BS Math 7th Semester

BS Mathematics (Semester Plan)

The BS Mathematics degree program is divided into eight semesters as follows.

Code

Course Title

CrHr

Pre-Req

 

Code

Course Title

CrHr

Pre-Req

Year 1 Semester 1

 

Year 1 Semester II

XXX***

UR-I

4(3+1)

None

 

MATH102

Calculus-II

3(3+0)

MATH101

MATH101

Calculus-I

3(3+0)

None

 

MATH103

Discrete Mathematics

3(3+0)

None

XXX***

UR-II

3(3+0)

None

 

XXX***

UR-IV

3(3+0)

None

XXX***

UR-III

3(3+0)

None

 

XXX***

UR-V

3(3+0)

None

XXX***

FR-I

3(3+1)

None

 

XXX***

FR-II

4(3+1)

None

Year 2 Semester III

 

Year 2 Semester IV

MATH201

Calculus-III

3(3+0)

MATH102

 

MATH211

Group Theory

3(3+0)

None

XXX***

FR-III

3(3+0)

MATH101

 

MATH271

Ordinary Differential Equations

3(3+0)

MATH101

MATH202

Number Theory

3(3+0)

None

 

MATH212

Linear Algebra

3(3+0)

None

XXX***

FR-IV

3(3+0)

None

 

XXX***

FR-V

3(3+0)

None

XXX***

UR-VI

3(3+0)

None

 

XXX***

FR-VI

3(3+0)

None

XXX***

UR-VII

3(3+0)

None

 

XXX***

FR-VII

3(2+1)

None

Year 3 Semester V

 

Year 3 Semester VI

MATH325

Vector and Tensor Analysis

3(3+0)

MATH212

 

MATH322

Real Analysis – II

3(3+0)

MATH321

MATH321

Real Analysis-I

3(3+0)

MATH101

 

MATH331

Numerical Analysis – I

4(3+1)

MATH321

MATH351

General Topology

3(3+0)

None

 

MATH332

Complex Analysis

3(3+0)

MATH321

MATH371

Partial Differential Equations

3(3+0)

MATH271

 

MATHxxx

Elective-II

3(3+0)/4(3+1)

MATHxxx

MATH313

Rings & Fields

3(3+0)

MATH211

 

MATHxxx

Elective-III

3(3+0)/4(3+1)

MATHxxx

MATHxxx

Elective-I

3(3+0)/4(3+1)

MATHxxx

 

 

 

 

 

Year 4 Semester VII

 

Year 4 Semester VIII

MATH471

Mathematical Modeling

3(3+0)

MATH271/MATH212

 

MATH472

Integral Equations

3(3+0)

MATH271

MATH424

Functional Analysis

3(3+0)

MATH212

 

MATH451

Differential Geometry

3(3+0)

MATH351

MATH431

Numerical Analysis-II

4(3+1)

MATH331

 

MATHxxx

Elective-IV

3(3+0)/4(3+1)

MATHxxx

MATHXXX/
MATH493

Elective/Report Writing & Presentation
Skills

3(3+0)/4(3+1)

MATHxxx

 

MATHxxx

Elective-V

3(3+0)/4(3+1)

MATHxxx


MATHXXX/
MATH499

Elective/Senior Design Project-I

3(0+9)/3(3+0)/4(3+1)

MATHxxx

 


MATHXXX/
MATH499

Elective/Senior Design Project-II

3(0+9)/3(3+0)/

4(3+1)

MATHxxx

                   

Note:   UR:          University Requirement Courses,    FR: Faculty Requirement Courses

University Requirement Courses (22 CrHr to be completed from this list)

 

These courses are related to general education category. Some of the courses such as Islamic studies, Pakistan studies and English are compulsory and must be studied by all students. Students will be required to complete certain courses and credit hours from this list as per the approved plan of the respective program.

 

Code

Title

CrHr

Pre-Requisite

ENG151

Functional English*

2(3+0)

None

 ENG253

Communication Skills*

3(3+0)

ENG151

ENG352 

Academics Reading and Writing

3(3+0)

ENG253

RS101

Islamic Studies *

3(3+0)

None

PS111

Introduction to Pakistan Studies *

3(3+0)

None

CS101

Introduction to Computing *

3(2+1)

None

BS121

Principles of Management

3(3+0)

None

BS131

Human Resources Management

3(3+0)

None

BS331

Organizational Theory and Behavior

3(3+0)

None

BS311

Entrepreneurship

3(3+0)

None

SWS101

Introduction to Sociology

3(3+0)

None

SWS231

Social Psychology

3(3+0)

None

SWS215 

Human Rights

3(3+0)

None

SWS201

Social Work and Human Behavior

3(3+0)

None

PHI101

Introduction to Logic

3(3+0)

None

STAT101

Introduction to Statistics

3(3+0)

None

STAT211

Basic Statistical Inference

3(3+0)

STAT101

PHI102

Introduction to Philosophy

3(3+0)

None

ECON111

Principles of Microeconomics

3(3+0)

None

BS261

Financial Accounting

3(3+0)

None

BS251

Financial Management

3(3+0)

None

*Compulsory (must be offered) and three courses of English language are also compulsory.

 

 

Faculty Requirement Courses (22 CrHr to be completed from this list)

As this program is offered by the Faculty of Physical and Numerical Sciences, therefore, there are certain courses which are mandatory to be offered to strengthen the fundamental scientific concepts of the students. Students will be required to complete 22 CrHr from the following list of Subjects.

Code

Title

CrHr

Pre-Requisite

STAT101

Introduction to Statistics

3(3+0)

None

STAT102

Probability and Statistics

3(3+0)

None ?

