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Introduction to Mathematical Analysis(Ⅱ)

Offered in 112-2

  • Serial Number

    39375

  • Course Number

    MATH2214

  • Course Identifier

    201 49660

  • No Class

  • 5 Credits

  • Preallocated

    DEPARTMENT OF MATHEMATICS

      Preallocated

    • DEPARTMENT OF MATHEMATICS

  • CHIUN-CHUAN CHEN
  • Schedule Location
    • Tue 2, 3, 4
      新102
    • Thu 3, 4
      新302

  • Type 3

  • 100 Student Quota

    NTU 90 + non-NTU 10

  • No Specialization Program

  • Chinese
  • NTU COOL
  • Core Capabilities and Curriculum Planning
  • Notes
  • Limits on Course Adding / Dropping

    • Restriction: within this department (including students taking minor and dual degree program)

  • NTU Enrollment Status

    Enrolled
    0/90
    Other Depts
    0/0
    Remaining
    0
    Registered
    0
  • Course Description
    這門課是數學系的重要課程，主要是讓學生熟悉數學分析的語言及更嚴謹的數學證明，也是更高階分析課程的基礎。我們上學期從實數基本性質、Cantor cardinal numbers及點集拓樸切入，引進極限、緊緻性、metric 的觀念，隨後介紹連續及uniform convergence。本學期將介紹fixed point theorem、Stone-Weierstrass theorem、 微分及其應用、隱函數定理、積分理論及Lebesgue定理等。如果時間允許，將略微講述基本的測度論。也會介紹Fourier series 理論。
  • Course Objective
    熟悉數學分析的基本觀念、工具、及操作嚴謹的證明。
  • Course Requirement
    週作業，期中考，期末考。 預備知識: calculus, linear algebra, Introduction to Mathematical Analysis I
  • Expected weekly study hours after class
  • Office Hour

    周一 16:00-17:00, 餘0班 舜傑助教 Office : 天數455 周二 15:00-16:00, 餘1班 行遠助教 Office : 天數445 週二 14:00-16:00, 餘2班 家豪助教 Office : 天數438

    *This office hour requires an appointment
  • Designated Reading
    1. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, 2nd Edition 2. Walter Rudin, Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics), McGraw-Hill Education; 3rd edition
  • References
    1. Jerrold E. Marsden and Michael J. Hoffman, Elementary Classical Analysis, 2nd 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
  • Grading
    25%

    homework and quiz

    35%

    midterm exam

    40%

    final exam

  • Adjustment methods for students
  • Course Schedule
    2/19-2/23Week 15. Uniform convergence 5-1. Contraction Mapping Theorem: Fredholm and Volterra equations 5-2. Bernstein's Theorem
    2/26-3/01Week 25-1. Existence and uniqueness of the solutions of an ODE 5-2. Applications of Bernstein's Theorem
    3/04-3/08Week 35-2. Revisit of Bernstein's Theorem: Law of Large Numbers 5-3. Stone-Weierstrass's Theorem
    3/11-3/15Week 45-3. Applications of Stone-Weierstrass's Theorem 5-4. Abel's and Dirichlet's tests
    3/18-3/22Week 55-5 Power series: radius of convergence, term by term differentiation 5-6 Cesaro and Abel summability: summation by parts, (C, 1) imples (Abel). Examples of a (C,2) summable series.
    3/25-3/29Week 66-1 Differentiation in R: Rolle's Theorem, Mean Value Theorem 6-2 Integration in R: upper and lower sum, Riemann integrable
    4/01-4/05Week 76-2 Integration in R: basic properties of Riemann integrals, Fundamental Theorem of Calculus
    4/08-4/12Week 86-3 Differentiation in R^n: definition, linearization, differentiable implies continuous, relation between differentiability and partial derivatives
    4/15-4/19Week 94/16: midterm examination 6-3 Chain rule
    4/22-4/26Week 106-4 Higher derivative and Taylor's expansion: 2nd derivative, Hessian matrix, higher derivatives, 1 variable Taylor expansion, Cauchy Mean Value Theorem
    4/29-5/03Week 116-4 Directional derivative, tangent plane, Taylor's expansion for several variables, L'Hopital's rule
    5/06-5/10Week 12Chapter 7. Inverse and Implicit Function Theorems 7-1 Solve a system of equations, linearization, proof of the Inverse Function Theorem
    5/13-5/17Week 137-2 Implicit Function Theorem and its proof, Lagrange multiplier Chapter 8. Integration 8-1. Riemann integrable, Riemann's condition, Darboux's Theorem
    5/20-5/24Week 148-2. Jordan content, Lebesgue (outer) measure, sets of measure zero, basic properties of measure zero sets
    5/27-5/31Week 158-3. Lebesgue's Theorem: measure zero of discontinuity set iff Riemann integrability Chapter 9. Fourier series 9-1. Wave equation, heat equation, and Fourier
    6/03-6/07Week 16June 4: final examination June 6: Convergence of the Fourier series
    Week 17