課程大綱

課程資訊

課程名稱

分析導論一
Introduction to Mathematical Analysis(Ⅰ)

開課學期

108-1

授課對象

社會科學院經濟學系

授課教師

王振男

課號

ECON5129

課程識別碼

323 U2030

班次

學分

5.0

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

備註

上課資訊依課號MATH2213訊息為主。限選修ECON課號，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

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1081ECON5129_

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課程概述

這門課是數學系的入門課程，主要是讓學生熟悉數學分析的語言，訓練學生更嚴謹的數學證明邏輯，也是更高階分析課程的基礎。為了要有更廣的觀點，我們會從基本的點集拓樸切入，引進極限的觀念，隨後介紹連續及微分，還有這些觀念的相關定理及應用，而後將介紹積分及相關的課題，如果時間允許，我們也會涉略基本的測度論。

課程目標

讓學生熟悉數學分析的語言，能夠使用分析的工具操作嚴謹的證明。

課程要求

週作業，期中考，期末考。

預期每週課後學習時數

Office Hours

每週二 13:30~15:30

指定閱讀

Advanced calculus with applications in statistics. 2nd Edition. Khuri, André I. 2003.
*原本指定Apostol的書，因為不容易取得正版，所以更換指定教科書。這本是電子書，從學校圖書館可以下載。選這本書的理由是除了基本理論外還加上應用，可以更了解分析如何實際應用。
網址：https://ntu.primo.exlibrisgroup.com/discovery/fulldisplay?docid=alma991027899169704786&context=L&vid=886NTU_INST:886NTU_INST&lang=zh-tw&search_scope=MyInstitution&adaptor=Local%20Search%20Engine&tab=LibraryCatalog&query=any,contains,advanced%20calculus%20for%20statistics&offset=0

參考書目

1. Mathematical Analysis. Second Edition. Tom M. Apostol.
2. Walter Rudin, Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics), McGraw-Hill Education; 3rd edition
3. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, W. H. Freeman, 2nd Edition
4. William R. Wade, An Introduction to Analysis, Prentice Hall, 4th Edition

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	期末考	30%
2.	期中考	30%
3.	作業	40%	週作業

課程進度

週次	日期	單元主題
第1週		Motivation, Sets, Functions, Domain of definition, Range, Injectivity, Surjectivity, Bijectivity, Finiteness, Countability, Q is countable.
第2週		Bounded above, Bounded below, Supremum, Infimum, Interior points, Open sets, Topological spaces, Characterization of open sets in R, Component intervals, Closed sets, Limit points, Characterization of closed sets, Bolzano-Weierstrass theorem.
第3週		Bolzano-Weierstrass theorem, Bisection argument, Cantor intersection theorem, Open covering, Lindelof theorem, Heine-Borel theorem, Compact sets, Equivalent conditions of compact sets in R^n.
第4週		Equivalent conditions of a compact set in R^n, Metrics, Metric spaces, Discrete metric, Relative metric, Balls, Interior points, Open sets, Closed sets, Equivalent metrics.
第5週		Characterization of open sets in relative metric, Characterization of closed sets in relative metric, Limit points, Characterization of closed sets using limit points, Compact sets (every open covering has a finite subcover), A closed subset of a compact metric space is compact.
第6週		Limit of a sequence, Convergence, Subsequences, Cauchy sequences, Completeness, Limits of a function, Continuous functions, Continuity of composite functions.
第7週		Continuity and open sets, Continuity and closed sets, Continuity and compact sets, Homeomorphism, Topological property, Bolzano theorem, Intermediate-value theorem, Connectedness, Two-valued functions, Continuity and connectedness.
第8週		Arcwise connected, Uniform continuity, A continuous function on a compact set is uniformly continuous, Contraction maps, Fixed-point theorem, Applications, Discontinuities.
第9週		Types of discontinuities for f(x), Strictly monotonic functions, The set of discontinuities of a monotonic function is countable, Derivative, The chain rule.
第10週		Derivative, Differentiability, Extremal points, Rolle's theorem, Generalized mean-value theorem, Intermediate-value theorem for derivatives, Derivatives of vector-valued functions, Partial derivatives, Differentiability of a multivariable function, Differentiability implies continuity.
第11週		Functions of bounded variation, Partition, Total variation, Additivity of total variation, Characterization of a function of bounded variation, Difference of two increasing functions, Curves, Length.
第12週		Rectifiable curves, Length of a rectifiable curve, Riemann-Stieltjes integrals, Partition, Refinement of a partition, Integration by parts, Change of variable, Continuously differentiable integrator.
第13週		Integrability based on the upper and lower Riemann sums, Comparison theorems, Sufficient conditions for the integrability with respect to integrators of bounded variation, Integration by parts (continuous functions as integrators).
第14週		Break (self study).
第15週		Integrators of bounded variation, Integrability on an subinterval, Integrability with respect to the total variation of a function of b.v., Mean-Value theorems for Riemann-Stieltjes integrals, Fundamental theorems of calculus.
第16週		Change of variable in a Riemann integral, Riemann-Stieltjes integrals depending on a parameter (continuity and differentiability), Interchange of the order of integration, Lebesgue's theorem, Sets of measure zero, Oscillation of a function.
第17週		Proof of Lebesgue's theorem: a function is Riemann integrable on [a,b] iff it is continuous almost everywhere.