課程資訊
課程名稱
分析一
Analysis(Honor Program)(Ⅰ) 
開課學期
109-1 
授課對象
理學院  數學系  
授課教師
蔡忠潤 
課號
MATH5232 
課程識別碼
221 U6540 
班次
 
學分
5.0 
全/半年
半年 
必/選修
選修 
上課時間
星期二2,3,4(9:10~12:10)星期四3,4(10:20~12:10) 
上課地點
天數102天數102 
備註
此課程研究生選修不算學分。
限學士班學生 且 限學士班二年級以上
總人數上限:30人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1091A1 
課程簡介影片
 
核心能力關聯
本課程尚未建立核心能力關連
課程大綱
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課程概述

This is a course of mathematical analysis which provides solid training for mathematics majored students and students who are really interested in mathematical analysis. It provides the completely rigorous training in fundamental contents of mathematical analysis.
Students are supposed to have the ability of writing rigorous mathematical arguments.

There are three main topics in this semester:
1. topology of metric space.
2. space of (single variable) functions.
3. multi-variable calculus. 

課程目標
Develop abstract and logical thinking.
Write rigorous mathematical statements and proofs. 
課程要求
This course is a rigorous introduction to mathematical analysis. The first semester will cover most of the material in the textbook by Pugh.

Course prerequisite:
1. freshman calculus (for math major)
2. linear algebra (for math major) 
預期每週課後學習時數
 
Office Hours
每週二 14:00~15:00 
參考書目
Walter Rudin, Principles of mathematical analysis. 3rd edition. 
指定閱讀
Charles Chapman Pugh, Real mathematical analysis. Second Edition.
You can download the PDF files with an NTU IP address.
link: https://link.springer.com/content/pdf/10.1007%2F978-3-319-17771-7.pdf 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
Homework 
35% 
You have two jokers: the lowest two grades will be discarded. 
2. 
Quiz 
15% 
9/22, 10/13, 12/01, 12/22 
3. 
Midterm 
25% 
11/10 
4. 
Final 
25% 
1/05 
 
課程進度
週次
日期
單元主題
第1週
9/15,9/17  9/15 期初小考(不計入學期成績)
metric space, continuity, topology. ref: 2.1, 2.2, 2.3 
第2週
9/22,9/24  9/22 第一次小考
completeness, compactness, connectedness. ref: 2.4, 2.5 
第3週
9/29  covering and compactness, Cantor set. ref: 2.7, 2.8
10/01 中秋節 
第4週
10/06,10/08  Cantor set. uniform convergence, integration and differentiation of sequence of functions, power series. ref: 2.8, 2.9, 4.1, 4.2 
第5週
10/13,10/15  10/13 第二次小考
Arzela-Ascoli theorem, Weierstrass approximation theorem. ref: 4.3, 4.4 
第6週
10/20,10/22  Stone-Weierstrass theorem, Picard's theorm for ODE, nowhere differentiable function. ref: 4.4, 4.5, 4.7 
第7週
10/27,10/29  Baire category theorem, space of continuous functions over non-compact spaces, \sigma-compact and hemi-compact, Urysohn lemma and Tietze extension, continuous dependence of ODEs on initial conditions. ref: 4.7, 4.8 
第8週
11/03,11/05  completion, paracompactness. derivatives and higher order derivatives in higher dimensions. ref: 2.10, 5.1, 5.2, 5.3 
第9週
11/10,11/12  11/10 期中考
inverse and implicit function theorem, constant rank theorem. ref: 5.4, 5.5 
第10週
11/17,11/19  自主學習週 
第11週
11/24,11/26  exterior algebra, differential forms, Stokes theorem. ref: 5.8, 5.9 
第12週
12/01,12/03  12/01 第三次小考
more on the Stokes theorem, Poincare lemma, Brouwer fixed point theorem, Perron-Frobenius theorem. ref: 5.9, 5.10 
第13週
12/08,12/10  Lebesgue measure, abstract outer measure, regularity. ref: 6.1, 6.2, 6.4 
第14週
12/15,12/17  product and slice, Lebesgue integral, monotone convergence theorem. ref: 6.5, 6.6 
第15週
12/22,12/24  12/22 第四次小考
dominated convergence theorem, Vitali covering lemma, density. ref: 6.7, 6.8 
第16週
12/29,12/31  Lebesgue's fundamental theorem of calculus, absolute continuity. ref: 6.9, 6.10 
第17週
1/05,1/07  1/05 期末考
differentiability of monotone function, Littlewood's Three Principles. ref: 6.10, Appendix F