学习微积分60年有感（IV）

现在可以说，我这一辈子几乎都是在与微积分打交道，分不开。苦苦研习60年，人都快学傻了。现在，人老了，还整天唠叨不停，真烦人。
1957年秋，我进入南京大学数学天文系学习。当时，何旭初教授主讲一年级微积分课程，指定教学参考书是原苏联菲赫金哥尔茨的《微积分学教程》（共计三卷9大本），内容几乎涵盖了上世纪前半期世界数学分析的全部成就。

在此期间，对基于（ε，δ）方法的极限论，顶礼膜拜，视为神圣，从不怀疑。

在20年之后，1978年，全国改革开放，轰轰烈烈。我看准了Keisler的无穷小微积分，观点全变了。普及无穷小微积分，而且坚持不断，矢志不移。这样，又一路走过了人生的40年。

明天就是新年了。我把Keisler关于微积分的肺腑之言献给广大读者，作为新年的一份礼物。

袁萌 12月31日

附：基础微积分教材的“后记”
（读者请注意：在下面的英语句子中，有些空格被编辑器“吃”掉了。）

There is one fact of basic importance that we state now as a theorem.

THEOREM (Standard Part Principle)

For every finite hyperreal number b, there is exactly one realnumberr that is infinitely close to b.

PROOF We first show that there cannot be more than one realnumberinfinitely close to b.Suppose r and sare real numbers such that r ≈band s ≈ b. Then r ≈ s, and sincer and s are real, r must be equal to s .Thus there is at most one realnumber infinitely close to b.

We now show that there is a real number infinitely close to b. Let Abetheset of all real numbers less than b. Then any real number between twoelementsof Ais an element of A. By the Completeness Axiom for the realnumbers, Ais aninterval. Since the hyperreal number bis finite. Amust be aninterval of theform(－∝, r ) or (－∝, r ]for somereal number r. Every realnumber s belongs to A,so s < b. Also, every realnumber t > r does not belong to A, so t ≥ b. This showsthat ris infinitely close to b.
It was pointedoutearlier that the Completeness Axiom does not qualify as a real statement.Forthis reason, the Transfer Principle cannot be used to transfer theCompletenessAxiom to the hyperreal numbers. In fact, the Completeness Axiom isnot true forthe hyperreal numbers. By a closed hyperreal interval, we mean aset ofhyperreal numbers of the form [a, b], the set of all hyperreal numbersxforwhich a≤ x ≤ b, where aandbare hyperreal constants. Open and half-openhyperreal intervals are defined ina similar way. When we say that theCompleteness Axiom is not true for thehyperreal numbers, we mean that there actuallyare sets A of hyperreal numberssuch that:
(a) wheneverxandyare in A, any hyperreal number between xand yis in A.
(b) A is notahyperreal interval.
Here aretwoquitefamiliar examples.
EXAMPLE1 ThesetAof all infinitesimals has property (a ) above but is not a hyperrealinterval.It has property (a ) because any hyperreal number that is betweentwoinfinitesimals is itself infinitesimal. We show that Ais not ahyperrealinterval. Acannot be of the form [a, ∞] or (a,∞) becauseeveryinfinitesimal is less than 1. Acannot be of the form [a, b] or (a, b],becauseif bis positive infinitesimal, then 2·b is a largerinfinitesimal.Acannot be of the form [a,b) or (a,b), because if bis positive andnotinfinitesimal, then b/2 is less than b but still positive and notinfinitesimal.
The set B ofallfinite hyperreal numbers is another example of a set that has property (a)above but is not an interval.
Here aresomeexamples that may help to illustrate the nature of the hyperreal numbersystemand the use of the Transfer Axiom.

EXAMPLE2 let fbethe real function given by the equation.
Its graph istheunit semicircle with center at the origin. The following two realstatementshold for all real numbers x:
By theTransferAxiom, these real statements also hold for all hyperreal numbers x.Thereforethe natural extension f* of f is given by the sameequation
The domain off*isthe set of all hyperreal numbers between －1 and 1. Thehyperreal graph of f*, shownin Figure E.1, can be drawn on paper by drawingthe real graph of f(x)andtraining an infinitesimal microscope on certain keypoints.
EXAMPLE3 letfbethe identity function on the real numbers, f(x)=x. By the Transfer Axiom,theequation f(x)= x is true for all hyperreal x. Thus the natural extensionf*offis defined, and f*(x )=x for all hyperreal x. Figure E.2 shows thehyperrealgraph of f*. Under a microscope, it has a 45°slope.
Here is anexampleof a hyperreal function that is not the natural extension of a realfunction.

