Course Introduction

• Prerequisite：calculus
• Recommended for： Sophomore、Junior
• Introduction：

The purpose of this course is to lay down the rigorous foundation for the study of real number system, real sequences, real series and real-valued functions. The course provides a bridge from “freshman” calculus to graduate courses that use analytic ideas, e.g. , real and complex analysis, partial and ordinary differential equations , numerical analysis, fluid mechanics , and differential geometry.

• Syllabus：

1. The real and Complex Number Systems
2. Basic Topology
3. Numerical sequences and series
4. Continuity
5. Differentiation
6. The Riemann-Stieltjes Integral
7. Sequences and Series of Functions
8. Some special Functions
9. Functions of Several Variables
10. Fundamental Theorems of Vector Calculus
11. Ordinary differential equations

• Reference：

1. W. Rudin: Principles of Mathematical Analysis ,Third Edition International Editions 1976
2. J. Marsden and M. Hoffman: Elementary classical Analysis,2nd edition,1993
3. W.R.Wade, an introduction to Calculus and Analysis I & II, Third Edition ,International Edition ,2004