MATH101

Calculus – I

3(3+0)

None

PHY101

Introductory Mechanics

3(3+0)/4(3+1)

None

MATH311

Linear Algebra

3(3+0)

None

MATH471

Mathematical Modeling

3(3+0)

None

MATH271

Ordinary Differential Equations

3(3+0)

None

MATH473

Operation Research

3(3+0)

None

MATH105

Mathematics for Chemistry

3(3+0)

 

PHY104

Introductory Electricity and
Magnetism

4(3+1)

None

PHY211

Classical Mechanics

3(3+0)

None

PHY212

Quantum Mechanics

3(3+0)

None

MATH241

Mathematical Statistics-I

3(3+0)

None

MATH242

Mathematical Statistics-II

3(3+0)

MATH241

MATH131

Computing Tools

3(2+1)

None

PHY203

Introductory Electronics

3(3+0)

None

MATH108

Basic Differential Equations

3(3+0)

None

CHEM105

Introductory Chemistry

3(2+1)

None

CS102

Programming Fundamentals

4(3+1)

None

 


Core Courses (65 CrHr):

The following courses are the core courses those are compulsory for every student registered in BS Mathematics degree program.

Code

Title

CrHr

Pre-Requisite

MATH101

Calculus-I

3(3+0)

None

MATH101

Calculus-II

3(3+0)

MATH101

MATH103

Discrete Mathematics

3(3+0)

None

MATH201

Calculus-III

3(3+0)

MATH102

MATH202

Number Theory

3(3+0)

None

MATH211

Group Theory

3(3+0)

None

MATH271

Ordinary Differential Equations

3(3+0)

MATH101

MATH212

Linear Algebra

3(3+0)

None

MATH325

Vector and Tensor Analysis

3(3+0)

MATH212

MATH321

Real Analysis-I

3(3+0)

MATH101

MATH351

General Topology

3(3+0)

None

MATH371

Partial Differential Equations

3(3+0)

MATH271

MATH313

Rings & Fields

3(3+0)

MATH211

MATH322

Real Analysis – II

3(3+0)

MATH321

MATH331

Numerical Analysis – I

4(3+1)

MATH321

MATH332

Complex Analysis

3(3+0)

MATH321

MATH471

Mathematical Modeling

3(3+0)

MATH271/MATH212

MATH424

Functional Analysis

3(3+0)

MATH212

MATH431

Numerical Analysis-II

4(3+1)

MATH331

MATH472

Integral Equations

3(3+0)

MATH271

MATH451

Differential Geometry

3(3+0)

MATH351

 

BS Statistics:   

BS in Statistics is a four-year degree program. This program is structured according to national and global benchmarks. Alumni of this program are seeking after fruitful vocations in the scholarly community and associations in Pakistan and abroad. This program is accredited by the HEC.

The educational modules of BS Statistics are partitioned into different classifications as appeared in the underneath table:

Sr #

Category

Description

Credit Hours

1

University Requirements

Courses under this category are compulsory which are completed by all bachelor students at KUST.

21

2

Faculty Requirement Courses

Basic courses from The Faculty of Physical and Numerical Sciences

25

3

Core Course

Core courses related to the degree program as per the HEC and KUST guidelines.

70

4

Technical Elective

In the senior year students are provided opportunity to choose area of specializations of their interest. The Institute offers various important latest courses for students, so they can choose market oriented courses for their electives.

12

5

Senior Design Project/Electives

In order to train students to tackle a project related real world problems, senior design project is offered that is spread over the last two semesters. 

6

 

Total

 

134

 

University Requirement Courses (21 CrHr to be completed from this list)

 

These courses are related to general education category. Some of the courses such as Islamic studies, Pakistan studies and English are compulsory and must be studied by all students. Students will be required to complete certain courses and credit hours from this list as per the approved plan of the respective program.

Code

Title

Cr Hrs

Pre-Requisite

General Electives Courses

ENG151

Functional English*

3(3+0)

None

 ENG253

Communication Skills*

3(3+0)

ENG151

ENG352 

Academics Reading and Writing

3(3+0)

ENG253

RS101

Islamic Studies *

3(3+0)

None

PS111

Introduction to Pakistan Studies *

3(3+0)

None

BS141

Principles of Management

3(3+0)

None

ECON101

Fundamental of Economics

3(3+0)

None

 *Compulsory (must be offered) and three courses of English language are also compulsory

Faculty Requirement Courses (25 CrHr to be completed from this list)

As this program is offered by the institute of Numerical Sciences, therefore, there are certain courses which are mandatory to be offered to strengthen the fundamental scientific concepts of the students. Students will be required to complete 25 CrHr from the following list of Subjects.

Code

Title

CrHr

Pre-Requisite

CS101

Introduction to Computing

4(2+1)

None

MATH101

Calculus-I

3(3 + 0)

 

MATH102

Calculus-II

3(3 + 0)

MATH101

MATH103

Discrete Mathematics

3(3 + 0)

 

MATH201

Calculus-III

3(3 + 0)

MATH103

ENV101

Introduction to Environmental Science

3(3 + 0)

 

MS245

Total Quality Management

3(3 + 0)

 

BTGE101

Introduction to Bioinformatics

3(3 + 0)

 

                                                                     

Core Courses (70 CrHr):

The following courses are the core courses those are compulsory for every student registered in BS Statistics degree program.

Code

Title

CrHr

Pre-Requisite

STAT101

Introduction to Statistics              

3(3 + 0)

 

STAT102

Introduction to Probability Distributions

3(3 + 0)

STAT101

STAT211

Applied Statistics

3(3 + 0)

STAT101

STAT201

Introduction to Regression and Analysis Experimental Design

3(3 + 0)

 

STAT221

Basic Statistical Inference

3(3 + 0)

STAT102

STAT222

Linear Algebra

3(3 + 0)

MATH101

STAT321

Probability and Probability Distribution- I

3(3 + 0)

STAT102

STAT311

Sampling Techniques-I

4(3 + 1)

STAT211

STAT331

Regression Analysis

4(3 + 1)

STAT201

STAT332

Design and Analysis of Experiments-I

4(3 + 1)

STAT201

STAT313

Statistical Packages

3(3 + 0)

 

STAT322

Probability and Probability Distribution- II

3(3 + 0)

STAT321

STAT312

Sampling Techniques-II

4(3 + 1)

STAT311

STAT334

Econometrics

4(3 + 1)

STAT331

STAT333

Design and Analysis of Experiments-II

4(3 + 1)

STAT332

STAT341

Population Studies

3(3 + 0)

STAT211

STAT421

Statistical Inference-I    

3(3 0 3)

STAT221

STAT431

Applied Multivariate Analyses

4(3 + 1)

STAT222

STAT411

Survey and Research Methods

3(3 + 0)

STAT312

STAT422

Statistical Inference-II   

3(3 + 0)

STAT421

STAT412

Non-parametric Methods

3(3 + 0)

 

 

 

Technical Electives

An important aspect of the Mathematics curriculum is the Technical Elective courses. Students are given choices towards the end of the program to choose specialization of their own interest.