Figure E.1

Figure E.2

EXAMPLE4 One hyperreal function, which we have already studied insomedetail, is the standard part function st(x). For real numbers the standardpartfunction has the same values as the identity function,

st(x) = x forall real x.
However,thehyperreal graph of st(x), shown in Figure E.3, is very different fromthehyperreal graph of the identity function f*. The domain of the standardpartfunction is the set of all finite numbers, while f* has domain R*. Thusforinfinite x, f*(x) = x, but st(x)is undefined. If xis finite but not real,f*(x)=x but st(x)≠ x. Under the microscope, aninfinitesimal piece of the graphof the standard part function is horizontal,while the identity function has a45° slope.
The standardpartfunction is not the natural extension of the identity function, and henceisnot the natural extension of any real function.

Figure E.3
The standardpartfunction is the only hyperreal function used in this course except for naturalextensionsof real functions.
We conclude withafew words about the construction of the real and the hyperreal numbers.BeforeWeierstrass, the rational numbers were on solid ground but the realnumbersweresomething new. Before one could safely use the axioms for the realnumbers, ithad to be shown that the axioms did not lead to a contradiction.This was doneby starting with the rational numbers and constructing astructure whichsatisfied all the axioms for the real numbers. Since anythingproved from theaxioms is true in this structure, the axioms cannot lead to acontradiction.
The idea istoconstruct real numbers out of Cauchy sequences of rational numbers.
DEFINITION
A Cauchy Sequence is a sequence <________> of numberssuchthat for every real ε > 0 there is an integer ____ such that thenumbers

are all within ε of each other.

Two Cauchy sequences

of rational numbers are called Cauchy equivalent, in symbols<_____>=<____>, if the differencesequence

converges to zero. (Intuitively this means that the two sequences havethesame limit.)

PROPERTIES OF CAUCHY EQUIVALENCE

(1)If <_______> =<______> and<______> = <_______>
then the sum sequences are equivalent.

(2)Under the same hypotheses, the product sequences are equivalent,

(3)if__ = __ for all but finitely many n, then

The set of real numbers is then defined as the set of all equivalenceclassesof Cauchy sequences of rational numbers. A rational number rcorresponds to theequivalence class of the constant sequence < r, r,r,…>. The sum of theequivalence class of <___ > and theequivalence class of <___ > is defined as the equivalence class of thesum sequence

The product is defined in a similar way. It can be shown that alltheaxioms for the real numbers hold for this structure.

Today the real numbers are on solid ground and the hyperreal numbersare anew idea. Robinson used the ultraproduct construction of Skolem to showthat theaxioms for the hyperreal numbers (for example, as used in this book)do not leadto a contradiction. The method is much like the construction of thereal numbersfrom the rationals. But this time the real number system is thestarting point.We construct hyperrealnumbers out of arbitrary (not justCauchy) sequences ofreal numbers.

By an ultraproduct equivalence we mean an equivalence relation= on thesetof all sequences of real numbers which have the properties of Cauchyequivalence(1)-(3) and also

(4)If each __belongs to the set {0,1} the <__>isequivalent to exactly one of the constant sequences <0,0,0,…>or<1,1,1,…>.

Given an ultraproduct equivalence relation, the set of hyperrealnumbers isdefined as the set of all equivalence classes of sequences of realnumbers. Areal number rcorresponds to the equivalence class of the constantsequence<r, r,r…>. Sums and products are defined as forCauchysequences. The natural extension f*of a real function f(x)is defined sothatthe image of the equivalence class of <__> istheequivalence class of <___>. It can be provedthatultraproduct equivalence relations exist, and that all the axioms for therealand hyperreal numbers hold for the structure defined in this way.

When hyperreal numbers are constructed as equivalence classes ofsequencesof real numbers, we can give specific examples of infinite hyperrealnumbers.The equivalence class of
<1,2,3,……n,……>
is a positive infinite hyperreal number. The equivalence class of
<1,4,9……n2……>
is larger, and the equivalence class of
<1,2,4……__……>
is a still larger infinite hyperreal number.
We can also give examples of nonzero infinitesimals. Theequivalenceclasses of
<1, 1/2,1/3……1/n, ……>,
<1,1/4,1/9……n–2,……>,
and <1,1/2,1/4……___________,……>,
are progressively smaller positive infinitesimals.

The mistake of Leibniz and his contemporaries was to identify alltheinfinitesimals with zero. This leads to an immediate contradictionbecausedy/dxbecomes 0/0. In the present treatment the equivalence classes of

<1, 1/2,1/3, ……, 1/n, ……>
and <0, 0, ……0, ……>
are different hyperreal numbers. They are not equal but merely havethesame standard part, zero. This avoids the contradiction and once againmakesinfinitesimals a mathematically sound method.

