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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 First fundamental form ： ウィキペディア英語版 First fundamental form In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R''3''. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space. The first fundamental form is denoted by the Roman numeral I, :$\backslash !\backslash mathrm(x,y)=\; \backslash langle\; x,y\; \backslash rangle.$ Let ''X''(''u'', ''v'') be a parametric surface. Then the inner product of two tangent vectors is :$$ \begin & where ''E'', ''F'', and ''G'' are the coefficients of the first fundamental form. The first fundamental form may be represented as a symmetric matrix. :$\backslash !\backslash mathrm(x,y)\; =\; x^T$ \begin E & F \\ F & G \endy ==Further notation== When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself. :$\backslash !\backslash mathrm(v)=\; \backslash langle\; v,v\; \backslash rangle\; =\; |v|^2$ The first fundamental form is often written in the modern notation of the metric tensor. The coefficients may then be written as $g\_$: : The components of this tensor are calculated as the scalar product of tangent vectors ''X''1 and ''X''2: :$g\_\; =\; X\_i\; \backslash cdot\; X\_j$ for ''i'', ''j'' = 1, 2. See example below. 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「First fundamental form」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.