 

Code

Title

Cr Hrs

Pre-Requisite

General Electives Courses

STAT413

Statistical Quality Control

3(3+0)

 

STAT414

Operations Research

3(3+0)

 

STAT423

Stochastic Processes

3(3+0)

 

STAT433

Robust Regression

3(3+0)

 

STAT442

Survival Analysis

3(3+0)

 

STAT443

Biostatistics

3(3+0)

 

STAT441

Reliability Analysis

3(3+0)

 

STAT432

Time Series Analysis and Forecasting

3(3+0)

 

STAT424

Decision Theory

3(3+0)

 

 

 

 

 

 

 

 

Course Code:  MATH101

Course Title:               Calculus-I

Credit Hours:              (3 0 3)

Pre-requisite(s):         None

 

Course Objectives:

 

 The objectives of this course are:

  1. To learn about limits and their central role in calculus.
  2. To learn about derivatives and their relationship to instantaneous rates of change.
  3. To understand practical applications of derivatives.
  4. To learn about integrals: their origin in the area problem and their relationship to derivatives.

 

Reading list:

  1. Anton, I. C. Bivens, S. Davis, “Calculus, Early Transcendental”, 11th edition, John Wiley, New York, 2016.
  2. Stewart, “Calculus”, 8th edition, Brooks/Cole, 2016.
  3. B. Thomas, A. R. Finney, “Calculus”, 14th edition, Pearson, USA, 2017.

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction to the course

L2

Functions and graphing techniques.

L3

Limit of a Function and Limit Laws, One sided Limits.

L4-L5

Continuity.

L7-L7

Limits Involving Infinity; Asymptotes.

L8-L9

Tangents and Derivatives at a Point, The Derivative by first principle.

L10

Left/Right Derivatives.

L11-L12

Differentiation Rules.

L13

Derivatives as a Rate of Change.

L14

Derivatives of Trig Functions.

L15

The Chain Rule, Implicit Differentiation.

L16

Derivatives of trigonometric and logarithmic functions.

L17

Derivatives of Inverse trigonometric and hyperbolic Functions

L18-L19

Related rates of change

L20-L21

Linearization and Differentials.

L22

MID EXAM

L23-L24

Extreme Values of Functions.

L25

The Mean Value Theorem.

L26-L27

Monotone Functions and the First Derivative Test.

L28-L29

Concavity and Curve Sketching.

L30-L32

Indeterminate Forms & L’Hopitals Rule.

L33-L35

Applied Optimization.

L36-L38

Anti-derivatives and techniques.

L39-L41

Area and Estimating with finite sums.

L42-L43

Definite Integral.

L44-L45

Fundamental Theorems of Calculus

L46-L48

Improper integrals.

Course Code: MATH102

Course Title:               Calculus-II

Credit Hours:              (3 0 3)

Pre-requisite(s):         MATH101

 

Course Objectives

The objectives of this course are:

  1. To develop greater depth of understanding of integration and its importance in scientific and engineering applications.
  2. To learn about series, including their convergence properties and their use in representing functions.
  3. To learn alternative coordinate systems which are natural for many problems and learn how calculus can be applied in these systems?

 

Reading list:

 

  1. Anton, I. C. Bivens, S. Davis, “Calculus, Early Transcendental”, 11th edition, John Wiley, New York, 2016.
  2. Stewart, “Calculus”, 8th edition, Brooks/Cole, 2016.
  3. B. Thomas, A. R. Finney, “Calculus”, 14th edition, Pearson, USA, 2017.

 

   
 

 

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction to the course

L2

Review of Integration

L3-L4

Area between curves

L5-L7

Volumes Using Cross Sections

L8-L9

Volumes Using Cylindrical Shells

L10-L11

Arc Length

L12-L14

Areas of Surfaces of Revolution

L15-L17

Moments and center of mass

L18-L19

Work done by the variable force

L20

MID EXAM

L21-L22

Sequences and related theorems for the limits of sequences

L23-L24

Infinite Series, sequence of partial sums

L25-L26

Integral Test

L27-L28

Comparison Tests

L29-L30

Root and Ratio Tests

L31-L33

Alternating Series, Absolute vs. Conditional Convergence

L34-L36

Series Convergence, Power Series

L37-L39

Conic Section in Cartesian coordinates

L40-L42

Parameterization of Plane Curves

L43-L45

Polar Coordinates

L46-L48

Areas and Lengths in Polar Coordinates, Pedal equations

 

Course Code:              MATH103

Course Title:               Discrete Mathematics

Credit Hours:              (3 0 3)

Pre-requisite(s):         None

 

Course Objectives

The objectives of this course are:

  1. To learn the basics of Discrete Mathematics.
  2. To help students in gaining the understanding of mathematical reasoning and to develop their
  3. To develop problem solving skills.
  4. To show students how discrete mathematics can be used in modern computer science

 

Reading list:

 

  1. H. Rosen, “Discrete Mathematics and its Applications”, 7th edition, McGraw-Hill Education, 2011.
  2. S. Epp, Discrete Mathematics with Applications, 4th edition, Cengage India, 2011.
  3. L. Biggs, “Discrete Mathematics”, 2nd edition, Oxford University press, 2013.

