ANALYSIS 1

Compulsory course. 6 credits.

 Theory:  4 credits  +  Exercises : 2 credits

I.  Overview

  This course provides : The real number. Sequences and series of real numbers. The continuity, the differentiation and the Riemann integral of function  of one-variable. Applications of Mathematica for differential and integral calculus.

II. Prerequisites : no

III. Contents

  Chapter 1 : Sets and maps :  Set, relations in a set, maps.

  Chapter 2 : Real numbers : Integers, rational numbers, irrational numbers and real numbers.  Least upper bound and greatest lower bound.

  Chapter 3 :  Sequences and series of real numbers :  sequences of real numbers,  limits of  sequences of real numbers, liminf and limsup, series of  real numbers,  Mathematica for sequences and series of real numbers.

  Chapter 4 : Continuous real functions :  continuity and its basis properties, the continuity of elementary real functions, properties of continuous real functions on an intervals, uniform  continuity.

  Chapter 5 : Differentiability : Limits of  real functions and their basic properties. Derivatives of real functions and their basic properties. Properties of differentiable real functions on an interval. High order derivatives. Taylor expansion. Applications of differentiability theory. Mathematica for differentiable calculus.

  Chapter 6 : Riemannian integral : Riemannian integral and its basic properties. Basic theorem of integral calculus. Improper integrals. Mahematica in integral calculus.

References

[1] M.I. Abell and J.P. Braschon,  Mathematica  by  example, Academic Press, New York, 1997.

[2] S.I. Grossman,   Calculus, Harcourt Brace College Publishers,  New York, 1992.

[3]   Duong Minh Duc , Analysis I  (Vietnamese). Statistics Publisher, Hochiminh City, 2005.

[4] W. Rudin, Principles of mathematical analysis, McGraw-Hill, New York, 1964.

[5]   S. Wolfram, Mathematica,  Cambridge University Press, 1996.