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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 D'Alembert operator ： ウィキペディア英語版 D'Alembert operator In special relativity, electromagnetism and wave theory, the d'Alembert operator (represented by a box: $\backslash Box$), also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist Jean le Rond d'Alembert. In Minkowski space in standard coordinates (''t'', ''x'', ''y'', ''z'') it has the form: : $$ \begin \Box & = \partial^\mu \partial_\mu = g^ \partial_\nu \partial_\mu = \frac - \frac - \frac - \frac \\ & = \frac - \nabla^2 = \frac - \Delta. \end Here $\backslash nabla^2$ is the 3-dimensional Laplacian and $g^$ is the inverse Minkowski metric with $g\_\; \backslash ,=\backslash ,\; 1$, $g\_\; \backslash ,=\backslash ,\; g\_\; \backslash ,=\backslash ,\; g\_\; \backslash ,=\backslash ,\; -1$, $g\_\; \backslash ,=\backslash ,\; 0$ for $\backslash mu\; \backslash ,\backslash neq\backslash ,\; \backslash nu$. Note that the μ and ν summation indices range from 0 to 3: see Einstein notation. We have assumed units such that the speed of light $c\; \backslash ,=\backslash ,\; 1$. Some authors also use the negative metric signature of (+ + + ) with $g\_\; \backslash ,=\backslash ,\; -1,\backslash ;\; g\_\; \backslash ,=\backslash ,\; g\_\; \backslash ,=\backslash ,\; g\_\; \backslash ,=\backslash ,\; 1$. Lorentz transformations leave the Minkowski metric invariant, so the d'Alembertian is a Lorentz scalar. The above coordinate expressions remain valid for the standard coordinates in every inertial frame. ==Alternate notations== There are a variety of notations for the d'Alembertian. The most common is the symbol $\backslash scriptstyle\backslash Box$ (Unicode: ): the four sides of the box representing the four dimensions of space-time and the $\backslash scriptstyle\backslash Box^2$ which emphasizes the scalar property through the squared term (much like the Laplacian). This symbol is sometimes called the quabla (''cf''. nabla symbol). In keeping with the triangular notation for the Laplacian sometimes $\backslash scriptstyle\backslash Delta\_M$ is used. Another way to write the d'Alembertian in flat standard coordinates is $\backslash scriptstyle\backslash partial^2$. This notation is used extensively in quantum field theory where partial derivatives are usually indexed: so the lack of an index with the squared partial derivative signals the presence of the D'Alembertian. Sometimes $\backslash scriptstyle\backslash Box$ is used to represent the four-dimensional Levi-Civita covariant derivative. The symbol $\backslash scriptstyle\backslash nabla$ is then used to represent the space derivatives, but this is coordinate chart dependent. 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「D'Alembert operator」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.