課程資訊
課程名稱
分析一
Analysis(Honor Program)(Ⅰ) 
開課學期
107-1 
授課對象
理學院  數學系  
授課教師
崔茂培 
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101天數101 
備註
此課程研究生選修不算學分。
限學士班學生
總人數上限:60人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1071MATH5232_ 
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
為確保您我的權利,請尊重智慧財產權及不得非法影印
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.

In the two-semester course, we plan to cover the following, not necessarily in the linear order.

I. The space: Metric space, open/compact/connected subsets ( we assume you are familiar with the topology of Euclidean space), topological space, continuous function, Lebesgue measure

II. Theory of functions: theory of continuous/differentiable/integrable functions, sequence, Lebesgue integral

III. Space of functions: Lp space, contraction mapping, convergence

IV. Other topics: Fourier series, Fourier transformation, some results in functional analysis. 

課程目標
A rigorous introduction to mathematical analysis. We will cover, in the first semester, most of the material in the textbook written by Marsden and Apostal, from the beginning topology of finite dimensional spaces, multivariable differential calculus, toward Fourier analysis. 
課程要求
1. 需要預習且要經常上台報告
3. 部分內容可能以閱讀, 分組討論的形式進行.
4. 準時參加所有考試.
5. 熟悉數理邏輯,有自學的能力,能掌握一般數學定理的嚴格證明。 
預期每週課後學習時數
 
Office Hours
 
參考書目
Apostol, Mathematical analysis. 2nd edition.
Protter and Morrey, A first course in real analysis. 2nd edition.
Rudin, Principles of mathematical analysis. 3rd edition.
John Lee, Introduction to Topological Manifolds, Second Edition
( Available at https://link.springer.com/content/pdf/10.1007%2F978-1-4419-7940-7.pdf
it can be downloaded from NTU IP) 
指定閱讀
Real Mathematical Analysis by Charles C. Pugh ( It can be downloaded at NTU with NTU IP
https://link.springer.com/content/pdf/10.1007%2F978-3-319-17771-7.pdf)
Apostol, Mathematical analysis. 2nd edition.
Protter and Morrey, A first course in real analysis. 2nd edition.
Rudin, Principles of mathematical analysis. 3rd edition.
John Lee, Introduction to Topological Manifolds, Second Edition
( Available at https://link.springer.com/content/pdf/10.1007%2F978-1-4419-7940-7.pdf
it can be downloaded from NTU IP) 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
Midterm 
35% 
 
2. 
Final 
35% 
 
3. 
Quiz and Homework 
30% 
 
 
課程進度
週次
日期
單元主題
第1週
9/11,9/13  9/11 2.1 metric space, 2.2 continuity
9/13 2.2 continuity,
2.3 The topology of a metric space

Video on Sep 11
https://bit.ly/2CV441d

Video on Sep 13
https://bit.ly/2x8Qge3

HW1 due Sep 18, 2018

 
第2週
9/18,9/20  9/18 Class starts from 9:10 am
2.4 compactness
2.5 Connectedness
2.6 Other Metric Space Concepts

9/20 TA session and Quiz

Video on Sep 18
https://bit.ly/2pvs6Xc 
第3週
9/25,9/27  9/25 2.7 Coverings 2.10 Completion
Ch2 (Introduction to Topological Manifolds) Topologies, Convergence and Continuity, Hausdorff Spaces

9/27 Ch2 (Introduction to Topological Manifolds) Bases and Countability, Manifolds
Ch4 Connectedness and Compactness 
第4週
10/02,10/04  10/02 Ch4 Function Spaces
4.1 Uniform Convergence 4.2 Power Series

10/4 4.3 Compactness and Equicontinuity
4.4 Uniform Approximation

Video on Oct 2, 2018
https://bit.ly/2Qn7748

Video on Oct 4, 2018
https://goo.gl/o6WrzC 
第5週
10/09,10/11  10/09 Contractions and ODEs + Catch Up

10/11 TA session and Quiz 
第6週
10/16,10/18  10/16 Ch6 Multivariable Calculus
6.1 Linear Algebra 6.2 Derivatives 6.3 Higher Derivatives

10/18 6.4 Implicit and Inverse Functions
6.5 The Rank Theorem
6.7 Multiple Integrals 
第7週
10/23,10/25  10/23 5.8 Differential Forms

10/25 5.6 The General Stokes Formula 
第8週
10/30,11/01  10/30 Midterm I
11/01 5.10 The Brouwer Fixed-Point Theorem 
第9週
11/06,11/08  11/06 11/08 Ch 3 from RMA
Section 2 Riemann integral 
第10週
11/13,11/15  Midterm break !!! No Class!!!
11/15日 本校校慶(停課一天) 
第11週
11/20,11/22  11/20 6.3 Meseomorphism 6.4 Regularity
11/22 6.5 Products and Slices 
第12週
11/27,11/29  11/27 6.6 Lebesgue Integrals
11/29 Italian Measure Theory
 
第13週
12/04,12/06  12/4 6.8 Vitali Coverings and Density Points
12/6 6.9 Calculus `a la Lebesgue 
第14週
12/11,12/13  12/11 6.10 Lebesgue’s Last Theorem 
第15週
12/18,12/20  Lecture note on Dec 18, 2018
https://goo.gl/bwk365
Lecture note on Dec 20, 2018
https://goo.gl/phLUQU 
第17週
1/01,1/03  1/01日 開國紀念日(放假日)

Final exam 9am-12:10 on Tuesday Jan 8, 2019.
Final exam topics:
https://goo.gl/Cr8AWs
Analysis Class video playlist
https://goo.gl/uBDiaC
Solution to quiz 3
https://goo.gl/GMNYNM
Solution to quiz 4
https://goo.gl/4C8Fdk
Solution to HW 7
https://goo.gl/zeadpA
Solution to HW 8
https://goo.gl/RYxDvM
Solution to HW 9
https://goo.gl/gn1PpF
Solution to HW 10
https://goo.gl/TDYGj2
Solution to HW 11
https://goo.gl/WktXdu
Solution to HW 12
https://goo.gl/xKz4CJ
Lecture note on Oct 25, 2018
https://goo.gl/JMxzkA
Lecture note on Nov 1, 2018
https://goo.gl/vkJynd
Lecture note on Nov 6, 2018
https://goo.gl/ANU6K6
Lecture note on Nov 8, 2018
https://goo.gl/mgN9Bx
Lecture note on Nov 20
https://goo.gl/nF7RCq
Lecture note on Nov 22, 2018
https://goo.gl/cMSbbS
Lecture note on Nov 27, 2018
https://goo.gl/FVAwrd
Lecture note on Dec 4, 2018
https://goo.gl/6v2DNd
Lecture note on Dec 6, 2018
https://goo.gl/GSp4SJ
Lecture note on Dec 11, 2018
https://goo.gl/fz6uKH
Lecture note on Dec 13, 2018
https://goo.gl/wbf62s
Lecture note on Dec 18, 2018
https://goo.gl/bwk365
Lecture note on Dec 20, 2018
https://goo.gl/phLUQU
Lecture note on Dec 25, 2018
https://goo.gl/xKnK8V
Lecture note on Jan 3, 2019
https://goo.gl/3JFdqT 
第18週
  Final exam on Jan 8, 2019
9am-12:10