 

   
 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1-L2

Introduction to the course, Set Theory(Set Operations, Venn Diagram, Algebra of sets, Counting Principle Classes of sets)

L3-L5

Logic, Propositional Equivalences, Predicates, Quantifiers

L6-L8

Nested Quantifiers, Methods of Proof

L9-L11

Algorithms, The Growth of Algorithms, Complexity of Algorithms

L12-L14

Proof Strategy, Sequences, Summations, Mathematical Induction

L15-L17

Recursive, Structural Induction, Recursive Algorithms

L18-L21

Program Correctness

L22-L23

Basics of Counting, The Pigeonhole Principal

L24

MID EXAM

L25-L27

Permutations, Combinations, Binomial Coefficients

L28-L29

Recurrence Relations, Solving Recurrence Relations

L30-L32

Generating Functions, Inclusion-Exclusion

L33-L35

Relations, n-ary Relations, Representing Relations ,Closures of Relations

L36-L38

Equivalence Relations, Partial Orderings

L39-L41

Introduction to Graphs, Graph Terminologies

L42-L43

Graph Isomorphism

L44-L46

Connectivity, Euler and Hamilton Paths

L47-L48

Planar Graphs

 

 

Course Code:  MATH201

Course Title:  Calculus-III

Credit Hours:               (3 0 3)

Pre-requisite(s):         MATH102

 

Course Objectives:

 

The objectives of this course are:

  1. To apply previously developed skills learned in Calculus to learn Multivariable Calculus and Vectors
  2. To cover Vectors, Partial Derivatives, Multiple Integrals and Vector Fields in order to prepare students for further study in technological disciplines.
  3. To cover relevant applications in science and engineering to illustrate the utility of learning these topics
  4. To use mathematical software, in problem solving, to allow the solution of more complex problems and provide visualization of the mathematical concepts in three dimensions.

 

Reading list:

  1. Anton, I. C. Bivens, S. Davis, “Calculus, Early Transcendental”, 11th edition, John Wiley, New York, 2016.
  2. Stewart, “Calculus”, 8th edition, Brooks/Cole, 2016.
  3. B. Thomas, A. R. Finney, “Calculus”, 14th edition, Pearson, USA, 2017.

 

 

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction to the course

L2-L3

3-Dimensional Coordinate Systems and Vectors.

L4

Dot Product.

L5

Cross Product.

L6

Lines and Planes in Space.

L7-L8

Cylinders and Quadric Surfaces

L9-L10

Curves and Tangents in Space.

L11-L12

Integrals of Vector Functions: Projectile Motion.

L13

Arc Length of Space Curves.

L14-L15

Functions of Several Variables.

L16

Partial Derivatives.

L17

The Chain Rule.

L18-L19

Directional Derivatives and Gradients.

L20-L21

Tangent Planes and Differentials.

L22

MID EXAM

L22-L23

Extrema and Saddle Points.

L24

Lagrange Multipliers.

L25-L26

Double and Iterated Integrals over Rectangles.

L27

Double Integrals over General Regions.

L28

Double Integrals in Polar Form.

L29-L30

Triple Integrals in Rectangular Coordinates.

L31-L32

Triple Integrals in Cylindrical Coordinates.

L33-L34

Line Integrals.

L35-L37

Vector Fields and Line Integrals: Work, Circulation and Flux.

L38-L40

Path Independence, Conservative Fields and Potential Functions.

L41-L42

Green’s Theorem in the plane.

L43-L44

Surface Area and Surface Integrals.

L45-L46

Divergence Theorem.

L47-L48

Stoke’s Theorem.

 

Course Code:    MATH241

Course Title:                  Mathematical Statistics-I

Credit Hours:                 (3 0 3)

Pre-requisite(s):            MATH101

 

Course Objectives:

 

  1. To learn how to handle data numerically and graphically
  2. To understand the basic principles of random variables and random processes needed in applications
  3. To learn discrete random variables and their probability distributions

 

Reading List:

  1. S. Mann, “Introductory Statistics”, 9th edition, John Wiley & Sons, 2016.
  2. T. Craig, J. McKean, R.V. Hogg, “Introduction to Mathematical Statistics”, 7th edition, Pearson Education, 2013.
  3. R. Sheldo, “Introductory Statistics”, 3rd edition, Oxford, 2010.
  4. R. Spiegel, J.L. Schiller, R.L. Sirinivasan, “Probability and Statistics”, 2nd edition, McGraw Hill. NY, 2000.
  5. M. Clark, and D. Cooke, “A Basic Course in Statistics” 4th edition, Arnold, London, 1998.
  6. T. Mclave, P.G. Benson,T. Snitch, “Statistics for Business & Economics” 9th edition, Prentice Hall, New Jersey, 2005.
  7. M. Chaudhry, S. Kamal, “Introduction to Statistical Theory” Parts I 6th edition, Ilmi Kitab Khana, Lahore, Pakistan, 1996.

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Review of Mathematical Statistics

L2-L4

The nature and scope of the statistics,

L5-L7

Organizing of data, classification of data and graphs,

L8- L10

Measures of central tendency: their properties, usage, limitations and comparison.

L11-L13

Continued…

L14-L16

Measures of dispersion: their properties, usage, limitations and comparison.

L17-L19

Continued…

L20-L22

Calculations for the ungrouped and grouped data, measures of skewness,

Kurtosis and distribution shapes.

L23

Mid Exam

L24-L25

Sets, probability concepts, permutation and combination.

L26-L28

Addition and multiplication rules, bivariate frequency tables, joint and marginal probabilities.

L29-L31

Conditional probability and independence, Bayes’ rule.

L32-L34

Random variables, properties of random variables, discrete and continuous

random variables

L35-L37

Percentile for continuous random variables, multivariate distributions, marginal distributions, conditional distributions.

L38-L40

Mathematical expectation, moments, Chebyshev’s theorem, moment generating

Functions

L41-L43

Relation between moments and cumulants, product moments, characteristic

Function, conditional expectations.

L44-L46

Probability distribution, discrete probability distributions, Bernoulli distribution, properties, binomial distribution,

L47-L48

Poisson and hypergeometric distributions, negative binomial and geometric

Distributions.

 

Course Code: MATH202

Course Title:               Number theory

Credit Hours:              (3 0 3)

Pre-requisite(s):        None

 

Course Objectives:

The objectives of this course are:

  1. To learn about divisibility and its properties.
  2. To learn about Congruencies its properties and applications.
  3. To understand Primitive Roots and Quadratic Residue.

