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科学网微分大观园之外文课本

文章来源:admin    时间:2019-08-03 09:02

  

微分概念在整个微积分体系中占有重要地位。理解微分概念是微积分教育的重要环节。在历史上,微分的定义经历了很长时间的发展。牛顿、莱布尼兹是微积分的主要创建人,他们的微积分可以称为第一代微积分,第一代微积分的方法是没有问题的,而且获得了巨大的成功,但是对微分的定义(即微分的本质到底是什么)的说明不够清楚;以柯西、维尔斯特拉斯等为代表的数学家在极限理论的基础上建立了微积分原理,可以称之为第二代微积分,并构成当前教学中微积分教材的主要内容。第二代微积分与第一代微积分在具体计算方法上基本相同,第二代微积分表面上解决了微分定义的说明, 澳门威尼斯人网站官网,但是概念和推理繁琐迂回。

当前,围绕微分定义问题,国内外学术界已经开始形成一些讨论,参与者从科学院院士,中青年数学工作者,以致在读博士硕士,当然也包括一些毫无话语权的“N无数学家”。但真理面前人人平等,只要我们抱着持之有故言之成理的科学态度,相信会引发深刻的思考。

为了使得微分定义的讨论更加深入,并且有充足的养料支撑,有必要将古今中外现行微积分学术著作中的微分定义详细调查。从今天起,我将在我所搜集整理的微积分定义逐次摘录在网上,方便大家讨论。在摘录的同时,将做一些简单的讨论。




【外文】

1

书名

 

Calculus(6th   Edition)20 

 

主编

 

James   Stewart

 

出版社

 

Brooks   Cole

 

If we use the traditional notation $y = f ( x )$to indicate that the independent variable is $x$ and the dependent variable is $y$then some common alternative notations for the derivative are as follows

$f ^ { \prime } ( x ) = y ^ { \prime } = \frac { d y } { d x } = \frac { d f } { d x } = \frac { d } { d x } f ( x ) = D f ( x ) = D _ { x } f ( x )$

The symbols  $D $ and  $d/dx $ are called differentiation operators because they indicate the operation of differentiationwhich is the process of calculating a derivative.

The symbol $d y / d x$which was introduced by Leibnizshould not be regarded as a ratio

for the time being);it is simply a synonym for $\mathrm{f}(\mathrm{x})$.Nonethelessit is a very useful and suggestive notationespecially when used in conjunction with increment notation.Refer-

ring to Equation 3.1.6instantaneous rate of change$= \lim _ { \Delta x \rightarrow 0 } \frac { \Delta y } { \Delta x } = \lim _ { x _ { 2 } - x _ { 1 } } \frac { f \left( x _ { 2 } \right) - f \left( x _ { 1 } \right) } { x _ { 2 } - x _ { 1 } }$),we can rewrite the definition of derivative in Leibniz notation in the form $\frac { d y } { d x } = \lim _ { \Delta x \rightarrow 0 } \frac { \Delta y } { \Delta x }$

If we want to indicate the value of a derivative $d y / d x$ in Leibniz notation at a specific num-ber $a$we use the notation $\left. \frac { d y } { d x } \right| _ { x = a } \quad$ or $\quad \frac { d y } { d x } ] _ { x - a }$

which is a synonym for$f ^ { \prime } ( a )$ .

参考文献:

[1]   James Stewart.Calculus(6th Edition)[M].Brooks Cole.Year,2007:126.


2

书名

 

Calculus   with Analytic Geometry  

 

主编

 

George   F.Simmons

 

出版社

 

The McGraw-Hill Companies

 

To explain Leibniz's notation,we begin with a function $y=f(x)$ and write the difference quotient $\frac { f ( x + \Delta x ) - f ( x ) } { \Delta x }$

in the form

$\frac { \Delta y } { \Delta x }$

where $\Delta y = f ( x + \Delta x ) - f ( x )$.Here $\Delta y$ is not just any change in$y$it is the specific change that results when the independent variable is changed from $x$ to $x + \Delta x$ .As we knowthe difference quotient $\frac { \Delta y } { \Delta x }$ can be interpreted as the ratio of the change in $y$ to the change in $x$ along the curve $y = f ( x )$and this is the slope of the secantFig2.9.Leibniz wrote the limit of this difference quotientwhich of course is the derivative $f ^ { \prime } ( x )$in the form $d y / d x$read"$dy$ over$d x$".In this notationthe definition of the derivative becomes

 $\frac{d y}{d x}=http://blog.sciencenet.cn/\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$  (1)

and this is the slope of the tangent in Fig.2.9.Two slightly different equivalent forms of  $dy/dx $ are  $\frac{d f(x)}{d x}$ and $\frac{d}{d x} f(x)$.

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