\lemmname 5.7.1 . [01UF] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes 93 original structured objects have an unresolved mathematical role; their permanent tags identify source occurrences only. Complete original source context · Original author HTML
\lemmname 5.7.1 .
Soit U U 𝐑 n \mathbf{R}^{n} h : U → 𝐑 h\colon U\rightarrow\mathbf{R} 𝒞 2 \mathscr{C}^{2}
( d ′ d ′′ h ) n = n ! det ( ∂ 2 ∂ x i ∂ x j ) d ′ x 1 ∧ d ′′ x 1 … d ′ x n ∧ d ′′ x n . (\mathop{\mathrm{d}^{\prime}\mathrm{d}^{\prime\prime}}h)^{n}=n!\det\big(\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\big)\mathop{\mathrm{d^{\prime}}}x_{1}\wedge\mathop{\mathrm{d}^{\prime\prime}}x_{1}\dots\mathop{\mathrm{d^{\prime}}}x_{n}\wedge\mathop{\mathrm{d}^{\prime\prime}}x_{n}.