教學大綱 Syllabus

週次

課程主題

課程內容與指定閱讀

教學活動與作業

1

1.2-1.11

The field, order axioms and geometric representation of real numbers, The intervals, integers, rational numbers, irrational numbers and upper bounds, maximum element least upper bound and the completeness axioms

第一章習題演練與講解;學習評量測驗

2

1.12-1.20

Properties of the integers deduced from the completeness axiom. The Archimedean property of the real number system. Rationalnumbers with finite decimal representation. Finite decimal approximations to rea numbers. Infinite decimal representation of real numbers. Absolute values and the triangle inequality. The Cauchy-Schwarz inequality. Plus and minus infinity and the extended real number system. Complex numbers.

第一章習題演練與講解;學習評量測驗

3

1.22-1.22

The imaginary unit. Absolute value of a complex number. Imposibility of ordering the complex numbers. Complex exponentials.  The argument of a complex number. Integral powers and roots of complex numbers. Complex logarithms. Complex powers. Complex sines and cosines. Infinity and the extended complexplane.

第一章習題演練與講解;學習評量測驗第一第一章習題演練與講解;學習評量測驗

4

2.2-2.15

Cartesian product of two sets.  One-to-one functions and inverses . Composite functions.  Finite and infinite sets. Countable and uncountable sets. Uncountability of the real-number system. Set algebra . Countable collections of countable sets

第二章習題演練與講解;學習評量測驗

5

3.2-3.11

Euclidean space R^n. Open balls and open sets in R^n . The structure of open sets in R^1. Closed sets. Adherent points. A cumulation points. Closed sets and adherent points. The Bolzano-Weierstrass theorem. The Cantor intersection theorem. The Lindelof covering theorem. The Heine-Borel covering theorem. . Compactness in R^n.

第三章習題演練與講解;學習評量測驗

6

3.12-3.16

Metric spaces. Point set topology in metric spaces. Compact subsets of a metric space. .Boundary of a set

第三章習題演練與講解;學習評量測驗

7

4.1-4.6

Convergent sequences in a metric space. Cauchy sequences. Complete metric spaces. Limit to a function. Limits of complex-valued functions.

第四章習題演練與講解;學習評量測驗

8

期中考

Chapter 1 and Chapter 2, Chapter 3, Chapther 4.1-4.6

9

4.7-4.12

Limits of vector-valued functions. Continuous functions. Continuity of composite functions. Continuous complex-valued and vector-valued functions. Examples of continuous functions . Continuity and inverse images of open or closed sets.

第四章習題演練與講解;學習評量測驗

10

4.13-4.23

Functions continuous on compact sets. Topological mappings. Bolzano's theorem. Conectedness. Components of a metric space. Arcwise connectedness, Uniform continuity and  compactsets . Fixed-point theorem for contractions. Discontinuities of real-valued functions. Monotonic functions.

第四章習題演練與講解;學習評量測驗

11

5.1-5.7

Definition of derivative. Derivatives and continuity. Algebra of derivatives. The chain rule. One-side derivatives and infinite derivatives. Functions with nonzero derivative.

第五章習題演練與講解;學習評量測驗

12

5.8-5.12

The Mean-Value Theorem for derivatives.Intermediate-value theorem for derivatives. Taylor's formula with remainder. Derivatives of vector-valued functions

第五章習題演練與講解;學習評量測驗

13

5.13-5.16

Partial derivatives. Diferentiation of functions of a complex variable. The Cauchy-Riemann equations

第五章習題演練與講解;學習評量測驗

14

6.2-6.8

Propositions of bounded variation, total variation, functions of bounded variation, continuous functions of bounded variation

第六章習題演練與講解;學習評量測驗

15

7.2—7.8

The definition of the Riemann-Stieltjes integral, Integration by parts, Change of variable, Reduction to a Riemann-Stieltjes integral

第七章習題演練與講解;學習評量測驗

16

期末考

Chapter 4-Chapter 7