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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 Vector Laplacian ： ウィキペディア英語版 Vector Laplacian In mathematics and physics, the vector Laplace operator, denoted by $\backslash scriptstyle\; \backslash nabla^2$, named after Pierre-Simon Laplace, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian. Whereas the scalar Laplacian applies to scalar field and returns a scalar quantity, the vector Laplacian applies to the vector fields and returns a vector quantity. When computed in rectangular cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied on the individual elements. ==Definition== The vector Laplacian of a vector field $\backslash mathbf$ is defined as :$\backslash nabla^2\; \backslash mathbf\; =\; \backslash nabla(\backslash nabla\; \backslash cdot\; \backslash mathbf)\; -\; \backslash nabla\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf).$ In Cartesian coordinates, this reduces to the much simpler form: :$\backslash nabla^2\; \backslash mathbf\; =\; (\backslash nabla^2\; A\_x,\; \backslash nabla^2\; A\_y,\; \backslash nabla^2\; A\_z),$ where $A\_x$, $A\_y$, and $A\_z$ are the components of $\backslash mathbf$. This can be seen to be a special case of Lagrange's formula; see Vector triple product. For expressions of the vector Laplacian in other coordinate systems see Nabla in cylindrical and spherical coordinates. 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「Vector Laplacian」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.