二元函数微分中值定理中值点的分析性质

讨论二元函数微分中值定理中值点的连续性及可导性问题,给出二元函数微分中值定理中值点连续及偏导数存在的充分务停,同时给出计算其偏导数的公式。     

广义二阶导数和二阶混合偏导数之间的关系

给出并证明了函数在一点处广义二阶可导的一个充分条件,分析了二元函数在一点的广义二阶导数和二阶混合偏导数之间的关系.     

二元函数极值的一种新判别方法

通常都是利用二阶偏导数来判别二元函数 z =f (x,y)的极值存在性 .本文将讨论如何利用一阶偏导数来判别二元函数的极值存在性 .我们知道 ,在利用二阶偏导数判别 z =f (x,y)的极值时存在着两方面的不便 :1°要计算三个二阶偏导数值 ;2°当 [fxx .fyy -f2xy]( x0 ,y0 ) =0时 ,不能确定极值是否存在 .下面我们受一元函数极值判别的启发 ,利用一元函数的性质 ,研究如何用一阶偏导数判别二元函数的极值 .设二元函数 z =f (x,y)在点 (x0 ,y0 )的 δ-邻域 B| ( x0 ,y0 ) ={ (x,y) | 0 <(x -x0 ) 2 (y -y0 ) 2 <δ}内有连续偏导数 ,(x,y)是该邻域…     

关于Pliicker曲面的两点说明

<正> 的几何图象就是Pliicker曲面。该曲面又叫做普吕克圆锥面或拟圆柱面。在讨论二元函数的极限、连续、偏导数、全微分以及它们之间的关系时,由于(1)式给出的函数在点(0,0)处的独特性质,它常被各种微积分教本作为典型例子所采用。该函数     

谈多元微分学中几个概念间的关系

学习多元函数微分学,一定要弄清连续、偏导数、全微分、方向导数之间的关系,并与一元函数中连续、可导、可微之间的关系比较,看看有何类似.有何区别,才能更好地掌握和使用这些基本概念.从教材中我们知道这几个基本概念间的关系(以二元函数为例)由下面定理给出:     

模糊数值函数的强可导性、连续性、局部平均McShane可积性

通过所提出的模糊数值函数的强可导性和平均 Mc Shane可积性等概念 ,对模糊数值函数的连续性从导数和积分两方面进行了讨论和刻画 ,也回答了强可导性与强导数可积之间的关系 ,给出两个充分必要条件。     

多元函数微分学中的反例构造技巧

文中给出了多元函数微分学的四个定义"连续、可偏导、可微、偏导数连续"之间关系的反例及其构造思路.     

一个有趣的二元函数

在一元函数中,“可导”和“可微”是等价的,“可微”的重要性似乎不大明显.在二元函数中情形就不一样了.“两个偏导数都存在”不能保证可微,甚至不能保证连续.也不能保证有极限;由可微却可以得到上述所有其它性质.还可以保证有方向导数.所以,对于二元函数,“可微”和“有两个偏导数”不等价,“可微”有重要的作用.     

一类二元函数连续与可微条件的归纳与推广

多元函数的可导性、连续性、可微性是多元函数微分学的重要内容.本文基于多元微分学教学实践,归纳了一类二元函数的可偏导、连续、可微的充分条件,并对此类二元函数做一般性推广,得到了推广函数连续与可微的充分条件.     

多元函数微分学中几个基本概念之间的关系

针对多元函数微分学中用以刻画函数局部性态的基本概念,给出连续、偏导数、可微、方向导数之间的关系图,采用证明和举反例的方式．深入分析这些概念之间的关系．     

On Regularity of Local Minimizers of one Class of Quadratic Nondifferentiable Functionals

We establish the partial Holder continuity of a minimizing functions in a neighborhood of the interface and partial Holder continuity of the derivatives up to the boundary of the interface. It is shown, that the singular sets are absent in the case of two spatial variables. Bibliography: 9 titles.     

Inequalities for Partial Moduli of Continuity and?Partial Derivatives

We obtain pointwise and integral type estimates of higher-order partial moduli of continuity in C via partial derivatives. Also, a Gagliardo–Nirenberg type inequality for partial derivatives in a fixed direction is proved. Our methods enable us to study the case when different partial derivatives belong to different spaces, including the space L ¹.     

Beats and periodicity for the forced harmonic oscillator

This note presents some simple examples illustrating the differences between the concepts of continuity, existence of partial derivatives and differentiability.     

Extension of Functions from Isotropic Nikol’skii–Besov Spaces and Their Approximation together with Derivatives

Mathematical Notes - Isotropic Nikol’skii–Besov spaces with norms whose definition, instead of the modulus of continuity of certain order of partial derivatives of functions of fixed...     

On functions with derivatives in a Lorentz space

We establish a sharp integrability condition on the partial derivatives of a Sobolev mapping to guarantee that sets of measure zero get mapped to sets of measure zero. This condition is sharp also for continuity and differentiability almost everywhere. Received: 6 January 1999 / Revised version: 5 May 1999     

(K_1,K_2)-拟正则映射的L_p可积性

对于Ｓｏｂｏｌｅｖ类Ｗ１，ｎｌｏｃ（Ω，Ｒｎ）的ｎ维（Ｋ１，Ｋ２）－拟正则映射，建立了其偏微商的Ｌｐ可积性结果，从而得到映射的Ｈ¨ｏｌｄｅｒ连续性；并且给出了其对一类拟线性椭圆型方程组先验估计的应用     

On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom), provided that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Hölder continuous.     

Stability of distributed-parameter systems with retarded argument

Lyapunov functions are used to investigate the stability of processes described by a system of linear partial differential equations with retarded argument (for example: magnetohydrodynamic processes, elastic vibrations in aircraft, etc.). Some equations of the system may not involve time derivatives (for example, the equation of continuity in incompressible fluid flow, and the equation for the magnetic induction vector in the theory of electromagnetic phenomena). Such equations also arise when the order of a partial differential equation is reduced by introducing new notation for the space derivatives. A method is developed for investigating the stability of processes described by a system of this kind, some of whose equations do not contain time derivatives. Two constructions of the Lyapunov functions, as different integral quadratic forms, are proposed. Sufficient conditions for stability, in the form of inequalities relating the coefficients of the system, are established. As an example, the stability of the vibrations of a stretched string in a viscoelastic medium due to a distributed control force is considered.     

The Dini condition and regularity of weak solutions of elliptic equations

Regularity results in domains of Euclidean n-space are established for generalized solutions of second order elliptic equations for which the coefficients of the differential operator and the nonhomogeneous term satisfy a Dini criterion. Generalized solutions are shown to be essentially classical solutions and a bound for the modulus of continuity of second order partial derivatives of the solution is established which yields Weyl's lemma as a corollary. A differentiability theorem is also established for the case the terms of the equation have further differentiability properties.     

On the best approximation of functions ofn variables

We propose a new approach to the solution of the problem of the best approximation, by a certain subspace for functions ofn variables determined by restrictions imposed on the modulus of, continuity of certain partial derivatives. This approach is based on the duality theorem and on the representation of a function as a countable sum of simple functions. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1352–1359, October, 1999.