A以矩形的面积概念过渡至圆的面积概念来阐述.doc

Guideline of Modern analysis for ACM 07 9Date：20070912Part I：Brief Introduction to Modern Analysis A以矩形的面积概念过渡至圆的面积概念来阐述1洞察一切初等的有限的直观的数学2用开放的心灵体味“分析数学是一门取极限的学问”B中学数学物理的通病：给出具体的函数表达式有碍数学物理的真正发展Part II：数系的发展1 数系的扩展源于新运算的需要2 实数系的建立源于极限运算的需要Richard Dedekind & Georg Cantor 1872技巧：基于算术基本定理来判定某些代数方程在有理数集中无解Part III：集合的概念1 枚举法2 描述法相关阅读材料1陈纪修於崇华金路：数学分析Part IV：数理逻辑初步相关阅读材料1 谢惠民恽自求易法槐钱定边：数学分析习题课讲义2 Manfred Stoll：Introduction to Real Analysis 如何看待教材中某些独具匠心的证明题的处理1尽量读透一个证明的要义在何处（why and why not are more important than just following the rigorous proof）2如果教材中的证明过于富于技巧大家不妨换个角度用自己的理解去证明（Everyone should have a personal viewpoint）Date 20070914中心问题：如何在十进制实数系中引入四则运算？今天将要解决的问题：1 在实数系中引入加法减法运算2 解释如何理解作为“数”来看2.999与3.000是一致的Part 1：十进制实数的表示及其全体的集合Part 2：实数的序关系Part 3：实数的加法运算相关阅读材料：华罗庚：高等数学引论：Page 5Part 4：实数的取负运算Part 5：实数的减法运算Part 6：实数系的真正创立Part 7：实数的绝对值运算Part 8：实数的三进制表示法补充思考题：如何实现数在不同进制之间的转换？相关阅读材料：Manfred Stoll：Introduction to Real Analysis：Pages 3034 依旧需要解决的问题：如何在十进制实数系中引入乘法除法运算？Date 20070917如何在实数系中引入乘法除法运算依然是目前亟须解决的中心问题1将十进制表示法革命到底：类似定义加法运算去定义乘法运算（You can have a try）2另辟蹊径：单调有界数列必有极限1.3 数列和收敛数列例：1，1/2，1/3，1/4，1/5，1/6，1/7，例：0.9，0.99，0.999，0.9999，0.99999，中心含义：指标越来越大误差越来越小收敛数列的概念（为定义数列极限而引进的-N语言是由德国著名数学家Karl Weierstrass所创立。Weierstrass是一位大器晚成的数学家，他与Cauchy，Bolzano一道为推动分析严密化运动做出了卓越的贡献）1 第一步确立误差标准2 第二步确立达到误差标准所需的起始指标例：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 确立误差标准2 确立达到误差标准所需的最小起始指标（这是更加深入的收敛速度问题）最常见的误差标准：c/log_a n (a1) & c/n (0) & c/an (a1)发散数列的精确定义（写出原命题的否命题在初学阶段作为更好理解数理逻辑的有效途径应该多尝试，这样的习惯对于解证明题是有帮助的）数列a_n以a为极限 v.s. 数列a_n不以a为极限 数列a_n是收敛数列 v.s. 数列a_n是发散数列1.4 收敛数列的性质定理：收敛数列的极限是唯一的概念：有上界，有下界，有界定理：收敛数列是有界的1.6 单调数列概念：单调数列定理：单调有界数列是收敛数列（实数系基本定理）（实数系具有完备性）应用：1 乘法除法的引入2 幂函数的引入3 对数函数的引入Dates 20070919 & 20070921实数系基本定理之一：单调有界数列是收敛数列0x_nB单调有上界数列必有落于B中的极限（缜密的逻辑）1 乘法除法的引入2 幂函数以及对数函数的引入Part I：最常见的误差标准：c/log_a n (a1) & c/n (0) & c/an (a1)例1：1/log_10 n General cases： c/log_a n (a1) 例2：1/n General cases： c/n (0)例3：1/2n General cases： c/an (a1)例4：(log_10 n)/n General cases： log_a n/n (a1,0) 例5：n/2n General cases： n/an (0,a1)Part II：基本定理定理1.6：夹逼定理（夹挤定理，两边夹定理）例：n1/n（Method 1：几何算术平均不等式 & Method 2：例5使然）例：a1/n (a0)定理1.7：极限的保序性例：0a_na implies a0定理1.5：极限的四则运算（予以传统方式地论证）例：设有_n0以及a0.则成立：(1+_n)a1.little trick: (1-|_n|)(a+1)(1-|_n|)a(1+_n)a(1+|_n|)a (1+|_n|)(a+1)平淡的观察：设有实数列a_n和实数a0. 求证：a_na 等价于 a_n/a1.LLP的讲义：定义数列收敛的-N语言不是唯一定理：Cauchy定理例：反复使用Cauchy定理例：迭代数列的收敛速度与发散速度（LLP06秋：数学分析习题课讲义：Page 30）例：谢惠民恽自求易法槐钱定边：数学分析习题课讲义：Page 35：例题2.4.2The same idea：Stolz定理例：Cauchy定理（There are no essential difference between Cauchy and Stolz）例：(1k+2k+nk)/n(k+1)1/(k+1)处理问题时建议Part I & Part II 联合使用Date 20709241 基于素数分布定理解决教材：Page 8：问题1.2：1相关阅读材料：数论概论：Chapter 13：by Joseph H. Silverman（知晓一些有关素数的知识还是饶有趣味的） 2 和积互化 & 差商互化 exp(a+b)=exp(a)exp(b)ln(ab)=ln(a)+ln(b)例：徐森林薛春华：数学分析：Page 23：例1.2.6Part III：三个基本常数 1圆周率=3.141 592 653 589 793 圆的周长与圆的直径的比率2自然对数底e=2.718 281 828 459 045 等分正数如何使各部分乘积最大？导函数与原函数一致论证引入欧拉常数的两种极限是一致的3欧拉常数=0.577 215 664 901 532 调和级数紧密联系Gamma函数（阶乘函数在非整数情形下的推广）有理数？无理数？Part IV：迭代数列的蜘蛛网工作法决定性现象（自由落体运动） v.s. 统计现象（掷硬币）单调现象 v.s. 