課程資訊
 課程名稱 分析一Analysis(Honor Program)(Ⅰ) 開課學期 104-1 授課對象 理學院  數學系 授課教師 余正道 課號 MATH5232 課程識別碼 221 U6540 班次 學分 5 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 天數101天數101 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 Ceiba 課程網頁 http://ceiba.ntu.edu.tw/1041analysis 課程簡介影片 核心能力關聯 本課程尚未建立核心能力關連 課程大綱 為確保您我的權利,請尊重智慧財產權及不得非法影印 課程概述 Chapter 1. Rn and its topology (7 weeks) x1.1 Introduction Metric on Rn, Schwartz inequality, limits, Cauchy sequence, series, completeness of Rn, countable and uncountable sets. Examples. x1.2 Topology of Rn 1. Open set, interior of a set, closed set, closure, accumulation points, boundary. Examples. 2. Continuous functions (Examples, including monotone), uniform convergence and power series. x1.3 Compact and connected sets. (1 week) 1. Compact sets, The Heine-Borel (Bolzano-Weierstrass) theorem and open covering. 2. Connected set, one-dimensional classi cation, path-connectedness, applications to continuous functions (Intermediate theorem). Examples. (xx1.1-1.3 for 3 weeks) x1.4 Metric space 1. Rn (in `p) 2. The space of continuous functions, Ascoli-Arzela theorem, Stone-Weierstrass theorems (compactness revisited). 3. Fixed point theorem. (a) Contraction maps. (b) Applications to O.D.E: existence and uniqueness, continuity on initial data, local stability in 2  2 system. (2 weeks) (c) Brouwer xed points (optional). 4. Baire category theorem and its applications. Chapter 2. Di erentials (6 weeks) x2.1 1. Overview (one-dimension) Rolle's theorem and mean-value theorem. Applications. 2. Linear transformation, Di erential (de nition) and example (with emphasis on geometry) 3. Partial derivatives and continuous P.D ! di erentials 4. The chain rule. The determinant of its di erential (for Rn ! Rn) x2.2 Higher order di erentials. Partial derivatives. Taylor expansions. x2.3 Implicit function theorem and its variations. Examples. x2.4 Local extrema and Lagrange multipliers. Applications. Examples. x2.5 Sard's theorem. Chapter 3. Riemann Integral of multi-variables (1.5 weeks) 1. De nition, change of variable formulas (for continuous function). Examples. 2. Surface integral and line integral. Stokes' theorem. (Applications) 3. Lebesgue's criterion for the existence of Riemann integral. (1 week, the next semester) 課程目標 待補 課程要求 待補 預期每週課後學習時數 Office Hours 每週一 15:30~16:30 參考書目 待補 指定閱讀 待補 評量方式(僅供參考)
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