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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 |
112-2 |
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授課對象 |
理學院 數學系 |
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授課教師 |
陳俊全 |
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課號 |
MATH2214 |
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課程識別碼 |
201 49660 |
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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 |
必帶 |
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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 |
新102新302 |
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備註 |
限本系所學生(含輔系、雙修生)
總人數上限：100人 |
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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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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 |
這門課是數學系的重要課程，主要是讓學生熟悉數學分析的語言及更嚴謹的數學證明，也是更高階分析課程的基礎。我們上學期從實數基本性質、Cantor cardinal numbers及點集拓樸切入，引進極限、緊緻性、metric 的觀念，隨後介紹連續及uniform convergence。本學期將介紹fixed point theorem、Stone-Weierstrass theorem、 微分及其應用、隱函數定理、積分理論及Lebesgue定理等。如果時間允許，將略微講述基本的測度論。也會介紹Fourier series 理論。 |
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課程目標 |
熟悉數學分析的基本觀念、工具、及操作嚴謹的證明。 |
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課程要求 |
週作業，期中考，期末考。
預備知識: calculus, linear algebra, Introduction to Mathematical Analysis I |
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預期每週課後學習時數 |
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Office Hours |
另約時間 備註： 周一 16:00-17:00, 餘0班 舜傑助教 Office : 天數455
周二 15:00-16:00, 餘1班 行遠助教 Office : 天數445
週二 14:00-16:00, 餘2班 家豪助教 Office : 天數438 |
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指定閱讀 |
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 |
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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 |
2/19-2/23 |
5. Uniform convergence
5-1. Contraction Mapping Theorem: Fredholm and Volterra equations
5-2. Bernstein's Theorem |
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第2週 |
2/26-3/01 |
5-1. Existence and uniqueness of the solutions of an ODE
5-2. Applications of Bernstein's Theorem
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第3週 |
3/04-3/08 |
5-2. Revisit of Bernstein's Theorem: Law of Large Numbers
5-3. Stone-Weierstrass's Theorem |
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第4週 |
3/11-3/15 |
5-3. Applications of Stone-Weierstrass's Theorem
5-4. Abel's and Dirichlet's tests |
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第5週 |
3/18-3/22 |
5-5 Power series: radius of convergence, term by term differentiation
5-6 Cesaro and Abel summability: summation by parts, (C, 1) imples (Abel). Examples of a (C,2) summable series. |
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第6週 |
3/25-3/29 |
6-1 Differentiation in R: Rolle's Theorem, Mean Value Theorem
6-2 Integration in R: upper and lower sum, Riemann integrable |
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第7週 |
4/01-4/05 |
6-2 Integration in R: basic properties of Riemann integrals, Fundamental Theorem of Calculus |
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第8週 |
4/08-4/12 |
6-3 Differentiation in R^n: definition, linearization, differentiable implies continuous, relation between differentiability and partial derivatives |
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第9週 |
4/15-4/19 |
4/16: midterm examination
6-3 Chain rule |
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第10週 |
4/22-4/26 |
6-4 Higher derivative and Taylor's expansion: 2nd derivative, Hessian matrix, higher derivatives, 1 variable Taylor expansion, Cauchy Mean Value Theorem |
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第14週 |
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