A以矩形的面積概念過渡至圓的面積概念來闡述

1、Guideline of Modern analysis for ACM 0711Date： 20070912Part I: Brief Introduction to Modern AnalysisA .以矩形的面積概念過渡至圓的面積概念來闡述1. 洞察一切初等的有限的直觀的數(shù)學(xué)2. 用開放的心靈體味“分析數(shù)學(xué)是一門取極限的學(xué)問”B.中學(xué)數(shù)學(xué)物理的通?。航o出具體的函數(shù)表達(dá)式有礙數(shù)學(xué)物理的真正發(fā)展Part II :數(shù)系的發(fā)展1. 數(shù)系的擴(kuò)展源于新運(yùn)算的需要2. 實(shí)數(shù)系的建立源于極限運(yùn)算的需要Richard Dedeki nd & Georg Can tor 1872技巧：基于算術(shù)基本定

2、理來判定某些代數(shù)方程在有理數(shù)集中無解Part III :集合的概念1. 枚舉法2描述法 相關(guān)閱讀材料1. 陳紀(jì)修於崇華金路：數(shù)學(xué)分析Part IV:數(shù)理邏輯初步相關(guān)閱讀材料1. 謝惠民惲自求易法槐錢定邊：數(shù)學(xué)分析習(xí)題課講義2. Manfred Stoll: Introduction to Real Analysis如何看待教材中某些獨(dú)具匠心的證明題的處理1. 盡量讀透一個(gè)證明的要義在何處 (why and why not are more important than just follow ing the rigorous proof)2 .如果教材中的證明過于富于技巧大家不妨換個(gè)角度用自己

3、的理解去證明(Every one should have a personal view poijtDate 20070914中心問題：如何在十進(jìn)制實(shí)數(shù)系中引入四則運(yùn)算?今天將要解決的問題：1. 在實(shí)數(shù)系中引入加法減法運(yùn)算2. 解釋如何理解作為“數(shù)”來看2.999與3.000是一致的Part 1：十進(jìn)制實(shí)數(shù)的表示及其全體的集合Part 2：實(shí)數(shù)的序關(guān)系實(shí)數(shù)的加法運(yùn)算Part 4：實(shí)數(shù)的取負(fù)運(yùn)算Part 5:實(shí)數(shù)的減法運(yùn)算Part 6:實(shí)數(shù)系的真正創(chuàng)立Part 7：實(shí)數(shù)的絕對(duì)值運(yùn)算Part 8:實(shí)數(shù)的三進(jìn)制表示法Part 3:相關(guān)閱讀材料：華羅庚：高等數(shù)學(xué)引論：Page 5補(bǔ)充思考題：如何實(shí)現(xiàn)數(shù)

4、在不同進(jìn)制之間的轉(zhuǎn)換？相關(guān)閱讀材料：Manfred Stoll: Introduction to Real Analysis： Pages 3034依舊需要解決的問題：如何在十進(jìn)制實(shí)數(shù)系中引入乘法除法運(yùn)算?Date 20070917如何在實(shí)數(shù)系中引入乘法除法運(yùn)算依然是目前亟須解決的中心問題1 .將十進(jìn)制表示法革命到底：類似定義加法運(yùn)算去定義乘法運(yùn)算（You can havea try）2. 另辟蹊徑：單調(diào)有界數(shù)列必有極限 1.3數(shù)列和收斂數(shù)列例：1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7,例：0.9, 0.99, 0.999, 0.9999, 0.99999,中心含義：指標(biāo)

5、越來越大誤差越來越小收斂數(shù)列的概念（為定義數(shù)列極限而引進(jìn)的-N語言是由德國著名數(shù)學(xué)家 KarlWeierstrass所創(chuàng)立。Weierstrass是一位大器晚成的數(shù)學(xué)家，他與Cauchy, Bolzano 一道為推動(dòng)分析嚴(yán)密化運(yùn)動(dòng)做出了卓越的貢獻(xiàn)）1. 第一步確立誤差標(biāo)準(zhǔn)2. 第二步確立達(dá)到誤差標(biāo)準(zhǔn)所需的起始指標(biāo) 例：1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7,例：1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64,1 .確立誤差標(biāo)準(zhǔn)2. 確立達(dá)到誤差標(biāo)準(zhǔn)所需的最小起始指標(biāo)（這是更加深入的收斂速度問題）最常見的誤差標(biāo)準(zhǔn)：c/log_a n (a>1)