周期现象 例题2.6.1：单调有界例题2.6.2：回旋振荡建议：基于(Maple,Matlab,Mathematica,C）数列的前20项的观察先归纳后证明相关阅读材料：苏州大学习题课讲义：Pages 4652 Leonhard Euler Leonhard Euler was born on April the 15th 1707 as the son of a Protestant minister in Basel (Switzerland). Already in his childhood he exhibited great mathematical talents, but his father wanted him to study theology and become a minister. In 1720 Euler began his studies at the University of Basel. There Euler met Daniel and Nikolaus Bernoulli, who noticed Eulers skills in mathematics. Paul Euler, Leonhards father, had attended Jakob Bernoullis mathematical lectures and respected his family. When Daniel and Nikolaus Bernoulli asked him to allow his son to study mathematics he finally agreed and Euler began to study mathematics.In 1727 Euler was called to St. Petersburg by Catherine I. and became professor of physics in 1730. Finally in 1733 he became professor of mathematics. His work was both in physics and mathematics. Euler was the first to publish a systematic introduction to mechanics in 1736: “Mechanica sive motus scientia analytice exposita” (Mechanics or motion explained with analytical science (that is, calculus). 1735 he lost much of his vision in the right eye because he had looked into the sun for too long.In 1733 he married Katharina Gsell, the daughter of the director of the academy of arts. They had thirteen children, of whom only three sons and two daughters survived. The descendants of these children, however, were in high positions in Russia in the 19th century.In the year 1741 Euler went to the Prussian Academy of Sciences in Berlin and became director of the mathematical class. His time in Berlin was very productive; however, he did not have an easy position because he was not much liked by the king. Therefore he returned to St. Petersburg in 1766, now ruled by Catherine II., where he would remain for the rest of his life.Also in that time Euler was very productive, though he very soon lost his vision completely. This was possible because he had an extraordinary memory and could calculate very well. It is reported that once he let his assistant calculate a series to 17 summands and noticed that his own result and the assistants result differed in the 50th digit. A recalculation showed that Euler was right!It has been calculated that it would take 50 years eight-hour work per day to copy all his works by hand. It was not till the year 1910 that a collection of his complete works was published and it took about 70 volumes. It is reported by Legendre that often he would write down a complete mathematical proof between the first and the second call for supper.In contrast to most intellectuals of his time he was conservative and a convinced Christian. There is a story, which is often told in books and on the web, saying that once at the court of Catherine the Great he met the French philosopher Denis Diderot, who was a convinced atheist and tried to convince the Russians of atheism, much to the annoyance of Catherine. Therefore she asked Euler to stop him. Euler thought about it and when Catherine invited Diderot to have a theological discussion with Euler, Euler said: “ (a+bn)/n=x, therefore God exists, answer!” Diderot, who knew almost nothing about algebra knew not what to answer and therefore returned to Paris. This story however is almost certainly an urban myth and Diderot knew enough algebra to answer Euler. However it is said that Euler published some other (not really serious) proofs of the existence of God, which may well be, since at that time people were wondering about the possibility to give an algebraic proof of the existence of God.When Euler died on 18th of September 1783 the mathematician and philosopher Marquis de Condorcet said “.et il cessa de calculer et de vivre” (and he st