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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 Liouville's formula ： ウィキペディア英語版 Liouville's formula In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system. The formula is named after the French mathematician Joseph Liouville. Jacobi's formula provides another representation of the same mathematical relationship. Liouville's formula is a generalization of Abel's identity and can be used to prove it. Since Liouville's formula relates the different linearly independent solutions of the system of differential equations, it can help to find one solution from the other(s), see the example application below. ==Statement of Liouville's formula== Consider the -dimensional first-order homogeneous linear differential equation :$y\text{'}=A(x)y\backslash ,$ on an interval of the real line, where for denotes a square matrix of dimension with real or complex entries. Let denote a matrix-valued solution on , meaning that each is a square matrix of dimension with real or complex entries and the derivative satisfies :$\backslash Phi\text{'}(x)=A(x)\backslash Phi(x),\backslash qquad\; x\backslash in\; I.$ Let :$\backslash mathrm\backslash ,A(\backslash xi)=\backslash sum\_^n\; a\_(\backslash xi),\backslash qquad\; \backslash xi\backslash in\; I,$ denote the trace of , the sum of its diagonal entries. If the trace of is a continuous function, then the determinant of satisfies :$\backslash det\backslash Phi(x)=\backslash det\backslash Phi(x\_0)\backslash ,\backslash exp\backslash biggl(\backslash int\_^x\; \backslash mathrm\backslash ,A(\backslash xi)\; \backslash ,\backslash textrm\backslash xi\backslash biggr)$ for all and in . 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「Liouville's formula」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.