課程名稱 |
分析導論一 Introduction to Mathematical Analysis(Ⅰ) |
開課學期 |
109-1 |
授課對象 |
社會科學院 經濟學研究所 |
授課教師 |
陳俊全 |
課號 |
ECON5129 |
課程識別碼 |
323 U2030 |
班次 |
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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 |
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備註 |
上課資訊依課號MATH2213訊息為主。限選修ECON課號,方可認定為經濟系選修課。 限學士班三年級以上 或 限碩士班以上 或 限博士班 總人數上限:20人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1091ECON5129_MA |
課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
課程大綱
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課程概述 |
這門課是數學系的入門課程,主要是讓學生熟悉數學分析的語言,訓練學生更嚴謹的數學證明邏輯,也是更高階分析課程的基礎。為了要有更廣的觀點,我們會從基本的點集拓樸切入,引進極限的觀念,隨後介紹連續及微分,還有這些觀念的相關定理及應用,而後將介紹積分及相關的課題,如果時間允許,我們也會涉略基本的測度論。 |
課程目標 |
讓學生熟悉數學分析的語言,能夠使用分析的工具操作嚴謹的證明。 |
課程要求 |
週作業,期中考,期末考。
預備知識: calculus, linear algebra |
預期每週課後學習時數 |
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Office Hours |
每週四 13:00~15:00 備註: 週四:王舜傑(天數455) |
指定閱讀 |
待補 |
參考書目 |
1. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, 2nd Edition
2. Walter Rudin, Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics), McGraw-Hill Education; 3rd edition
3. Mathematical Analysis. Second Edition. Tom M. Apostol.
4. William R. Wade, An Introduction to Analysis, Prentice Hall, 4th Edition |
評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
homework and quiz |
25% |
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2. |
midterm exam |
35% |
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3. |
final exam |
40% |
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週次 |
日期 |
單元主題 |
第1週 |
9/15,9/17 |
0. Introduction -problems arising from calculus; new topics:
0.1.real numbers and completeness
0.2 what is infinity?
0.3 topology of the Euclidean space: Riemann integral and compactness
0.4 uniform convergence of functions
0.5 differentiation in R^n
0.6 Solve a system of non-linear equations:- Inverse and Implicit Function Theorems
0.7 Lebesgue's Theorem for integrals
0.8 Fourier series
1. The real number system and the Euclidean space
1.1 Sets and Functions:
- power set of A, product of A and B
- domain, target, range of a function, one-to-one, onto
1.2 Origin of number concept
- Piraha people in the Amazon rainforest
- Research on infants
1.3 Number system: natural numbers, integers, rational numbers |
第2週 |
9/22,9/24 |
1.4 Ordered Fields
- addition axioms, multiplication axioms and order axioms
- sequence and limit: uniqueness of limits, sandwich lemma, limits of a
sum and a product
- Cauchy sequence
- Axiom of completeness |
第3週 |
9/29,10/01 |
- Basic properties of Cauchy sequences
- Axioms of a complete ordered field |
第4週 |
10/06,10/08 |
1-5 Construction of a complete ordered field
1-5-1 three approaches: infinite decimals, Cauchy sequences and Dedekind cuts
1-5-2 Cauchy sequence approach:
- S=the set of all rational Cauchy sequences
- an equivalence relation on S and the corresponding equivalence classes
- addition and multiplication on the equivalence classes |
第5週 |
10/13,10/15 |
1-5-2 Cauchy sequence approach:
- order on the equivalence classes
- Cauchy sequences in the space of the equivalence classes
-the equivalent classes together with the addition, multiplication and order forms a complete ordered field |
第6週 |
10/20,10/22 |
-Theorem: There exists a "unique" complete ordered field, called the real number system.
- Monotone sequence property (MSP)
-sup, inf and the least upper bound property (LUBP)
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第7週 |
10/27,10/29 |
-Theorem: The three versions of completeness (CSP)+(AP), (MSP) and (LUBP) are equivalent.
1-6 limsup and liminf |
第8週 |
11/03,11/05 |
- more properties and applications of limsup and liminf
1-7 Cantor's theory of infinity
- Definition of card A=card B and card A<card B
- finite, countable and uncountable
- an infinite subset of a countable set is countable
- card N = card Q < card R = card RxR=card P(N), Cantor's diagonal method
- card A < card P(A)
- existence of an algebraic number |
第9週 |
11/10,11/12 |
- Schroder-Bernstein Theorem
- continuum hypothesis: Godel and Cohen
1-8 Some "paradoxes" about real numbers
- a number of all knowledge
- Pi is a normal number?
Borel's theorem: Almost every real number is normal.
- Richard's paradox
1-9 Complex numbers
1-10 Euclidean space
- norm, metric, inner product, Schwarz's inequality
Chapter 2 Topologies of Metric Spaces
2-1 Metric space: definition and examples
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第10週 |
11/17,11/19 |
Midterm examination
2-2 Open sets and interior of a set
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第11週 |
11/24,11/26 |
2-3 Closed sets, accumulation points, closure of a set
2-4 Boundary of a set
2-5 Sequences and limits
2-6 Completeness of a metric space
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第12週 |
12/01,12/03 |
Chapter 3 Compact sets
3-1. Examples: the difference between I= [0,1] and I=(0,1]; consider continuous function on I
3-2 Sequentially compact: bisection process and bounded sequence; Heine-Borel Theorem
3-3 Open cover and compact:
- examples
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第13週 |
12/08,12/10 |
- compact implies bounded and closed; counterexample
- totally bounded;
- Bolzano-Weierstrass Theorem
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第14週 |
12/15,12/17 |
- compact iff totally bounded and complete in a metric space
3-4 Path-connected and connected
- path connected implies connected
Chapter 4 Continuous maps
4-1 Continuity
- limit at a point
- continuous at a point and on the whole domain
- continuity defined by sequential limits |
第15週 |
12/22,12/24 |
- continuity characterized by preimages of open and closed sets
- continuity for +,-,×,÷, and f(g(x))
4-2 Images of compact and connected sets
4-3 Real-valued functions
- Maximum-minimum theorem
- Intermediate value theorem
4-4 Uniform continuity |
第16週 |
12/29,12/31 |
Chapter 5 Uniform convergence of functions
-Motivations
5-1 Pointwise and uniform convergence
- examples
- uniform convergence implies pointwise convergence
- uniform convergence iff sup ρ(f_k,f) → 0
- Theorem: The limit function of an uniformly convergent sequence
of continuous functions is continuous.
5-2 Cauchy criterion and M test
- Cauchy criterion and uniform convergence
- examples: uniformly convergence of series of functions |
第17週 |
1/05,1/07 |
5-3 Integration and differentiation of sequences and series of
functions
- Theorem: uniform convergence implies convergence of the
integrals
- Theorem : pointwise convergence of the functions and uniform
convergence of their derivatives together imply differentiability of
the limit function
5-4 The space of continuous functions
- completeness property
- equicontinuity
- Arzela-Ascoli Theorem |
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