翻訳と辞書 Words near each other ・ Lappa, Rethymno ・ Lappach ・ Lappajärvi ・ Lappalanjärvi ・ Lapja ・ Lapka ・ Lapko ・ Laplace (disambiguation) ・ Laplace Affair ・ Laplace distribution ・ Laplace equation for irrotational flow ・ Laplace expansion ・ Laplace expansion (potential) ・ Laplace formula ・ Laplace functional ・ Laplace invariant ・ Laplace Island ・ Laplace Island (Antarctica) ・ Laplace Island (Western Australia) ・ Laplace limit ・ Laplace no Ma ・ Laplace number ・ Laplace operator ・ Laplace operators in differential geometry ・ Laplace plane ・ Laplace pressure ・ Laplace principle (large deviations theory) ・ Laplace transform ・ Laplace transform applied to differential equations ・ Laplace's demon Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Laplace invariant ： ウィキペディア英語版 Laplace invariant In differential equations, the Laplace invariant of any of certain differential operators is a certain function of the coefficients and their derivatives. Consider a bivariate hyperbolic differential operator of the second order :$\backslash partial\_x\; \backslash ,\; \backslash partial\_y\; +\; a\backslash ,\backslash partial\_x\; +\; b\backslash ,\backslash partial\_y\; +\; c,\; \backslash ,$ whose coefficients :$a=a(x,y),\; \backslash \; \backslash \; b=c(x,y),\; \backslash \; \backslash \; c=c(x,y),$ are smooth functions of two variables. Its Laplace invariants have the form :$\backslash hat=\; c-\; ab\; -a\_x\; \backslash quad\; \backslash text\; \backslash quad\; \backslash hat=c-\; ab\; -b\_y.$ Their importance is due to the classical theorem: Theorem: ''Two operators of the form are equivalent under gauge transformations if and only if their Laplace invariants coincide pairwise.'' Here the operators :$A\; \backslash quad\; \backslash text\; \backslash quad\; \backslash tilde\; A$ are called ''equivalent'' if there is a gauge transformation that takes one to the other: :$\backslash tilde\; Ag=\; e^A(e^g)\backslash equiv\; A\_\backslash varphi\; g.$ Laplace invariants can be regarded as factorization "remainders" for the initial operator ''A'': :$\backslash partial\_x\backslash ,\; \backslash partial\_y\; +\; a\backslash ,\backslash partial\_x\; +\; b\backslash ,\backslash partial\_y\; +\; c\; =\; \backslash left\backslash$ (\partial_x + b)(\partial_y + a) - ab - a_x + c ,\\ (\partial_y + a)(\partial_x + b) - ab - b_y + c . \end\right. If at least one of Laplace invariants is not equal to zero, i.e. :$c-\; ab\; -a\_x\; \backslash neq\; 0\; \backslash quad\; \backslash text\; \backslash quad$ c- ab -b_y \neq 0, then this representation is a first step of the Laplace–Darboux transformations used for solving ''non-factorizable'' bivariate linear partial differential equations (LPDEs). If both Laplace invariants are equal to zero, i.e. :$c-\; ab\; -a\_x=0\; \backslash quad\; \backslash text\; \backslash quad$ c- ab -b_y =0, then the differential operator ''A'' is factorizable and corresponding linear partial differential equation of second order is solvable. Laplace invariants have been introduced for a bivariate linear partial differential operator (LPDO) of order 2 and of hyperbolic type. They are a particular case of ''generalized invariants'' which can be constructed for a bivariate LPDO of arbitrary order and arbitrary type; see Invariant factorization of LPDOs. ==See also== * Partial derivative * Invariant (mathematics) * Invariant theory 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「Laplace invariant」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.