Proposition 3.5 (Stokes’ formula) [0359] 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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Proposition 3.5 (Stokes’ formula)
Let ( 𝒞 , m ) ({\mathscr{C}},m) ℝ {\mathbb{R}} n n η ′ ∈ A c n − 1 , n ( 𝒞 ) \eta^{\prime}\in A_{c}^{n-1,n}({\mathscr{C}}) η ′′ ∈ A c n , n − 1 ( 𝒞 ) \eta^{\prime\prime}\in A_{c}^{n,n-1}({\mathscr{C}})
∫ ( 𝒞 , m ) d ′ η ′ = ∫ ∂ ( 𝒞 , m ) η ′ , ∫ ( 𝒞 , m ) d ′′ η ′′ = ∫ ∂ ( 𝒞 , m ) η ′ . \int_{({\mathscr{C}},m)}d^{\prime}\eta^{\prime}=\int_{\partial({\mathscr{C}},m)}\eta^{\prime},\quad\int_{({\mathscr{C}},m)}d^{\prime\prime}\eta^{\prime\prime}=\int_{\partial({\mathscr{C}},m)}\eta^{\prime}.