轮换对称性在积分中的应用

在某些积分的计算过程中,若积分区域具备轮换对称性,则可以简化积分的计算过程.本文讨论了利用轮换对称性简化二重积分,三重积分,第一,二类曲线积分,第一,二类曲面积分的计算方法.(以下都在积分存在下予以讨论)     

对称法求积分

积分计算是高等数学的基本运算 ,巧妙地利用对称性解积分题 ,常能化难为易 ,简化计算 ,收到事半功倍的效果 ,本文拟就此方法作一探讨。　　一　利用函数奇偶性利用被积函数的奇偶性和积分区间关于原点的对称性简化计算 ,是积分运算中经常使用的方法。例 1　求积分 I =∫1- 12 x2 +xcosx1 +1 -x2 dx解　本题中虽然积分区间关于原点对称 ,但被积函数不具奇偶性 ,但通过拆项 ,可利用奇偶性来简化积分运算。原积分 I =∫1- 12 x21 +1 -x2 dx +∫1- 1xcosx1 +1 -x2 dx △ I1+I2 .因为 xcosx1 +1 -x2 是奇函数 ,而 2 x21 +1 -x2 是偶函数 ,所以 …     

关于对称性在积分计算中的应用补遗

《高等数学研究》杂志第 4卷第 1期介绍了对称性在二重积分、三重积分、第一型曲线积分和第一型曲面积分计算中的应用 ,其方法可参见该期杂志 P2 4-2 7。除以上应用外 ,本文还要介绍对称性在第二型曲线积分和第二型曲面积分计算中的应用。一、对称性在第二型曲线积分计算中的应用定理 1　设分段光滑的平面曲线 L关于 x轴对称 ,且 L在上半平面的部分 L1与在下半平面的部分 L2 的方向相反 ,则( 1 )若 P( x,y)关于变量 y是偶函数 ,则∫LP( x,y) dx =0( 2 )若 P( x,y)关于变量 y是奇函数 ,则∫LP( x,y) dx =2 ∫L1P( x,y) dx图 1证 :由 L …     

对称性在积分计算中的应用

一、引言 在积分的计算中充分利用积分区域的对称性及被积函数的奇、偶性,往往可以简化计算,达到事半功倍的效果.近年来,在全国研究生入学考试数学试题中不乏涉及对称性的积分试题.本文拟系统地介绍有关内容并举出相关例子.为简化叙述,我们假定以下涉及到的积分都是存在的,有关函数均满足通常的条件.     

对称性在积分计算中的应用

阐述了对称性在在多元函数积分下的性质,并借助于实例说明对称性在重积分、曲线积分和曲面积分计算中的应用.     

浅谈对称性在积分计算中的应用

本文讨论对称性在积分计算中的一些灵活运用.     

使用对称性计算一类有限元基函数的二重积分

本文使用对称性计算一类有限元基函数在非规则区域的二重积分.通过两个算例验证对称性技巧在重积分计算上所带来的极大便利性.本文的内容进一步说明了使用对称性进行重积分计算在其他学科的应用价值.     

对称性及其在曲面积分计算中的应用

讨论了曲面积分中的奇偶对称性和轮换对称性问题,并通过具体例子说明了对称性在曲面积分计算中的作用.     

多元函数积分的简化计算

利用多元函数积分区域的对称性,可通过对被积函数以及积分区域的变换来简化多元函数积分的计算.     

关于分段函数定积分的计算

通过例题分别讨论分段连续函数定积分的计算及分段有界非连续函数定积分的计算,旨在于进一步丰富高等数学的教学内容,提高学生的计算能力.     

对称性在重积分及曲面积分中的应用

在积分区域具有某种对称性时,给出重积分及曲面积分所具有的相应性质,并通过例题给出这些性质在重积分及曲线、曲面积分中的应用方法.     

Symmetries of nonlinear hyperbolic systems of the Toda chain type

We consider hyperbolic systems of equations that have full sets of integrals along both characteristics. The best known example of models of this type is given by two-dimensional open Toda chains. For systems that have integrals, we construct a differential operator that takes integrals into symmetries. For systems of the chosen type, this proves the existence of higher symmetries dependent on arbitrary functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 344–355, May, 2008.     