                               

Reading list:

  1. W. Stein,” Elementary Number Theory”, Springer-Verlag,2017
  2. K.H Rosen, “Elementary Number Theory and its Applications”, 6th edition, Pearson, 2010.
  3. T. Koshy, “Elementary Number Theory with Applications”, 2nd edition, Academic Press, 2007.
  4. D.M. Burton, “Elementary Number Theory”, 7th edition, McGraw-Hill, 2010.

 

 

   
 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1-L3

Introduction to the course, Well-ordering  principle , Mathematical induction  Divisibility ,Transitivity of divisibility ,Division of linear combination of integers and other related properties of divisibility  

L3-L4

The Division Algorithm, Proof and consequences of Division Algorithm   

L5-L6

Representation of integers in different base systems. Arithmetic in non-decimal systems.  Negative bases 

L7-L9

Prime Numbers, Prime divisors, infinitude of primes, Upper bound for a prime factor of composite integers, Sieve of Eratosthenes , Prime number theorem(without proof)

L10-L12

Greatest Common Divisor, Properties of GCD(theorems) ,GCD of more than two integers , The Euclidean Algorithm for finding GCD,

L13-L15

Fibonacci numbers, GCD of successive Fibonacci numbers , Lame’s Theorem The Fundamental Theorem of Arithmetic, Least common multiple, Theorem relating GCD and LCM 

L16-L18

Fermat factorization, Fermat number, Prime divisors of Fermat number

L19-L21

Linear Diophantine Equation, Solution in integers for Linear Diophantine Equation and its Applications.

L22-L24

Congruencies, equivalence relation and partition classes, Congruencies as equivalence relation. Complete system of residues modulo m, A reduced residue system modulo m

L25

MID EXAM

L26-L28

Linear Congruencies, Criterion for existence of solution and method of solution for linear congruencies, System of linear congruencies in one variable , Chinese Reminder theorem .

L29-L31

System of linear congruencies in two or more than two  variables ,Matrix method for solution of the system ,Applications of congruencies , Divisibility Tests, The Perpetual Calendar and Hashing Functions       

L32-L33

Wilson's Theorem and Fermat's Little Theorem, Pseudo-prim, Euler phi-function ,Euler Theorem

L34-L36

Arithmetic function, Multiplicative functions, Euler phi-function as Multiplicative function , Mobius function, Mobius inversion formula

L37-L38

Functions for The Sum and Number of Divisors, Perfect Numbers and Mersenne Primes

L39-L41

Order of an Integer and primitive Roots, Incongruent Primitive roots of a positive integer ,Primitive roots for Primes

L42-L43

Lagrange's Theorem, The Existence of Primitive Roots

L44-L45

Index Arithmetic ,Quadratic Residue, Legendre symbol

L46-L48

Euler's criterion, Gauss Lemma, law of quadratic reciprocity , Jacobi symbol

 

Course Code: MATH211

Course Title:               Group Theory

Credit Hours:              (3 0 3)

Pre-requisite(s):         None

 

Course Objectives:          

  1. To learn group structures
  2. To learn some fundamental results and techniques of group theory
  3. To classify groups (not all) up to isomorphism

 

Reading List:

  1. B. Bhattacharya, S.K. Jain and S.R. Nagpaul, “Basic Abstract Algebra”, 2nd ed., Cambridge University Press, 1995.
  2. S. Dummit and R.M. Foote, “Abstract Algebra”, 3rd ed., Addison-Wesely, 2004.
  3. B. Fraleigh, “A First Course in Abstract Algebra”, 7th ed., Pearson, 2002.
  4. A. Gallian, “Contemporary Abstract Algebra”, 7th ed., Brooks/Cole, 2010.
  5. F. Humphreys, “A course in Group Theory”, Oxford University Press, 1996.
    1. Majeed, “Theory of Groups”, Ilmi Kitab Khana, 2012.
 

 

   
 

Lecture-wise distribution of the course contents

Lecture #

Topics

L1

Introduction to the course

L2-L3

Operations, binary operations, usual and unusual operations (including modular arithmetic), semigroup, monoid

L4-L6

Group,  Caley’s table for finite groups,  elementary properties of groups,  order of a group, order of an element

L7-L9

Group of symmetries ( e.g., equilateral triangle, square, rectangle etc)

L10-L12

Subgroup with examples, subgroup test, finite subgroup test

L13-L15

Subgroup lattice, product of subgroups, direct product of groups

L16-L18

Generators and defining relations, cyclic groups and their properties

L19-L21

Cosets, index of subgroup, Lagrange’s Theorem,  its converse and consequences

L22-L23

normalizer and centralizer of a subset of a group, center of a group

L24

MID EXAM

L25-L27

Normal subgroups, factor groups

L28-L30

Group homomorphism, kernel and image

L31-L33

Properties elements/subgroups under homomrphism

L34-L36

Injective and surjective homomorphism, endomorphism,  isomorphism,

L37-L39

The Fundamental Theorem of homomorphism and its applications

L40-L42

2nd and 3rd isomorphism theorems

L43-L45

Permutations, Cycles in group of permutations and their properties, even and odd permutations

L46-L48

Symmetric and alternating subgroups, Caley’s Theorem

 

Course Code: MATH271

Course Title:  Ordinary Differential Equations

Credit Hours: (3 0 3)

Pre-requisite(s):         MATH101

 

Course Objectives: 

The course objectives are:

  1. To learn elementary analytical solution techniques for the solution of ordinary differential equations (ODEs).
  2. To understand the solution structure of linear ODEs in terms of independent homogeneous solutions and non-homogeneous solutions.
  3. To understand by exposure to examples how systems and phenomena from science and engineering can be modeled by ODEs.
  4. To learn how solution of different models can be used to analyze or predict a system’s behavior.

