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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 Mean curvature ： ウィキペディア英語版 Mean curvature In mathematics, the mean curvature $H$ of a surface $S$ is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was introduced by Sophie Germain in her work on elasticity theory.〔Marie-Louise Dubreil-Jacotin on (Sophie Germain )〕 It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which by the Young–Laplace equation have constant mean curvature. ==Definition== Let $p$ be a point on the surface $S$. Each plane through $p$ containing the normal line to $S$ cuts $S$ in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated (always containing the normal line) that curvature can vary, and the maximal curvature $\backslash kappa\_1$ and minimal curvature $\backslash kappa\_2$ are known as the ''principal curvatures'' of $S$. The mean curvature at $p\backslash in\; S$ is then the average of the principal curvatures , hence the name: :$H\; =\; (\backslash kappa\_1\; +\; \backslash kappa\_2).$ More generally , for a hypersurface $T$ the mean curvature is given as :$H=\backslash frac\backslash sum\_^\; \backslash kappa\_.$ More abstractly, the mean curvature is the trace of the second fundamental form divided by ''n'' (or equivalently, the shape operator). Additionally, the mean curvature $H$ may be written in terms of the covariant derivative $\backslash nabla$ as :$H\backslash vec\; =\; g^\backslash nabla\_i\backslash nabla\_j\; X,$ using the ''Gauss-Weingarten relations,'' where $X(x)$ is a smoothly embedded hypersurface, $\backslash vec$ a unit normal vector, and $g\_$ the metric tensor. A surface is a minimal surface if and only if the mean curvature is zero. Furthermore, a surface which evolves under the mean curvature of the surface $S$, is said to obey a heat-type equation called the mean curvature flow equation. The sphere is the only embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true when the condition "embedded surface" is weakened to "immersed surface".〔http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102702809〕 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「Mean curvature」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.