REAL ANALYSIS II

Course Code:

MATH 353

Course Type:

Area Elective

P:

3

Lab:

0

Laboratuvar Saati:

0

Credits:

3

ECTS:

7

Prerequisite Courses:

REAL ANALYSIS I

Course Language:

English

Course Objectives:

This course is the continuation of Real Analysis I and together with Real Analysis I they constitutes the pillar of many topics in mathematics such as complex analysis, differential equations, differential and integral calculus, and differential geometry. It is impossible to assimilate these areas of mathematics without having this basic knowledge of analysis. The aim of the course is to equip students with this basic knowledge.

Course Content:

Riemann Integral. Riemann Integrable Functions. The Fundamental Theorem of Calculus. The Darboux Integral. Sequences of functions. Pointwise and Uniform Convergence. Interchange of Limits. The Exponential and LogarithmicFunctions. The Trigonometric Functions.

Course Methodology:

1: Lecture, 2: Problem Solving

Course Evaluation Methods:

A: Written examination, B: Homework

Vertical Tabs

Course Learning Outcomes

Learning Outcomes	Program Learning Outcomes	Teaching Methods	Assessment Methods
1) A sound understanding of Riemann integral	2,3,4	1	A,B
2) Learn the class of Riemann integrable functions.	2,3,4	1	A,B
3) Aquire the knowledge of Darboux integral and its equivalence to Riemann integral of functions	2,3,4	1	A,B
4)Aquire the knowledge of Darboux integral and its equivalence to Riemann integral of functions	2,3,4	1	A,B
5) Learn sequences of functions.	2,3,4	1	A,B
6) Apply these ideas to obtain rigorous definitions most important analytic functions	2,3,4	1	A,B

Course Flow

Week	Topics	Study Materials
1	Partitions and Tagged Partitions, Riemann Sum, Riemann integrability	Bartle, Sherbert Chapter 7-1
2	Some Properties of the Integral, Boundedness Theorem	Bartle, Sherbert Chapter 7-1
3	Riemann Integrable Functions, Cauchy Criterion, Squeeze Theorem	Bartle, Sherbert Chapter 7-2
4	Classes of Riemann Integrable Functions	Bartle, Sherbert Chapter 7-2
5	Additivity Theorem	Bartle, Sherbert Chapter 7-2
6	The Fundamental Theorem of Calculus, Substitution Theorem	Bartle, Sherbert Chapter 7-3
7	Lebesgue’s integrability Criterion, Composition Theorem	Bartle, Sherbert Chapter 7-3
8	The Product Theorem, Integration by Parts, Taylor’s Theorem with the Remainder	Bartle, Sherbert Chapter 7-3
9	The Darboux Integral, Upper and Lower Sums, Upper and Lower Integrals	Bartle, Sherbert Chapter 7-4
10	Darboux integrable functions Darboux Integrability Criterion, Continuous and Monotone Functions	Bartle, Sherbert Chapter 7-4
11	Equivalence of Riemann and Darboux integrals, Sequences of Functions, Pointwise and Uniform Convergence	Bartle, Sherbert Chapter 7-4, 8
12	Cauchy Criterion for Uniform Convergence	Bartle, Sherbert Chapter 8
13	The Exponential and Logarithmic Functions	Bartle, Sherbert Chapter 8
14	The Trigonometric Functions	Bartle, Sherbert Chapter 8

Recommended Sources

RECOMMENDED SOURCES

Textbook

Robert G. Bartle,Donald R. Sherbert, Introduction to Real Anlaysis, Fourth Edition, John Wiley & Sons, Inc.(2011),ISBN-13: 978-0471433316ISBN-10: 9780471433316.

https://sciencemathematicseducation.files.wordpress.com/2014/01/0471433314realanalysis4.pdf

Additional Resources

Stephen Abbott, Understanding Analysis, Springer, 2. Edition (2015)

Material Sharing

Documents
Assignments
Exams

Assessment

IN-TERM STUDIES	NUMBER	PERCENTAGE
Mid-terms	2	70
Quizzes
Assignments	3	30
Total		100
CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE		40
CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE		60
Total		100

COURSE CATEGORY

Expertise/ Field Courses

Course’s Contribution to Program

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry)					x
2	Acquiring fundamental knowledge on fundamental research fields in mathematics					x
3	Ability form and interpret the relations between research topics in mathematics					x
4	Ability to define, formulate and solve mathematical problems					x
5	Consciousness of professional ethics and responsibilty			x		x
6	Ability to communicate actively			x
7	Ability of self-development in fields of interest				x
8	Ability to learn, choose and use necessary information technologies			x
9	Lifelong education			x

ECTS

ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION
Activities	Quantity	Duration (Hour)	Total Workload (Hour)
Course Duration (14x Total course hours)	14	4	56
Hours for off-the-classroom study (Pre-study, practice)	14	6	84
Mid-terms (Including self-study)	1	15	15
Quizzes	-	-	0
Assignments	3	1	3
Final examination (Including self-study)	1	20	20
Total Work Load			178
Total Work Load / 25 (h)			7,12
ECTS Credit of the Course			7

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