課程大綱

課程資訊

課程名稱

分析導論二
Introduction to Mathematical Analysis(Ⅱ)

開課學期

108-2

授課對象

社會科學院經濟學系

授課教師

王振男

課號

ECON5130

課程識別碼

323 U2040

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

上課教室及資訊依課號MATH2214訊息為主。限選修ECON課號之課程，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
總人數上限：20人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1082ECON5130_b

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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。分析二的重點為多變數的分析，反函數及隱函數定理，多重積分都是重要的觀念。我們也將學習到一些分析有趣的應用。

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

修過分析導論一。

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.
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
*Mathematical Analysis. Second Edition. Tom M. Apostol.

參考書目

1. Walter Rudin, Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics), McGraw-Hill Education; 3rd edition
2. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, W. H. Freeman, 2nd Edition
3. 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.	作業	60%	週作業
2.	期中考	0%	因疫情取消
3.	期末考	40%

課程進度

週次	日期	單元主題
第1週	2/18,2/20	Classes suspended
第2週	2/25,2/27	Classes suspended
第3週	3/03,3/05	Taylor's formula, Sequences, Convergence, Limit superior, Limit inferior, Series, Inserting and removing parentheses, Alternating series, Absolutely and conditionally convergence, Rearrangements of series, Riemann's theorem on conditionally convergent series, Tests for convergence, Comparison test, Limit comparison test.
第4週	3/10,3/12	Ratio test, Root test, Integral test, Summation by parts, Dirichlet's test, Abel's test, Partial sums of the geometric series on the unit circle, Rearrangement of a series, Riemann's theorem on conditionally convergent series, Double sequences, Double series.
第5週	3/17,3/19	Convergence of a double series, Iterated series, Cauchy product of two series, Mertens theorem, Cesaro summability, Infinite products, Convergence of an infinite product.
第6週	3/24,3/26	Convergence of an infinite product, Cauchy's criterion, Absolute convergence, Riemann zeta function, Euler's product, Pointwise convergence, Uniform convergence, Continuity and uniform convergence, Cauchy's criterion, Infinite series of functions, Convergence, Weierstrass M-test.
第7週	3/31,4/02	Dirichlet test for uniform convergence, Integrability and uniform convergence, Integration term by term, Differentiability and uniform convergence.
第8週	4/07,4/09	Convergence in mean, Incompleteness of L^2[a,b] defined by the Riemann integrals, Power series, Radius of convergence, Disk of convergence, Integration term by term of a power series, Differentiation term by term of a power series, Taylor's series, Analytic functions.
第9週	4/14,4/16	Multiplication of power series, Abel's limit theorem, Tauber's theorem, Sequential compactness in a metric space, Total boundedness.
第10週	4/21,4/23	The equivalence of compactness and sequential compactness for a metric space, Arzela-Ascoli theorem, Existence of the initial value problem, Weierstrass approximation theorem.
第11週	4/28,4/30	Functions of multivariable, Continuity, Differentiability, Mean-value theorem, Taylor's formula, Inverse function theorem.
第12週	5/05,5/07	Proof of inverse function theorem, Metric space of matrices, Application of inverse function theorem, Implicit function theorem, Continuity of the simple root.
第13週	5/12,5/14	Proof of ImFT, Geometric interpretation of ImFT, Method of Lagrange's multiplier, Multiple Riemann integrals.
第14週	5/19,5/21	Jordan content, Jordan measurable sets, Change of variables, Lebesgue's outer measure, G_\delta, F_\sigma sets.
第15週	5/26,5/28	Measurable sets, \sigma-algebra, Borel algebra, Countable additive, Continuity of measure, Caratheodory characterization.
第16週	6/02,6/04	Proof of Caratheodory's characterization, Non-measurable sets, Zermelo's axiom of choice, Measurable functions, Simple functions.
第17週	6/09,6/11	Egorov's theorem, Convergence in measure, Lebesgue integrals, Monotone convergence theorem, Fatou's lemma, Dominated convergence theorem, Bounded convergence theorem.