 

Reading list:

  1. W. E. BoyceR. C. DiPrima, “Elementary Differential Equations and Boundary Value Problems, 10th edition”, John Wiley & Sons, Inc., 2012.
  2. D. G. Zill, M. R. Cullen, “Differential Equations with Boundary-Value Problems”, 10th edition, Brooks/Cole, 2013.
  3. R. L. Borrelli, C. S. Coleman, “Differential Equations: A Modeling Perspective”, 2nd edition”, John Wiley & Sons Inc., 2004.
  4. R. K. Nagle, E. B. Saff, A. D. Snider, “Fundamentals of Differential Equations, 9th  edition”, Pearson Education Inc., 2017.
  5. E. Kreyszig, “Advanced Engineering Mathematics”, 10th edition, John Wiley & Sons Inc., 2010.
  6. M. Colomb, M. Shanks, “Elements of Ordinary Differential Equations”, McGraw-Hill, 2001.

 

   
 

Lecture-wise distribution of the course contents

Lecture #

Topic

L1

Introduction to the course

L2

Differential Equations, Classification, Examples.

L3-L4

Some Basic Models; Direction Fields.

L5

Well-Posedness, Separable Differential Equations.

L6

Homogenous Differential Equations.

L7-L8

Exact Differential Equations, Integrating factor in different cases.

L9

Linear Differential Equations.

L10

Differential equations reducible to linear DEs.

L11-L13

Modeling with First Order Linear Equations: Growth and Decay, carbon dating, Newton’s law of cooling

L14

Modeling with First Order Linear Equations: Series Circuits

L15

Modeling with First Order Non-Linear Equations: Logistic Population Model

L16-L17

Higher Order Linear Differential Equations, IVP, BVPs with different possible solutions, Linear Dependence and Linear Independent functions, The Wronskian.

L18-L19

Solutions of Linear Homogeneous Equations with constant coefficients:, Roots of the Characteristic equation.

L20

Reduction of Order method.

L21

MID EXAM

L22-L23

Non-homogeneous Equations; Method of Undetermined Coefficients.

L24

Method of Variation of Parameters.

L25-L26

Cauchy Euler differential equation, Legendre differential equation.

L27-L29

Modeling with 2nd Order linear differential equations: Spring mass system (Free Undamped Motion, Free Damped Motion and Driven Motion).

L30-L31

Review of Power series.

L32-L33

Series Solutions of Second Order Linear ODEs with Non-constant Coefficients; Solution Near an Ordinary Point.

L34-L36

Singular points, Series Solutions Near a Regular Singular Point.

L37-L39

 Laplace Transform and Solution of Initial Value Problems.

L40-L42

The Convolution Integral.

L43-L45

System of Linear first order differential equations: Preliminary theory, Homogenous Linear system with constant coefficients.

L46-L48

Qualitative approach for system of linear first order equations

Course Code: MATH 242

Course Title:  Mathematical Statistics-II

Credit Hours: (3 0 3)

Pre-requisite(s):         241

 

Course Objectives:

  1. To learn continuous random variables and their probability distributions
  2. To learn sampling concepts and methods
  3. To help students in making predictions and decisions.

Reading List:

  1. T. Craig, J. McKean and R.V Hogg, “Introduction to Mathematical Statistics”, 7th edition, Pearson Education, 2013.
  2. A. Garaybill, A.M Mood & D.C. Boes, “Introduction to Theory of Statistics” 3rd edition, McGraw Hill Co, 1974.
  3. Miller and M. Miller, “John E. Freund's Mathematical statistics with Applications”, 8th edition, Pearson Education, 2014.
  4. B. Ash, “Basic Probability Theory, Dover”, Dover Books on Mathematics, 2008.

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction to Mathematical statistics II

L2-L4

Continuous random variables, probability density function and its properties

 

L5-L7

Continuous distributions: rectangular distribution, exponential distribution

L8- L10

Gamma distribution

L11-L13

Beta distribution, relation between gamma and beta distribution with normal

Distribution

L14-L16

Bivariate discrete random variables

L17-L19

Bivariate continuous random variables, conditional distributions, independence

of random variables

L20-L22

Numerical examples on continuous random variables, conditional distributions,

independence of random variables

L23-L24

Covariance of bivariate random variables, variance of the linear combination of

random variables

L25

MID EXAM

L26-L28

Correlation and independence, moment generating functions

L29-L31

Differences between partial and multiple correlation.

L32-L34

The chi square table, introduction of F distribution

L35-L37

Some properties of F distribution, the F tables of area, tests based on F

Distribution

L38-L40

Introduction to sampling theory. Sampling distribution and properties. limiting

L41-L43

Some examples on sampling distribution for illustration,

L44-L46

Transformation of variables reasons and importance, transforming to uniform

Distribution

L47-L48

Statistical hypothesis and tests

 

Course Code: MATH212

Course Title:               Linear Algebra

Credit Hours: (3 0 3)

Pre-requisite(s):         None

 

Course Objectives

The objectives of this course are:

  1. To provide students with a solid grounding in concepts and methods of linear algebra to help them to develop the ability to formulate and solve problems using these techniques.
  2. To improve their ability to reason abstractly and understand proofs and know the geometric approaches which help them in visualizing an idea.
  3. To know some applications which illustrates the power of linear algebra to explain fundamental principles and simplify calculations in engineering, computer science, mathematics, physics, biology, economics and statistics.

Reading list:

  1. C. Lay, S. R. LayJ. J. McDonald, “Linear Algebra and Its Applications”, 5th edition, Pearson Education, 2015.
  2. A. Beezer, “A First Course in Linear Algebra”, 3rd edition, Congruent Press, 2012.
  3. Strang, “Introduction to Linear Algebra”, 4th edition, Wellesley Press, 2009.
  4. K. Nicholson, “Linear Algebra and Applications, 7th edition, McGraw-Hill, 2013.
  5. Anton, “Elementary Linear Algebra”, 9th edition”, John Wiley & Sons Inc., USA, 2005.

 

 

   
 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction to the course

L2-L3

System of Linear Equations, Row Reduction and Echelon Forms, Vector Equations,The Matrix Equation Ax=b.

L4-L6

Solution sets of Linear systems, Applications of Linear systems.

