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.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 2. Differentiable mappings, matrix representation, conditions for differentiability, Taylor's theorem and higher derivatives, maxima and minima 3. Contraction principle, inverse function theorem, implicit function theorem, rank theorem. Lagrange multipliers, existence theorem for ODE 4. Integration: integrable functions, volume, measure zero, Lebesgue's theorem, improper integral, Fubini's theorem, change of variables, polar, spherical, cylindrical coordinates, interchanges of limiting operations 5. Fourier analysis: Fourier series, inner product spaces, orthogonal families, completeness and convergence theorems, functions of bounded variation, Fejer's theorem

課程學習目標 Course Objectives

  • 多變數微分、範數空間
  • 反函數、隱函數定理
  • 多變數積分
  • 課程進度 Course Outline

    週次 Week進度說明 Progress Description
    1Normed Vector Space
    2Differentiation for Functions of Several Variables
    3Differentiation for Functions of Several Variables
    4Inverse Function Theorem
    5Implicit Function Theorem
    6Rank Theorem
    7Taylor Theorem for Functions of Several Variables
    8Midterm 1
    9Iterated Integrals
    10Primitive Mapping、Partitions of Unity
    11Differential Forms
    12Integrals of Differential Forms
    13Change of Variables
    14Simplices and Chains
    15Stoke's Theorem
    16Midterm 2
    17Closed and Exact Forms
    18Final Exam
     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? 否
    Is there any industry specialist invited in this course? How many times? No
    2.本課程是否規劃含校外實習(並非參訪)? 否
    Are there any internships involved in the course? How many hours? No
    3.本課程是否可歸認為學術倫理課程?
    Is this course recognized as an academic ethics course? In the course how many hours are regarding academic ethics topics?
    4.本課程是否屬進入社區實踐課程?
    Is this course recognized as a Community engagement and Service learning course? Which community will be engaged?