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課程名稱 |
分析導論二
Introduction to Mathematical Analysis(Ⅱ) |
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開課學期 |
109-2 |
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授課對象 |
社會科學院 經濟學系 |
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授課教師 |
陳俊全 |
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課號 |
ECON5130 |
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課程識別碼 |
323 U2040 |
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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 |
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備註 |
上課教室及資訊依課號MATH2214訊息為主。限選修ECON課號之課程，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 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1092ECON5130_MA_2 |
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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 |
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課程大綱
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課程概述 |
這門課是數學系的基本課程，主要是讓學生熟悉數學分析的重要知識，當作使用高階數學的基礎。這學期內容將包擴 uniform convergence functions, Stone-Weierstrass theorem, Cesaro and Abel summability, multi-variable differential calculus, inverse and implicit function theorems, multiple Riemann integrals and Fubini's theorem, Lebesgue's theorem, and Fourier series. |
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課程目標 |
讓學生熟悉數學分析的重要基礎觀念、相關應用、及能夠操作嚴謹的證明。 |
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課程要求 |
mathematical analysis I |
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預期每週課後學習時數 |
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Office Hours |
備註： (二)王舜傑 (天數455) (三)李雙言 (天數103) (四)吳逸安 (天數455) |
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指定閱讀 |
待補 |
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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 |
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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. |
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 |
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第1週 |
2/22,2/26 |
0. Introduction
1. Uniform convergence of functions and space of continuous functions
1-1. Basic properties: uniform convergence, differentiation and integration of series of functions, the space of continuous functions |
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第2週 |
3/01,3/05 |
1-1. Basic properties: Arzela-Ascoli Theorem
1-2. Contraction mapping principle: fixed point, iteration method, contraction mapping principle |
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第3週 |
3/08,3/12 |
1-2. Contraction mapping principle: local existence of ODE
1-3. Stone-Weierstrass Theorem: Bernstein's theorem, probabilistic interpretation of Bernstein's theorem |
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第4週 |
3/15,3/19 |
1-3. Stone-Weierstrass Theorem: proof of Stone-Weierstrass Theorem, applications |
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第5週 |
3/22,3/26 |
1-4 Cesaro and Abel summability: power series, radius of convergence, term-by-term differentiation of power series, root test and ratio test, Cesaro 1-summable and 2-summable, Abel summable, examples, "convergent" implies "Cesaro 1-summable" and "Abel summable", "Cesara 1-summable" implies "Abel summable" |
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第6週 |
3/29,4/02 |
1-4 Cesaro and Abel summability: "Cesaro 1-summable plus O(1/k)" implies "convergent in the usual sense"
1-5 Dirichlet and Abel tests: Dirichlet test and alternating series |
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第7週 |
4/05,4/09 |
spring break
1-5 Dirichlet and Abel tests: Abel's partial summation formula, Dirichlet's test, Abel's test |
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第8週 |
4/12,4/16 |
2. Differentiable mappings
2-1 Derivatives: linear mapping and definition of total derivative, mean value theorem for one variable, differentiable implies continuous, partial derivative, directional derivative, matrix representation of a total derivative, Jacobian matrix
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第9週 |
4/19,4/23 |
2-1 Derivatives: conditions for differentiability, tangent direction of a path, tangent plane, gradient
2-2 Chain rule and product rule: proof of the chain rule
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第10週 |
4/26,4/30 |
Midterm examination
2-2 Chain rule and product rule: Mean Value Theorem |
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第11週 |
5/03,5/07 |
2-3: Higher order derivatives: 2nd order derivative and bilinear map, Hessian matrix, symmetry of 2nd order derivative
2-4: Taylor's Theorem |
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第12週 |
5/10,5/14 |
2-4 Taylor's Theorem: multi-variable case and its proof
2-5 Maximum, minimum, and Hessian |
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第13週 |
5/17,5/21 |
3 Inverse and Implicit Function Theorems
3-1. Inverse Function Theorem |
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第14週 |
5/24,5/28 |
3-2 Implicit Function Theorem and applications
3-3 Lagrange multipliers |
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第15週 |
5/31,6/04 |
3-4 Continuity method
4 Integration
4-1 Riemann Integrals
4-2 Improper Integrals
4-3 Measure Zero and Lebesgue’s Theorem |
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第16週 |
6/07,6/11 |
4-3 Measure Zero and Lebesgue’s Theorem: Proof of Lebesgue’s Theorem
5 Fourier Series
5-1 Waves
5-2 Fourier series |
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第17週 |
6/14,6/18 |
5-3 Pointwise convergence
5-4 Fourier sine and cosine series
5-5 Change of interval
5-6 exp(inx)
5-7 Inner product space: general Fourier series and convergence in the mean |
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