Zarqa University
Faculty of Science
Department: Math.
Course title: Real Analysis(2)
(0301411)
Instructor: Dr Jamila Jawdat
Lecture’s time: 2 – 3 S/T/T
Semester: 1st 2015- 2016
Office Hours: 1 – 2 S/T/T
Course description:
In this course, Riemann integrals, Riemann- Stieltjes integrals, and functions
of bounded variation are studied. On the other hand, the course deals with the
n
n
n-dimensional Euclidean space R and differential calculus on R : partial
and directional derivatives, the Jacobian matrix along with some related
theorems.
Aims of the course:
This course extends the course "Real Analysis(1)" in which theorems of Calculus are
analyzed and proved.
Intended Learning Outcomes: (ILOs)
A.
Knowledge and Understanding
A1. Concepts and Theories:
A2. Contemporary Trends, Problems and Research:

To study Riemann integral, its properties and a generalization: the RiemannStieltjes integral.

To study functions of bounded variation and the relations with R- and R-S
integrals.

To study differentiability of functions on R : partial and directional
derivatives of vector functions.

To study the total derivative of vector functions and the Jacobian Matrix.

To study some related theorems: the inverse function theorem and the implicit
function theorem.
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B.
Subject-specific skills
B1. Problem solving skills:
B2. Modeling and Design:
B3. Application of Methods and Tools:.
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C.
Critical-Thinking Skills
C1. Analytic skills: Assess
C2. Strategic Thinking:
C3. Creative thinking and innovation:
D.
General and Transferable Skills (other skills relevant to employability and personal development)
D1. Communication:
D2. Teamwork and Leadership:
Course structures:
Week
1
Credit
ILOs
Hours
3
Topics
Teaching
Procedure
Assessment
methods
Some revision on the concept of
bounded sets, bounded functions and
their supremum and infimum.
 Partitions, lower and upper sums and
their properties.
Presentations
and
discussions
Quizzes and
homeworks
2
 Definition of The Riemann integral
for bounded functions on bounded
intervals.
 Examples on Riemann integrable
and non integrable functions.
 Riemann criterion for integrability
(Riemann Condition).
3
 Some classes of integrable functions
 Properties of Riemann integrals.
4
 (continue) Properties of Riemann
integrals.
 Mean Value Theorem for integrals.
5
 Riemann integral as a limit of
Riemann sums.
The First Exam
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6
7
 The integral as a function
The Fundamental Theorem of
Calculus.
Integration by substitution and by
parts.
Definitions of Riemann-Stieltjes
(R-S) integrals and examples.
Riemann condition for R-S
integrability.
8
Some classes of R-S integrable
functions.
Properties of Riemann integrals.
9
Two theorems that are used to
calculate R-S integrals.
Mean Value Theorem for R-S
integrals.
Functions of Bounded Variation:
10
.Definitions and some examples.
Properties of Bounded Variation
functions and related theorems.
The
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Second Exam
 Total Variation Functions and some
properties.
 Jordan decomposition theorem.
12
 Riemann-Stieltjes integrals
with integrators are functions of
bounded variation.
n
13
(Differential Calculus on R )
The n-Dimensional Euclidean Space
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n
R.
n
Differentiability on R ; partial and
directional derivatives.
Differentiability of vector functions
n
m
f: R R .
The Jacobian matrix and total
derivatives of vector functions.
The Inverse Function Theorem & the
Implicit Function Theorem.
14
References:
A.
Main Textbook:
Introduction to Mathematical Analysis, by S.A. Douglass.
B.
Supplementary Textbook(s):
a. The Elements of Real Analysis, by Bartle
b. Mathematical Analysis, by Apostol.
c. Principles of Real Analysis, by S.L. Gupta and Nishra Rani.
d. Mathematical Analysis, by S. Shirali and H. L. Vasudeva
Assessment Methods:
Methods
Grade
25%
25%
1st exam
2nd exam
Date
19. 11. 2015
24. 12. 2015
Final Exam
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