L7

Linear Independence.

L8-L10

Introduction to Linear Transformations, The Matrix of a Linear Transformations.

L11-L13

Linear Models in Business, Science and Engineering.

L14-L16

Matrix Operations, The Inverse of a Matrix, Characterizations of Invertible matrices, Matrix Factorizations.

L17-L18

Applications to Computer Graphics.

L19-L20

Subspaces of .

L21

Dimension and Rank.

L22

MID EXAM

L23-L25

Introduction to Determinants, properties, Cramer’s Rule, Voume, and Linear Transformations.

L26-L28

Vector spaces and subspaces, Null spaces, Column spaces, and Linear Transformations

L29-L31

Linear Independent Sets, Bases, Coordinate system, The Dimension of a vector space, Rank

L32-L33

Change of Basis.

L34-L36

Applications to Difference equations and Markov Chain.

L37-L39

 Eigenvalues and Eigenvectors, The Characteristic Equation, Diagonalization.

L40-L41

Discrete Dynamical Systems.

L42-L43

Applications to Differential Equations.

L44-L45

Inner product, Length and Orthogonality, Orthogonal sets, Orthogonal Projections

L46

The Gram-Schmidth Process

L47-L48

Least-Squares Problems

 

 

 

 

Course Code:  MATH321

Course Title:               Real Analysis-I

Credit Hours: (3 0 3)

Pre-requisite(s):         MATH101

Course Objectives

The objectives of this course are:

  1. To understand and deal with the real number system.
  2. To be able to construct proofs regarding limit and continuity.
  3. To learn basic properties of functions on R.
  4. To learn the elementary theory of differentiation.

Reading list:

 

   
  1. L. Brabenec, “Introduction to Real Analysis”, PWS Publishing Co, USA 1997
  2. D. Gaughan, “Introduction to Analysis”, 5th edition, Brooks/Cole, 1997.
  3. G. Bartle, D. R. Sherbert, “Introduction to Real Analysis”, 4th edition, John Wiley & Sons Inc, 2011.
  4. H. Protter, “Basic Elements of Real Analysis”, Springer Verlag, New York, 1998
  5. C Malik, S. Arora, “Mathematical Analysis”, Wiley Eastern Ltd., 2009.
 

Lecture-wise distribution of the Contents

Lecture #

Topic                                            

L1

Introduction to the course

L2 - L3

Real number system and extended real number system. Axioms for a Field. Related theorems.

L4 - L6

Neighbourhoods, limit point, boundedness and related theorems

L7 - L8

The Bolzano–Weierstrass Theorem. More theorems on bounded infinite sets

L9 - L11

Supremum and infimum, completeness properties of the real numbers and related theorems

L12

Convergence of sequences

L13 - L15

More theorems on Convergence of sequences

L16 – L17

Sub-sequences and related proofs

L18 - L20

Cauchy sequences, Cauchy’s Convergence Criterion etc.

L21 - L22

Monotone sequences and related theorems

L23

MID EXAM

L24 - L26

Limits of functions and their properties

L27 – L28

Cauchy’s first theorem on limits and related problems

L29 – L30

Cauchy’s second theorem on limits and related problems

L31 - L33

Continuous functions and their properties, discontinuity

L34 - L36

 Properties of continuous functions on closed bounded intervals

L37 - L39

Uniform Continuity

L40 – L 41

Derivatives in one variable and related theorems

L42 - L44

The mean value theorems: Rolle’s, Cauchy’s and Lagrange’s Mean Value etc.

L45 - L46

Taylor’s Theorem and its extension

L47 – L48

More theorems of differentiation on open and closed intervals

 

 

Course Code:  MATH351

Course Title:               General Topology

Credit Hours:              (3 0 3)

Pre-requisite(s):         None

 

Course Objectives

 

The objectives of this course are:

a.      To introduce the elementary properties of topological and metric spaces with structures defined on them.

b.      To introduce the maps between topological spaces.

c.       To understand connected and compact spaces 

Reading list:

1.      S. A. Morris, “Topology without Tears”, EMS Publishers, 2016.

2.      J. Munkres, “Topology”, 2nd edition”, Prentice Hall, 2000.

3.      L. A. Steen, J. A. Seebach, “Counter Examples in Topology”, Dover, 1995.

4.      S. Willard, “General Topology”, Dover Publications, 1970.

5.      S. Lipsschutz, “General Topology”, McGraw-Hill, 1965.

 

 

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction to the course

L2-L4

Topology of the Line and Plane

L5

Topological spaces.

L6-L7

Closet sets, closure of a set and related results.

L8-L10

Interior, exterior and boundary of a set and related results.

L11-L13

Neighborhoods and neighborhood systems, Accumulation points , Coarser and finer topologies.

L14

Subspaces, relative topologies.

L15-L17

Base for a topology, Sub-bases, Local bases and related theorems.

L18-L20

Continuous functions, Continuity at a point,open and closed functions.

L21

MID EXAM

L22-L23

Homeomorphism and related theorems.

L24-L26

Metrics, Distance between sets, diameters, open spheres.

L27-L29

Metric topologies, metric spaces, Properties of metric topologies, Metrization problem, Isometric metric spaces, Euclidean m-space.

L30-L32

Convergence and Continuity in metric spaces.

L33-L35

T1-spaces, Hausdorff spaces, Regular spaces, Normal spaces and related theorems.

L36-L38

Urysohn’s lemma and metrization theorem, Completely regular spaces.

L39-L41

Covers, Compact sets, subset of compact sets, Finite intersection property.

L42-L43

Locally compact spaces, Compactness in metric spaces.

L44-L46

Product topology, Base for a finite product topology.

L47-L48

Separated sets, Connected sets, Connected spaces, Connectedness on the real line , Components, Locally connected spaces, Paths, Arc-wise connected sets.

 

 

 

Course Code:  MATH 371

Course Title:  Partial Differential Equations

Credit Hours: (3 0 3)

Pre-requisite(s):         MATH271

 

Course Objectives

The objectives of this course are:

  1. To learn the theory behind the three important classes of partial differential equations of applied mathematics, that is, the Diffusion equation, the Wave equation, and Laplace’s equation.
  2. To apply analytical methods to solve these PDEs and be able to interpret the results.

