基本素養 Basic Literacy

核心能力 Competence

掌握連續與逼進的能力
Ability to handle continuity phenomena and approximations.
處理數字及符號運算的能力
Ability to operate with symbols and digits.
透視形像與空間的能力
Ability to visualize space, shape, and images.
利用計算機處理數學的能力
Ability of using computer to do mathematics.
應用隨機理論的能力
Ability to apply stochastic mathematics.
處理大量數據的能力
Ability to handle large amount of data.
處理離散型數學的能力
Ability to do mathematics relating to discrete models.
獨立研究的能力
Ability to do independent studies of mathematics.
使用計算機的能力
Abilities to use computer as a tool.

課程概述 Course Description

這是一個學年的課程。介紹微分、積分及函數性質,重點為定理的證明。我們研究的實數系和基本的拓撲結構,然後討論序列和級數。接下來,我們討論函數的連續性,微分、積分。最後,我們將談論函數所形成的序列和級數。
1.Real number system, Euclidean n-space 2.Point set topology: open and closed sets, boundary of a set, sequence and series 3.Compact and connected sets: Heine-Borel and Bolzano-Weierstrass theorems, path-connected and connected sets 4.Continuous mappings: boundedness of continuous functions on compact sets, intermediate-value theorem, uniform continuity 5.Differentiation: matrix representation, Taylor’s theorem, maxima and minima 6.Uniform convergence of sequence of functions: pointwise and uniform convergences, integration and differentiation of series, Arzela-Ascoli theorem, fixed point theorem, Stone-Weierstrass theorem, Dirichlet and Able test

課程學習目標 Course Objectives

  • 建立學生基礎分析知識及技巧熟練
  • 基本拓撲學及賦距空間之認識
  • 高維度空間及抽象空間上微積分之應用
  • 課程進度 Progress Description

    進度說明 Progress Description
    1Introduction of real number system
    2Introduction of real number system
    3Countability, completeness
    4Sequence, Cluster points, Limit inferior and superior
    5Normed vector space, inner product spaces and metric spaces
    6Open set, closed set, boundary
    7sequence and completeness
    8Compactness, Connectedness
    9Continuity, Operations on continuous maps
    10Images of comapact sets, connected sets and path connected sets under continuous maps
    11Uniform continuity
    12Differentiation and Integration of functions of one variable
    13Pointwise and uniform convergence
    14Series of functions and the Weierstrass M-test
    15Integration and Differentiation of series
    16Space of continuous functions, Arzela-Ascoli theorem
    17Stone-Weierstrass theorem, contraction mapping principle
    18the existence and uniqueness of the solutions to ODEs
     以上每週進度教師可依上課情況做適度調整。The schedule may be subject to change.

    有關課程其他調查 Other Surveys of Courses

    1.本課程是否規劃業界教師參與教學或演講?
    Is there any industry specialist invited in this course? How many times?
    2.本課程是否規劃含校外實習(並非參訪)?
    Are there any internships involved in the course? How many hours?
    3.本課程是否可歸認為學術倫理課程? 否
    Is this course recognized as an academic ethics course? In the course how many hours are regarding academic ethics topics? No
    4.本課程是否屬進入社區實踐課程? 否
    Is this course recognized as a Community engagement and Service learning course? Which community will be engaged? No