Fall 2022 MATH 2111
Advanced Calculus 1 高等微積分(一)

Syllabus 課程大綱

宣導事項

教學進度

  • 9/5/2022(星期一): Introduction and overview.
  • 9/7/2022(星期三): Definition of real numbers.
  • 9/12/2022(星期一): Introduction to topology on vector spaces.
  • 9/14/2022(星期三): Open and closed subsets.
  • 9/19/2022(星期一): Neighborhoods, boundaries, interiors, and closures.
  • 9/21/2022(星期三): Cluster points and Bolzano-Weirstrass Theorem.
  • 9/26/2022(星期一): Compact subsets, Heine-Borel Theorem.
  • 9/28/2022(星期三): More on compact subsets, connected subsets.
  • 10/3/2022(星期一): Properties on connected subsets.
  • 10/5/2022(星期三): Connected subsets on R, introduction to metric spaces.
  • 10/10/2022(星期一): Metric spaces and sequences.
  • 10/12/2022(星期三): No class. (台灣國慶日)
  • 10/17/2022(星期一): Sequences and some basic properties.
  • 10/19/2022(星期三): Midterm Exam 1.
  • 10/24/2022(星期一): Subsequences and completeness.
  • 10/26/2022(星期三): Completeness of real numbers.
  • 10/31/2022(星期一): limsup and liminf.
  • 11/2/2022(星期三): More on limsup and liminf, some examples.
    <1i>11/7/2022(星期一): Double sequence and some applications.
  • 11/9/2022(星期三): Sequences of functions and uniform convergences.
  • 11/14/2022(星期一): Sequences of functions, other convergence modes, introduction to continuous functions.
  • 11/16/2022(星期三): Continuous functions and connected sets, intermediate value theorem.
  • 11/21/2022(星期一): Continuous functions and compact sets, extreme value theorem.
  • 11/23/2022(星期三): Uniform continuity, fixed point and contraction principle.
  • 11/28/2022(星期一): Berstein polynomials, Weirstrass approximation theorem.
  • 11/30/2022(星期三): Stone approximation theorem.
  • 12/5/2022(星期一): Urysohn's lemma and Tietze extension theorem.
  • 12/7/2022(星期三): Midterm 2.
  • 12/12/2022(星期一): Equivalence of Urysohn'a lemma and Tietze extension theorem, Arzela-Ascoli Theorem.
  • 12/14/2022(星期三): Review of differentiation, Rolle's Theorem and mean value theorem.
  • 12/19/2022(星期一): Taylor's theorem.
  • 12/21/2022(星期三): Uniform convergence and differentiations, introduction to Riemann-Stieltjes integration.
  • 12/26/2022(星期一): Basic properties of Riemann-Stieltjes integration.
  • 12/28/2022(星期三): More properties of Riemann-Stieltjes integration, other definitions of integrations.

    作業

  • Homework 1 (截止日9/14/2022):
    Section 6 (pp.42-43): A, B, F, G, I, K, M (replace "2" by "n" and "square root" by "nth root").
    Section 7: A, D, G, K, L.
    Prove that LUBP is equivalent to GLBP.
  • Homework 2 (截止日 9/28/2022， 9/26 演習課 抽題小考):
    Section 9 (pp. 68-68): G, H, I, J, M, O, Q.
    Section 10 (p72: C, D, F, G.
  • Homework 3 (截止日 10/5/2022):
    Section 11 (pp. 79-80): C, D, H, I, K, L, P, Q.
    Section 12 (pp. 85-86): A, B, E, F, I.
  • Homework 4 (截止日 10/17/2022，當日抽題小考):
    Download Here
  • Homework 5 (截止日 11/2/2022):
    Section 15: C, D, F, L, O.
    Section 16: A, C, I, K, Q.
  • Homework 6 (截止日 11/9/2022，當日抽題小考):
    Section 18 (pp. 127-128): A(bc), B, F, G(bd).
    Section 19 (p 135): E, I, K, O.
    Prove that {m{nx}}={mnx} for all m,n in N and x in R.
  • Homework 7 (截止日 11/23/2022，於11/21/2022演習課抽題抽題小考):
    Section 17: A, K, D, M, F, G, P, S.
    Section 20: E, H, I, L, O, Q.
    Prove that on a normed vector space (V, || . ||), the function x -> ||x|| is continuous.
  • Homework 8 (截止日 11/30/2022):
    Section 22 (pp. 157-158): C, F, I, J, Q, R.
    Section 23 (p 164): D, G, I, L, M.
  • Homework 9 (截止日 12/14/2022):
    Section 24: (pp. 173-171): A, B, F, J, I, R.
    Section 26: (pp. 191-192): B, C, F, H, J, K, M, Q, R.
    For a closed subset A, prove that f(x)=d(x,A) defined in class is a continuous function.
    Note: 第二次期中考範圍至 Section 25 H.
  • Homework 10 (截止日 12/21/2022，於12/19/2022演習課抽 Section 27 之習題小考):
    Section 27: C, H, I, P, U.
    Section 28: F, G, H.
  • Homework 11 (截止日 12/28/2022):
    Section 28 (p208): M, P.
    Section 29 (pp. 222-224) : C, E, G, I, J, S.
  • Homework 12 (截止日 1/3/2023 17:00 @老師信箱):
    Section 29: B, L, M, R, S, T.

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