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課程名稱 |
高等微積分優一
Honors Advanced Calculus (Ⅰ) |
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開課學期 |
100-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
王金龍 |
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課號 |
MATH2207 |
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課程識別碼 |
201 49470 |
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班次 |
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學分 |
4 |
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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 |
1.需修過微積分及線性代數，且分數都達B以上。
2.高微優可抵必修高微。
總人數上限：50人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1001hac |
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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 |
Advanced Calculus is generally regarded as the most basic and solid training for mathematics majored students. In contrast to Calculus, it provides the completely rigorous training in "writing proofs". Also instead of intuition, it is based on general topology and real number system to develop the mathematical analysis. It is the foundation to all analytic methods.
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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 textbooks written by Apostal and Rudin, from the beginning topology of finite dimensional spaces, multivariable differential calculus, Fourier transforms, toward Lebesgue integrals and Stokes theorem.
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課程要求 |
1. 準時上課 (含習題課), 無法上課須事先請假.
2. 準時繳交作業.
3. 部分內容可能以閱讀, 報告的形式進行.
4. 準時參加小考, 期中考以及期末考. |
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預期每週課後學習時數 |
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Office Hours |
每週四 12:20~13:20 |
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指定閱讀 |
Apostol: Introduction to Mathematical Analysis (main textbook)
Rudin: Principle of Mathematical Analysis |
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參考書目 |
Spivak: Calculus on Manifolds
Courant and John: INtroduction to Calculus and Analysis (I and IIA, IIB) |
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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. |
作業 |
20% |
每週指派. |
2. |
小考 |
20% |
隔週考. |
3. |
期中考 |
30% |
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4. |
期末考 |
30% |
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週次 |
日期 |
單元主題 |
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第1週 |
9/13,9/15 |
Ch.2 + Ch.3 Basic set theory and point set topology |
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第2週 |
9/20,9/22 |
Ch.3 Compactness + Ch.4 Limits and continuity |
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第3週 |
9/27,9/29 |
Ch.4 Connectedness + Ch.5 Derivatives |
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第4週 |
10/04,10/06 |
Ch.6 Functions of bounded variations + Ch.7 Riemann-Stieltjes integral |
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第5週 |
10/11,10/13 |
Ch.7 Riemann-Stieltjes integral (continued) |
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第6週 |
10/18,10/20 |
Ch.7 Lebesgue criterion + Ch.8 Infinite series and products |
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第7週 |
10/25,10/27 |
Ch.8 Double series/products (continued) + Ch.9 Space-filling curves |
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第8週 |
11/01,11/03 |
Ch.9 Sequence of functions - uniform convergence |
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第9週 |
11/08,11/10 |
11/08: Midterm exam Ch.1 - Ch.9 + 11/10: Ch.10 Power series |
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第10週 |
11/15,11/17 |
11/15: University holiday + 11/17: Ch.10 Lebesgue integral |
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第11週 |
11/22,11/24 |
Ch.10 Levi, Fatou, Lebesgue's convergence theorems |
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第12週 |
11/29,12/01 |
Ch.10 Lebesgue measurable functions and sets |
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第13週 |
12/06,12/08 |
Ch.11 L2 theory and Fourier series |
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第14週 |
12/13,12/15 |
Ch.11 (continued) Fejer theorem + Fourier integrals |
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第15週 |
12/20,12/22 |
Ch.12 Multivariable differential calculus |
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第16週 |
12/27,12/29 |
Ch.13 Implicit function theorem |
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第17週 |
1/03,1/05 |
Lagrange multiplier and Introduction to calculus of variations |
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