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課程名稱 |
分析一
Analysis(Honor Program)(Ⅰ) |
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開課學期 |
107-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
崔茂培 |
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課號 |
MATH5232 |
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課程識別碼 |
221 U6540 |
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班次 |
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學分 |
5.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期二2,3,4(9:10~12:10)星期四3,4(10:20~12:10) |
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上課地點 |
天數101天數101 |
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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 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1071MATH5232_ |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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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.
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. |
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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. |
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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。 |
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預期每週課後學習時數 |
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Office Hours |
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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) |
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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) |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Midterm |
35% |
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2. |
Final |
35% |
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3. |
Quiz and Homework |
30% |
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週次 |
日期 |
單元主題 |
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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
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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 |
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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 |
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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 |
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第5週 |
10/09,10/11 |
10/09 Contractions and ODEs + Catch Up
10/11 TA session and Quiz |
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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 |
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第7週 |
10/23,10/25 |
10/23 5.8 Differential Forms
10/25 5.6 The General Stokes Formula |
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第8週 |
10/30,11/01 |
10/30 Midterm I
11/01 5.10 The Brouwer Fixed-Point Theorem |
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第9週 |
11/06,11/08 |
11/06 11/08 Ch 3 from RMA
Section 2 Riemann integral |
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第10週 |
11/13,11/15 |
Midterm break !!! No Class!!!
11/15日 本校校慶(停課一天) |
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第11週 |
11/20,11/22 |
11/20 6.3 Meseomorphism 6.4 Regularity
11/22 6.5 Products and Slices |
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第12週 |
11/27,11/29 |
11/27 6.6 Lebesgue Integrals
11/29 Italian Measure Theory
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第13週 |
12/04,12/06 |
12/4 6.8 Vitali Coverings and Density Points
12/6 6.9 Calculus `a la Lebesgue |
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第14週 |
12/11,12/13 |
12/11 6.10 Lebesgue’s Last Theorem |
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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 |
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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 |
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第18週 |
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Final exam on Jan 8, 2019
9am-12:10 |
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