U)\alpha\in A^{p,p}(U) and β∈Aq,q​(U)\beta\in A^{q,q}(U) are symmetric with p+q=n−1p+q=n-1 and that the intersection of the supports of α\alpha and β\beta is compact. Then we have

∫σα∧d′​d′′​β−β∧d′​d′′​α=∫∂σα∧d′′​β−β∧d′′​α.\int_{\sigma}\alpha\wedge d^{\prime}d^{\prime\prime}\beta-\beta\wedge d^{\prime}d^{\prime\prime}\alpha=\int_{\partial\sigma}\alpha\wedge d^{\prime\prime}\beta-\beta\wedge d^{\prime\prime}\alpha.

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