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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク De Gua's theorem ： ウィキペディア英語版 De Gua's theorem __notoc__ De Gua's theorem is a three-dimensional analog of the Pythagorean theorem and named for Jean Paul de Gua de Malves. If a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces. :$A\_^2\; =\; A\_^2+A\_^2+A\_^2$ ==Generalizations== The Pythagorean theorem and de Gua's theorem are special cases (''n'' = 2, 3) of a general theorem about ''n''-simplices with a right-angle corner. This, in turn, is a special case of a yet more general theorem, which can be stated as follows.〔Theorem 9 of James G. Dowty (2014). Volumes of logistic regression models with applications to model selection. 〕 Let ''P'' be a ''k''-dimensional affine subspace of $\backslash mathbb^n$ (so $k\; \backslash le\; n$) and let ''C'' be a compact subset of ''P''. For any subset $I\; \backslash subseteq\; \backslash$ with exactly ''k'' elements, let $C\_I$ be the orthogonal projection of ''C'' onto the linear span of $e\_,\; \backslash ldots,\; e\_$, where $I\; =\; \backslash$ and $e\_1,\; \backslash ldots,\; e\_n$ is the standard basis for $\backslash mathbb^n$. Then :$\backslash mbox\_k^2(C)\; =\; \backslash sum\_I\; \backslash mbox\_k^2(C\_I),$ where $\backslash mbox\_k(C)$ is the ''k''-dimensional volume of ''C'' and the sum is over all subsets $I\; \backslash subseteq\; \backslash$ with exactly ''k'' elements. This theorem is essentially the inner-product-space version of Pythagoras’ theorem applied to the ''k''th exterior power of ''n''-dimensional Euclidean space. De Gua's theorem and its generalisation (above) to ''n''-simplices with right-angle corners correspond to the special case where ''k'' = ''n''−1 and ''C'' is an (''n''−1)-simplex in $\backslash mathbb^n$ with vertices on the co-ordinate axes. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「De Gua's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.