6、 & c/n a ( a>0) & c/an (a>1)這樣的習(xí)慣對(duì)于解證明題是有幫助的）發(fā)散數(shù)列的精確定義（寫出原命題的否命題在初學(xué)階段作為更好理解數(shù)理邏輯的 有效途徑應(yīng)該多嘗試，V.S.數(shù)列a_n不以a為極限V.S.數(shù)列a_n是發(fā)散數(shù)列數(shù)列a_n以a為極限數(shù)列a_n是收斂數(shù)列1.4收斂數(shù)列的性質(zhì)定理：收斂數(shù)列的極限是唯一的 概念：有上界，有下界，有界 定理：收斂數(shù)列是有界的1.6單調(diào)數(shù)列概念：單調(diào)數(shù)列定理：單調(diào)有界數(shù)列是收斂數(shù)列（實(shí)數(shù)系基本定理）（實(shí)數(shù)系具有完備性）應(yīng)用：1. 乘法除法的引入2. 幕函數(shù)的引入3. 對(duì)數(shù)函數(shù)的引入Dates 20070919 &a

7、mp; 20070921實(shí)數(shù)系基本定理之一：單調(diào)有界數(shù)列是收斂數(shù)列0. x_n B單調(diào)有上界數(shù)列必有落于B中的極限（縝密的邏輯）1.乘法除法的引入2幕函數(shù)以及對(duì)數(shù)函數(shù)的引入Part I：最常見的誤差標(biāo)準(zhǔn)：c/log_a n (a>1) & c/nA a (o>0) & c/aAn (a>1)例1：例2：例3：例4：例5：1/log_10 n 1/n 1/2An (log_10 n)/n n /2AnGen eral cases Gen eral cases Gen eral casesGen eral cases Gen eral casesc/log_a n

8、 (a>1)c/nA a( a>0)c/an (a>1)log_a n/nAa(a>1, a>0) nA o/an (a>0,a>1)Part II :基本定理定理1.6：夾逼定理（夾擠定理，兩邊夾定理）例：nA1/n （Method 1:幾何算術(shù)平均不等式 & Method 2：例5使然）例：a1/n (a>0)定理1.7：極限的保序性例：0 三 a_na imp lies a = 0定理1.5：極限的四則運(yùn)算(予以傳統(tǒng)方式地論證)例：設(shè)有£_n0以及a>0.則成立：(1+ Ln)Aa 1.little trick: (

9、1-|_n|) A(a+1)三(1-|_n|) Aa三(1+_n)Aa三(1+| _n|) Aa 三(1+l _n|) A(a+1)平淡的觀察：設(shè)有實(shí)數(shù)列a_n和實(shí)數(shù)a>0.求證：a_na等價(jià)于a_n/a 1.LLP的講義：定義數(shù)列收斂的 -N語言不是唯一定理： Cauchy 定理例：反復(fù)使用Cauchy定理例:迭代數(shù)列的收斂速度與發(fā)散速度(LLP06 秋:數(shù)學(xué)分析習(xí)題課講義：Page 30例：謝惠民惲自求易法槐錢定邊：數(shù)學(xué)分析習(xí)題課講義：Page 35例題2.4.2The same idea Stolz 定理例: Cauchy 定理(There are no essential dif

10、ferenee between Cauchy and StOlz例：(1你+2你+nk)/n (k+l) 1/(k+1)處理問題時(shí)建議Part I & Part II聯(lián)合使用Date 20709241.基于素?cái)?shù)分布定理解決教材：Page 8問題1.2： 1相關(guān)閱讀材料：數(shù)論概論：Cha pter 13: by Jose ph H. Silverman(知曉一些有關(guān)素?cái)?shù)的知識(shí)還是饒有趣味的)2 .和積互化&差商互化exp( a+b)=ex p( a)ex p(b)ln( ab)=l n(a)+l n(b)例：徐森林薛春華：數(shù)學(xué)分析：Page 23例1.2.6Part III :三

