Syllabus for MATH01 510 – Real Analysis I

Dr. Abdul Hassen

E-mail           Hassen@rowan.edu               Phone          (856)256 4500 ext 3888

 

CATALOG DESCRIPTION: The theoretical treatment of the foundations of calculus covering the real number system, elementary set theory, number sequences and series, continuity, differentiation, Riemann integration, and sequences & series of functions

OBJECTIVES:         The purpose is to provide students an opportunity for a rigorous treatment of analysis.  The intention is to effect a transition from elementary calculus where a considerable part of the effort is necessarily devoted to mastering the technical aspects of differentiation and integration.  The emphasis thus shifts to the development of concepts and methods of proof.  The student should be adequately prepared by this course to continue with other branches of analysis such as complex variable theory and topology.

 

PREREQUISITE:    Real Analysis I (Math 01 510)

TEXT: Goldberg, Richard, METHODS OF REAL ANALYSIS, 2^nd edition, John Wiley & Sons, 1976

 

CONTENT:   We will cover the following topics from the text. (Lecture notes are available at WebCT)

1.         Sets, relations, Functions, Equivalence Relations

 

2.         Properties of the Real Numbers

 

2.1       The complete ordered field definition

2.2       Denseness of the rationals and irrationals

2.3       Countable and uncountable sets

2.4       Absolute value

2.5       Least upper bound, greatest lower bound and the completeness property

 

3.         Sequences of Real Numbers

 

3.1       Sequences and their limits

3.2       Cauchy sequences, subsequences, monotone sequences

3.3       Limit superior and limit inferior

 

4.         Series of Real Numbers

 

4.1       Convergence and divergence

4.2       Series with nonnegative terms

4.3       Alternating series

4.4       Conditional convergence and absolute convergence

4.5       Rearrangement of series

 

5.         Continuity

 

5.1       Definitions and elementary theorems

5.2       Uniform continuity

5.3       Relative and absolute extrema

5.4       Intermediate value property

 

6.         Differentiability

 

6.1       Definition and elementary properties

6.2       Conditions for differentiability

6.3       Derivatives of composite functions, chain rule

6.4       Mean value theorems

6.5       L'Hospital's rule

 

7.         Riemann Integration

 

7.1       Definition of Riemann integrability

7.2       Elementary properties of the Riemann integral

7.3       Riemann condition for integrability

7.4       Fundamental theorem of Calculus

 

8.         Sequences and Series of Functions

 

8.1       Pointwise convergence and sequences of functions

8.2       Uniform convergence of sequences of functions

8.3       Consequences of uniform convergence

8.4       Convergence and uniform convergence of series of functions

8.5       Differentiation and integration of series of functions

 

GRADING POLICY:    Students will be graded based on midterm and final exams (60% of the total grade), projects (20% of total grade), and assignments (20% of the total grade). The dates for the tests will be announced in class at least a week in advance. 

Numerical grades will be converted to letter grades by the following scale.

A (-)= 90 to 100, B(-,+)= 80 to 89, C(-,+)= 70 to 79, D(-,+)= 60 to 69, F= 0 to 59

HOMEWORK      These will be given in class and the due dates will be announced. I suggest that you get a separate notebook for homework assignments