λ‐symmetries,nonlocal transformations and first integrals to a class of Painlevé–Gambier equations

In this work, we consider a class of Painlevé–Gambier equations that model the motion of chain ball drawing with constant force in the frictionless surface. λ‐symmetries, first integrals, integrating factors, nonlocal transformations and local transformations are derived by using the some recent studies that are proposed by Muriel and Romero. Copyright © 2012 John Wiley & Sons, Ltd.     

On symmetries of stochastic differential equations

This note can be considered as a supplement to article [8]. Its purpose is twofold. First, to show that symmetries of Itô stochastic differential equations form a Lie algebra. Second, to provide more precise formulation of the relation between symmetries of SDEs and symmetries of the associated Fokker–Planck equation. Relation between first integrals of SDEs and symmetries of the associated Fokker–Planck equation is also considered.     

Algebraic properties of first integrals for systems of second-order ordinary differential equations of maximal symmetry

Symmetries of the first integrals for scalar linear or linearizable secondorder ordinary di?erential equations (ODEs) have already been derived and shown to exhibit interesting properties. One of these is that the symmetry algebra sl(3, IR) is generated by the three triplets of symmetries of the functionally independent first integrals and its quotient. In this paper, we first investigate the Lie-like operators of the basic first integrals for the linearizable maximally symmetric system of two second-order ODEs represented by the free particle system, obtainable from a complex scalar free particle equation, by splitting the corresponding complex basic first integrals and its quotient as well as their associated symmetries. It is proved that the 14 Lie-like operators corresponding to the complex split of the symmetries of the functionally independent first integrals I₁, I₂ and their quotient I₂/I₁ are precisely the Lie-like operators corresponding to the complex split of the symmetries of the scalar free particle equation in the complex domain. Then, it is shown that there are distinguished four symmetries of each of the four basic integrals and their quotients of the two-dimensional free particle system which constitute four-dimensional Lie algebras which are isomorphic to each other and generate the full symmetry algebra sl(4, IR) of the free particle system. It is further shown that the (n + 2)-dimensional algebras of the n + 2 first integrals of the system of n free particle equations are isomorphic to each other and generate the full symmetry algebra sl(n + 2, IR) of the free particle system.     

Generalized symmetries and integrals of motion for the Hénon-Heiles model

Generalized symmetries of the Hénon-Heiles model are examined for the purpose of finding parameter values at which integrals of motion exist. In these instances, the integrals of motion can be calculated by applying the divergence theorem.     

A relationship between λ‐symmetries and first integrals for ordinary differential equations

In this paper, we provide some geometric properties of λ‐symmetries of ordinary differential equations using vector fields and differential forms. According to the corresponding geometric representation of λ‐symmetries, we conclude that first integrals can also be derived if the equations do not possess enough symmetries. We also investigate the properties of λ‐symmetries in the sense of the deformed Lie derivative and differential operator. We show that λ‐symmetries have the exact analogous properties as standard symmetries if we take into consideration the deformed cases.     

Variational symmetries of Euler and non-Euler functionals

We develop a unified approach to the investigation of invariant properties of Euler and non-Euler functionals and establish a relationship of variational symmetries with first integrals of a given evolution operator equation of second order with respect to t. In addition, we investigate the properties of the generators of divergence symmetries.     

Invariants of Isospectral Deformations and Spectral Rigidity

We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace–Beltrami operator on a compact Riemannian manifold with Robin boundary conditions. Given a Kronecker invariant torus Λ of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on Λ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with Λ. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. These invariants are obtained from the first Birkhoff invariant of the microlocal monodromy operator associated to Λ. As an application we prove spectral rigidity of the Robin boundary conditions in the case of Liouville billiard tables of dimension two in the presence of a (?/2?)² group of symmetries.     

第一类曲面积分的一种解法

运用第一类曲线积分方法解决一类特殊的第一类曲面积分问题,并举例说明此方法的简便性.