11、個(gè)基本常數(shù)1.圓周率 n=3.141 592 653 589 793 圓的周長與圓的直徑的比率2. 自然對(duì)數(shù)底 e=2.718 281 828 459 045 等分正數(shù)如何使各部分乘積最大？導(dǎo)函數(shù)與原函數(shù)一致論證引入歐拉常數(shù)的兩種極限是一致的3 .歐拉常數(shù) Y0.577 215 664 901 532 調(diào)和級(jí)數(shù)緊密聯(lián)系Gamma函數(shù)(階乘函數(shù)在非整數(shù)情形下的推廣) 有理數(shù)？無理數(shù)？Part IV :迭代數(shù)列的蜘蛛網(wǎng)工作法決定性現(xiàn)象（自由落體運(yùn)動(dòng)）V.S.統(tǒng)計(jì)現(xiàn)象（擲硬幣）單調(diào)現(xiàn)象V.S.周期現(xiàn)象例題2.6.1 :單調(diào)有界例題2.6.2:回旋振蕩建議：基于（Maple,Matlab,Mathem

12、atica,C）數(shù)列的前20項(xiàng)的觀察先歸納后證明相關(guān)閱讀材料：蘇州大學(xué)習(xí)題課講義：P ages 4652Leon hard EulerLeon hard Euler was born on April the 15th 1707 as the son of a Protestant min ister in Basel (Switzerla nd). Already in his childhood he exhibited great mathematical tale nts, but his father wan ted him to study theology and become

13、a mini ster. I n 1720 Euler bega n his studies at the Uni versity of Basel. There Euler met Daniel and Nikolaus Berno ulli, who no ticed Euler's skills in mathematics. P aul Euler, Leon hard's father, had atte nded Jakob Berno ulli's mathematical lectures and res pected his family. When

14、Dan iel and Nikolaus Berno ulli asked him to allow his son to study mathematics he fin ally agreed and Euler bega n to study mathematics.In 1727 Euler was called to St. P etersburg by Catheri ne I. and became pro fessor of p hysics in 1730. Fin ally in 1733 he became pro fessor of mathematics. His w

15、ork was both in p hysics and mathematics. Euler was the first to p ublish a systematic in troductio n to mecha nics in 1736:“ Mecha nicasive motus scientia analytice exposita” (Mechanics or motion explained withan alytical scie nee (that is, calculus). 1735 he lost much of his visi on in the right e

16、ye because he had looked into the sun for too long.In 1733 he married Kathari na Gsell, the daughter of the director of the academy of arts. They had thirtee n childre n, of whom only three sons and two daughters survived. The desce ndants of these childre n, however, were in high po siti ons in Rus

17、sia in the 19th cen tury.In the year 1741 Euler went to the Prussia n Academy of Scie nces in Berli n and became director of the mathematical class. His time in Berli n was very p roductive; however, he did not have an easy po siti on because he was not much liked by the king. Therefore he retu rned

18、 to St. Petersburg in 1766, now ruled by Catheri ne II., where he would remai n for the rest of his life.Also in that time Euler was very p roductive, though he very soon lost his visio n comp letely. This was po ssible because he had an extraordi nary memory and could calculate very well. It is rep

19、 orted that once he let his assista nt calculate a series to 17 summa nds and no ticed that his own result and the assista nt's result differed in the 50th digit. A recalculati on showed that Euler was right!It has bee n calculated that it would take 50 years eight-hour work per day to copy all

20、his works by han d. It was not till the year 1910 that a collect ion of his comp lete works was p ublished and it took about 70 volumes. It is rep orted by Lege ndre that ofte n he would write dow n a compi ete mathematical p roof betwee n the first and the sec ond call for supper.In con trast to mo

21、st in tellectuals of his time he was con servative and a convin ced Christia n. There is a story, which is ofte n told in books and on the web, say ing that once at the court of Catheri ne the Great he met the French p hilos op her Den is Diderot, who was a convin ced atheist and tried to convince t

22、he Russia ns of atheism, much to the annoyance of Catheri ne. Therefore she asked Euler to stop him. Euler thought about it and whe n Catheri ne in vited Diderot to have a theological discussi on with Eu ler, Euler said:(a+bh )/n=x,therefore God exists, an swer!” Diderot, who knew almost no thi ng a

23、boutalgebra knew not what to an swer and therefore returned to P aris. This story however is almost certai niy an urba n myth and Diderot knew eno ugh algebra to an swer Euler. However it is said that Euler p ublished some other (not really serious) proofs of the existe nee of God, which may well be, since at that time people were won deri ng about the p ossibility to give an algebraic proof of the existe nee of God.When Euler died on 18th of Sep tember 1783 the mathematicia n andp hilos op her Marquis de Con dorcet saidet iicessa de calcul