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Professor Law,Chun-Kong-ADVANCED CALCULUS (I)




Prologue 導論  Introduction
Chapter 1 實數集合  The set of real numbers
Field axioms;  order axioms;  completeness axiom;  supremum and infimum.
Chapter 2 數列的收斂性  Convergent sequences
Limit of a sequence; properties of limits;  monotone convergence theorem, Bolzano-Weierstrass theorem, nested interval theorem and Cauchy sequences; subsequences
Chapter 3 函數的極限與連續性  Limits and continuity of functions
Limit of a function and its properties; continuity of a function and its properties; extreme value theorem; intermediate value theorem; uniform continuity; examples of continuous functions; Rolle’s theorem and mean value theorem.
Chapter 4 黎曼積分理論  Riemann integrals
Upper sum and lower sum; integrable functions; integrability condition; continuous and monotone functions; Riemann sum; Darboux theorem, Fundamental Theorem of Calculus; properties of integrals; inequalities.
Chapter 5 泰勒展開式定理和冪級數  Taylor expansion theorem and power series
Taylor expansion theorem; Taylor series and Maclaurin series; uniform convergence; power series and its properties.


  1. W.R. Wade, An Introduction to Analysis, fourth edition, Pearson, 2010. ISBN 0-13-615370-4
  2. P.M. Fitzpatrick, Advanced Calculus, second edition, Thomson Brook/Cole, 2006. ISBN 0-534-37603-7.


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