For more information about the ideas touched on in this epilogue, see theinstructor’s supplement, Foundations of Infinitesimal Calculus,which has as elf-containedtreatment of ultraproducts and the hyperreal numbers.

FOR FURTHER READING ON THE HISTORY OF THE CALCULUS SEE:
The History of the Calculus and its Conceptual Development: Carl c.Boyer,Dover, New York, 1959.
Mathematical Thought from Ancient to Modern Times; Morris Kline,OxfordUniv.
Press, New York, 1972.
No-standard Analysis: Abraham Robinson, North-Holland, Amsterdam,London,1966.

FOR ADVANCED READING ON INFINITESIMAL ANALYSIS SEE NO-STANDARD ANALYSISBYABRAHAM ROBINSON AND:
Lectures on Non-standard Analysis; M. Machover and J.Hirschfeld,Springer-Verlag, Berlin, Heidelberg, New York, 1969.
Victoria Symposium on Nonstandard Analysis; A. Hurd and p.Loeb,Springer-Verlag, Berlin, Heidelberg, New York, 1973.
Studies in Model Theory; M. Morley, Editor, Mathematical AssociationofAmerica, Providence, 1973.
Applied Nonstandard Analysis: M. Davis, Wiley, New York, 1977.
Introduction to the Theory of Infinitesimals: K.D Stroyan and W.A.J.Luxemburg, Academic Press, New York and London, 1976.
Foundations ofInfinitesimal Stochastic Analysis: K.D.（全文完）

学习微积分60年有感（IV）相关推荐

用Julia学习微积分：这有一份高赞数学教程 | 附习题+代码
晓查发自凹非寺量子位报道 | 公众号 QbitAI 以快速简洁闻名Julia,本身就是为计算科学的需要而生.用它来学习微积分再合适不过了,而且Julia的语法更贴近实际的数学表达式,对没学过 ...
[网络安全学习篇60]：文件上传
引言:我的系列博客[网络安全学习篇]上线了,小编也是初次创作博客,经验不足:对千峰网络信息安全开源的视频公开课程的学习整理的笔记整理的也比较粗糙,其实看到目录有300多集的时候,讲道理,有点怂了,所以 ...
00后大学生在数学真理阳光下学习微积分
在上世纪30年代,著名数学家哥德尔证明了模型论紧致性定理.紧接着,1960年,美国数学家鲁宾逊依据哥德尔紧致性定理,创立了现代微积分.这就是数学真理的阳光. 2018年高考,今天结束了.速许多00后考 ...
00后学习微积分，推荐访问袁萌专栏
事实表明,自2012年11月起,袁萌连续发表微积分科普文章达2000余篇,是国内微积分科普第一人. 微积分有两种,一是菲氏极限论微积分,二是现代模型论微积分,两者是不同时代的数学理论. 考虑到,00后 ...
英译汉翻译器大显神通，学习微积分不犯愁
下载70MB的英译汉翻译软件,安装完毕,即可使用. 实际上,Keisler微积分教材,英语语言使用规范,英译汉翻译器工作顺利,读者学习现代微积分不用犯愁. 不过,袁萌建议读者使用英语学习微积分,原汁原 ...
PyTorch深度学习：60分钟闪电战
使用PYTORCH进行深度学习:60分钟的闪电战本教程的目标: 全面了解PyTorch的Tensor库和神经网络. 训练一个小型神经网络对图像进行分类请确保您有 torch 和 torchvisi ...
第四单元用python学习微积分（二十七）积分-部分分式-分部积分
本文内容来自于学习麻省理工学院公开课:单变量微积分-分部积分-网易公开课开发环境准备:CSDN 目录一.多项式部分分式方法求积分 1.效果 2.步骤 (1) 长除法 (2) 分解因式 (fac ...
第一单元用python学习微积分（三）求导四则运算及三角函数（上）- 三角函数
本文内容来自学习麻省理工学院公开课:单变量微积分-求导四则运算及三角函数导数-网易公开课开发环境准备:CSDN 目录一.需要用到的公式: 二.求导特殊三角函数 1. 2. 三.三角函数基础公 ...
第一单元用python学习微积分（五）隐函数微分法和逆函数导数（下）- 反函数
本文内容来自学习麻省理工学院公开课:单变量微积分-隐函数微分法和逆函数导数-网易公开课和麻省理工学院公开课:单变量微积分习题课-除法法则-网易公开课开发环境准备:CSDN 目录一.反函数 1.定 ...

学习微积分60年有感（IV）

学习微积分60年有感（IV）相关推荐

最新文章

热门文章