若是通識課程評價，請用 [通識] 分類，勿使用 [評價] 分類 標題範例：[通識] A58 普通心理學丙 林以正 (看完後請用ctrl+y刪除這兩行) ※ 本文是否可提供臺大同學轉作其他非營利用途？（須保留原作者 ID） （是／否／其他條件）： 哪一學年度修課： ψ 授課教師 (若為多人合授請寫開課教師，以方便收錄) 陳俊全 λ 開課系所與授課對象 (是否為必修或通識課 / 內容是否與某些背景相關) 數學系 δ 課程大概內容 第1週 9/15,9/17 0. Introduction -problems arising from calculus; new topics: 0.1.real numbers and completeness 0.2 what is infinity? 0.3 topology of the Euclidean space: Riemann integral and compactness 0.4 uniform convergence of functions 0.5 differentiation in R^n 0.6 Solve a system of non-linear equations:- Inverse and Implicit Function The orems 0.7 Lebesgue's Theorem for integrals 0.8 Fourier series 1. The real number system and the Euclidean space 1.1 Sets and Functions: - power set of A, product of A and B - domain, target, range of a function, one-to-one, onto 1.2 Origin of number concept - Piraha people in the Amazon rainforest - Research on infants 1.3 Number system: natural numbers, integers, rational numbers 第2週 9/22,9/24 1.4 Ordered Fields - addition axioms, multiplication axioms and order axioms - sequence and limit: uniqueness of limits, sandwich lemma, limits of a sum and a product - Cauchy sequence - Axiom of completeness 第3週 9/29,10/01 Basic properties of Cauchy sequences Axioms of a complete ordered field 第4週 10/06,10/08 1-5 Construction of a complete ordered field 1-5-1 three approaches: infinite decimals, Cauchy sequences and Dedekind cuts 1-5-2 Cauchy sequence approach: - S=the set of all rational Cauchy sequences - an equivalence relation on S and the corresponding equivalence classes - addition and multiplication on the equivalence classes 第5週 10/13,10/15 1-5-2 Cauchy sequence approach: - order on the equivalence classes - Cauchy sequences in the space of the equivalence classes -the equivalent classes together with the addition, multiplication and order f orms a complete ordered field. 第6週 10/20,10/22 -Theorem: There exists a "unique" complete ordered field, called the real numb er system. - Monotone sequence property (MSP) -sup, inf and the least upper bound property (LUBP) 第7週 10/27,10/29 -Theorem: The three versions of completeness (CSP)+(AP), (MSP) and (LUBP) are equivalent. 1-6 limsup and liminf 第8週 11/03,11/05 - more properties and applications of limsup and liminf, 1-7 Cantor's theory of infinity - Definition of card A=card B and card A < card B - finite, countable and uncountable - an infinite subset of a countable set is countable - card N = card Q < card R = card RxR=card P(N), Cantor's diagonal method - card A < card P(A) - existence of an algebraic number 第9週 11/10,11/12 - Schroder-Bernstein Theorem - continuum hypothesis: Godel and Cohen 1-8 Some "paradoxes" about real numbers - a number of all knowledge - Pi is a normal number? Borel's theorem: Almost every real number is normal. - Richard's paradox 1-9 Complex numbers 1-10 Euclidean space - norm, metric, inner product, Schwarz's inequality Chapter 2 Topologies of Metric Spaces 2-1 Metric space: definition and examples 第10週 11/17,11/19 Midterm examination 2-2 Open sets and interior of a set 第11週 11/24,11/26 2-3 Closed sets, accumulation points, closure of a set 2-4 Boundary of a set 2-5 Sequences and limits 2-6 Completeness of a metric space 第12週 12/01,12/03 Chapter 3 Compact sets 3-1. Examples: the difference between I= [0,1] and I=(0,1]; consider continuou s function on I 3-2 Sequentially compact: bisection process and bounded sequence; Heine-Borel Theorem 3-3 Open cover and compact: - examples 第13週 12/08,12/10 - compact implies bounded and closed; counterexample - totally bounded; - Bolzano-Weierstrass Theorem 第14週 12/15,12/17 - compact iff totally bounded and complete in a metric space 3-4 Path-connected and connected - path connected implies connected Chapter 4 Continuous maps 4-1 Continuity - limit at a point - continuous at a point and on the whole domain - continuity defined by sequential limits 第15週 12/22,12/24 - continuity characterized by preimages of open and closed sets - continuity for +,-,?,?, and f(g(x)) 4-2 Images of compact and connected sets 4-3 Real-valued functions - Maximum-minimum theorem - Intermediate value theorem 4-4 Uniform continuity 第16週 12/29,12/31 Chapter 5 Uniform convergence of functions -Motivations 5-1 Pointwise and uniform convergence - examples - uniform convergence implies pointwise convergence - uniform convergence iff sup ρ(f_k,f) → 0 - Theorem: The limit function of an uniformly convergent sequence of continuous functions is continuous. 5-2 Cauchy criterion and M test - Cauchy criterion and uniform convergence - examples: uniformly convergence of series of functions 第17週 1/05,1/07 5-3 Integration and differentiation of sequences and series of functions - Theorem: uniform convergence implies convergence of the integrals - Theorem : pointwise convergence of the functions and uniform convergence of their derivatives together imply differentiability of the limit function 5-4 The space of continuous functions - completeness property - equicontinuity - Arzela-Ascoli Theorem 第18週 1/12 Final Exam (這些是 Ceiba 上寫的） Ω 私心推薦指數(以五分計) ★★★★★ 老師：★★★★（有趣老師） 喜歡看鬼滅：-★★★★★（老師說國中生才看鬼滅） 整體：★★★★ η 上課用書(影印講義或是指定教科書) 1. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, 2 nd Edition 2. Walter Rudin, Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics), McGraw-Hill Education; 3rd edition 3. Mathematical Analysis. Second Edition. Tom M. Apostol. 4. William R. Wade, An Introduction to Analysis, Prentice Hall, 4th Edition μ 上課方式(投影片、團體討論、老師教學風格) 都寫黑板，老師超有趣，可以去查俊全語錄，但常常遲到，可以睡晚一點XD 。其他就， 我覺得老師很神，上課每次的證明好像都記在腦子裡，好像沒看過他帶任何筆記，每次都 只有帶咖啡# σ 評分方式(給分甜嗎？是紮實分？) 1. homework and quiz 25% 2. midterm exam 35% 3. final exam 40% 我覺得是扎實分，期中平均78，期末平均49，但應該是有調分的。 ρ 考題型式、作業方式 考題我放在考試版了，會考上課證明跟作業，考前一週上的內容也是必考。BTW, 期中超 簡單，但期末直接大暴死QQ，不知道是不是每屆都這樣，你各位自己注意啊。 作業大概 2/3 簡單、1/3 難，每週大概會花4-5小時寫作業，有時候更久。 ω 其它(是否注重出席率？如果為外系選修，需先有什麼基礎較好嗎？老師個性？ 加簽習慣？嚴禁遲到等…) 一起寫在總結OuO Ψ 總結 我覺得是外系的要想清楚，自己為啥要修這門課吧，我當初是想了解一些，常在論文中看 到的名詞跟概念。一學期下來雖然有學到，但其實我覺得有點不合時間成本，如果只是要 了解那些概念，自己去看書可能會比較快，但不可否認地，上課還是學到很多，算是有其 他意外的收穫，例如：找 bound 的技巧、norm的一些等價概念......。總之，是有收穫 的！ --

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