复合函数求偏导数法则的证明一般书中都有毛病

本文指出了一般书中求复合函数偏导数法则的证明中的一个普遍毛病,并提出了改正的方法     

方向导数的计算公式的注记

方向导数本质上也是函数的一种变化率.利用向量的Schmidt正交化方法进行坐标变换,将方向导数转换为对新变量的偏导数,再结合多元复合函数的求导法则,给出方向导数计算公式的一种新的证明.     

梯度与等高线的教学体会

根据多元函数的偏导数可以确定等高线凹侧的一个法向量是梯度,该结果进一步揭示了梯度的几何意义,可以作为高等数学教学中方向导数一节的补充     

浅谈导数学定义在导数计算的中地位与作用

不少学生学习了求导公式后 ,往往对导数定义不太重视。其实 ,导数的定义不仅是导数的原始基本概念 ,而且它在求极限、求导数的计算及证明中都有着重要的、甚至是不可替代的作用。本文仅就导数定义在导数计算中的地位与作用问题谈点粗浅的认识 ,以期学生对此问题引起重视。一、在分段函数求导计算中的情形对分段函数分段点的导数的计算 ,必须按定义求 ,不能套公式。例 1　设 f ( x) =e|x|,求 f′( x)。[错解 ]　因为 f ( x) =ex,　　 x≥ 0e- x,　 x <0 ,所以 ,f′( x) =ex,　　 x≥ 0-e- x,　 x <0[辨析 ]　 x=0是分段点 ,而对分段点的导数 …     

小议导数与原函数

本文从分析导数与原函数定义入手，探讨求原函数较求导数困难的原因．     

Stokes问题有限元逼近中求导数的提取法

1.引言 我们知道Poisson方程和平面弹性问题的解的导数的近似值可以通过所谓提取公式得到,而不必对近似解直接求导数.这样我们可以得到具有与近似解本身同阶精度的导数的近似值.这一方法已被用于基于插值误差的后验误差估计及相应的自适应有限元方法中本文将这一方法应用于Stokes问题的有限元逼近,从Stokes方程的解的     

多元微分学中几个重要概念间的因果关系

通过一些反例说明多元函数的极限、连续、偏导数存在、方向导数存在、可微等概念之间的关系。     

应用导数新解一道新编综合题

利用导数求最值,可以化难为易、变特法为通法,我们要从中得到启迪,对该法熟练地加以运用.下面利用导数知识对《数学通讯》2002年第1期P36的综合题28加以解答.并对两种解法加以比较.     

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．     

从一首古诗引入偏导数概念

将苏轼的诗《题西林壁》引入高等数学课堂教学,结合一元函数导数的几何意义,利用＂峰＂和＂岭＂的不同含义,从苏诗切入到多元函数偏导数概念的引入和偏导数的几何意义的讲解,以图改善教学手段,丰富授课方式并增加数学知识的人文气息.     

Symmetric and quantum symmetric derivatives of Lipschitz functions

The symmetric derivative of a real valued function f at the real number x is defined to be

     

利用导数证明一类函数不等式

本文给出一个关于函数不等式的命题，用之证明了一类函数不等式．     

Pre-Schwarz导数意义下区域之间的距离

Lehto曾用Schwarz导数定义了边界多于一点的两个单连通区域的Mbius等价类之间的"距离",并猜测它是一个距离.但最近Bozin和Markovic否定了这一猜想.一个自然的问题就是:在Pre-Schwarz导数意义相应情况如何?用Pre-Schwarz导数给出了边界多于一点的两个单连通区域的仿射等价类之间的"距离",并证明了这样定义的"距离"是一个伪距离,即使将其限制在由具有解析边界的单连通区域的仿射等价类空间上也是如此.     

Existence Results for the Higher-Order Weighted Caputo-Fabrizio Fractional Derivative

By the definition of the higher-order fractional derivative, we explore the central properties of the higher-order Caputo-Fabrizio fractional derivative and integral with a weighted term. Furthermore, by dint of Schaefer''s fixed point theorem, $\alpha$-$\psi$-Contraction theorem, etc., we establish the existence of solutions for nonlinear equations. We also give three examples to make our main conclusion clear.     

Numerical solution of fractional oscillator equation

We focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval. This type of equation includes a complex form of left- and right-sided fractional derivatives. Its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations. Here we elaborated two numerical schemes being dependent on a fractional order of the equation. The results of numerical calculations are compared with analytical solutions. Then we illustrate convergence and stability of our schemes.     

关于时标上的适应Nabla分数阶导数

本文研究了时标上的适应Nabla分数阶导数的问题.利用时标理论,获得了关于适应Nabla分数阶导数的若干重要性质.这些结果推广并改进了文献[9,10]中的有关结论以及一般Nabla导数的性质.     

On Conformable Delta Fractional Derivative on Time Scales

In this paper, we introduce and investigate the concept of conformable delta fractional derivative on time scales. By using the theory of time scales, we obtain some basic properties of the conformable delta fractional derivative. Our results extend and improve both the results in [9] and the usual delta derivative.     

On connections between approximate second-order directional derivative and second-order Dini derivative for convex functions

For a real-valued convex functionf, the existence of the second-order Dini derivative assures that of the limit of the approximate second-order directional derivativef (x ₀;d, d) when 0⁺ and both values are the same. The aim of the present work is to show the converse of this result. It will be shown that upper and lower limits of the approximate second-order directional derivative are equal to the second-order upper and lower Dini derivatives, respectively. Consequently the existence of the limit of the approximate second-order directional derivative and that of second-order Dini derivative are equivalent.Dedicated to Professor N. Furukawa of Kyushu University for his 60th birthday.     

New approach to a generalized fractional integral

The paper presents a new fractional integration, which generalizes the Riemann-Liouville and Hadamard fractional integrals into a single form. Conditions are given for such a fractional integration operator to be bounded in an extended Lebesgue measurable space. Semigroup property for the above operator is also proved. We give a general definition of the fractional derivatives and give some examples.