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
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
 
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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-Arzela 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)
   
課程進度
週次
日期
單元主題