Reading list:           

  1. Richard Haberman, “Applied Partial Differential Equations with Fourier Series and Boundary Value Problems”, 5th edition, Pearson Prentice Hall, 2012.
  2. Amaranath , “An Elementary Course in Partial Differential Equations”, 4th edition”,

Jones and Barlett learning Massachusetts, 2009.

  1. Sankarra Rao, “Introduction to Partial Differential Equations” 3rd edition, Prentice Hall, 2010.
  2. Walter A. Strauss, “Partial Differential Equations: An Introduction”, 2nd edition, Willey, 2007.
  3. David Logan, “Applied Partial Differential Equations”, 3rd edition”, Springer, 2014.
  4. Donald Greenspan, “Introduction to Partial Differential Equations”, 1st edition, Dover Publications, 2000.
   

 

 

 

Lecture-wise Distribution of the Contents

Lecture #

Topics

L1-L3

Introduction to the course, Differential equations, PDEs; Order, linearity of PDEs, IVPs and BVPs, Well-posedness, Mathematical Models,

L4-L6

Formation of PDEs, Elimination of arbitrary constants and arbitrary functions

L7-L9

PDEs solvable like ODEs, Linear 2nd Order PDEs, Classification of PDEs,

L10-L12

Heat Equation: 1D Derivation & Boundary Conditions, Steady State Diffusion equation(Laplace equation), Wave Equation: 1D Derivation

L13-L15

 

Characteristics of PDEs, Canonical forms of different types of PDEs, 

L16-L18

working examples, General solutions of PDEs using method of characteristics

L19-L21

Heat  equation/Wave equation and Laplace equation in cylindrical and spherical coordinates, Fourier Series and term-by-term operations

L22-L24

Method of Separation of Variables, Working Examples with Heat Equation, Wave equation with homogeneous boundary conditions

L25

MID EXAM

L25-L27

Working Examples with the Heat Equation, Laplace equation, Wave equation with non-homogeneous boundary conditions

L28-L30

PDE’s in 2D: Vibration of a Rectangular Membrane, Vibration of a Circular Membrane, Bessel Functions,

L31-L33

Non-homogeneous Problems: Heat flow with sources and Non-homogenous BCs

L34-L36

Integral transform methods, Fourier Transform and its properties, Infinite Domain Problems

L37-L39

Fourier Transform Solutions of PDEs, Working Examples

L40-L42

Laplace transform, its properties, Semi-Infinite Domain Problems:

L43-L45

Laplace transform solutions of PDEs, working examples

46-48

Hankel transform, its properties, Hankel transform solutions of PDEs, working examples

 

 

 

Course Code: MATH313

Course Title:               Rings and Fields

Credit Hours:              (3 0 3)

Pre-requisite(s):         MATH 211

 

Course Objectives:

  1. To learn construction of rings
  2. To study structural properties of rings
  3. To understand familiar concepts, such as fractions, factorization, primness, divisibility etc., in general setting of rings.

 

Reading list:

  1. P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, “Basic Abstract Algebra”, 2nd ed., Cambridge University Press, 1995.

 

  1. D.S. Dummit and R.M. Foote, “Abstract Algebra”, 3rd ed., Addison-Wesely, 2004.

 

  1. J.B. Fraleigh, “A First Course in Abstract Algebra”, 7th ed., Pearson, 2002.

 

  1. J. A. Gallian, “Contemporary Abstract Algebra”, 7th ed., Brooks/Cole, 2010.

 

  1. S. Lang, “Algebra”, 3rd ed., Springer, 2005

 

  1. C. Musili, “Introduction to rings and Modules”, 2nd ed., Narosa Publishing House, 2009.

 

Lecture-wise distribution of the course contents

Lecture #

Topics

L1

Introduction to the course

L2-L3

Rings: examples and basic notions

L4-L6

Units and their properties,  idempotent and nilpotent elements in a ring

L7-L9

Some important rings: rings of continuous functions, matrix rings, polynomial rings, power series rings, Laurent rings, Boolean rings, endomorphism ring, group ring

L10-L12

Opposite rings, direct product of rings, characteristic of a ring

L13-L15

Division ring, field, zero divisors, integral domains and their properties

L16-L18

Subrings, subring test, examples and properties, center of a ring

L19-L21

Motivation for an ideal, left, right and two sided ideals, their examples and properties

L22-L23

Ideals in a field, annihilators, radical of an ideal

L24

MID EXAM

L25-L27

Ideal generated by a subset, finitely generated ideal, ideals in a commutative ring with unity, principal ideal

L28-L30

Factor rings, constructing examples, further structural properties, ideals in factor rings

L31-L33

Prime ideals, maximal ideals, examples and important characterizations, local rings

L34-L36

Ring homomorphism, kernel and image, monic and kernel, fundamental theorem of homomorphism and its consequences

L37-L39

Field of fractions, examples and construction

L40-L42

Factorization of polynomials over a field, irreducible polynomials

L43-L45

Division in domains,  Euclidean domains

L46-L48

 Principal ideal domains , factorization domains, unique factorization domains

 

 

Course Code: MATH131

Course Title:               Computing Tools         

Credit Hours:               (2 3 3)

Pre-requisite(s):         None

 

Course Objectives:

The objectives of this course are: 

  1. To learn about arrays, cell and structure.
  2. To learn how to plot 2 and 3D functions with in Computing Tools (e.g, Matlab/ Maple/ Mathematica).
  3. To learn about symbolical processing Tools (e.g, Matlab/ Maple/ Mathematica)
  4. To learn how to construct user define functions and do simple programming in Computing Tools (e.g, Matlab/ Maple/Mathematica)

 

Reading list:

  1. J. Palm III, “A Concise Introduction to MATLAB”, McGraw-Hill, 2008.
  2. Attaway, “A Practical Introduction to Programming and Problem solving”, 2nd edition, Elsevier Inc, 2011.
  3. R. Hunt, R.L. Lipsman and J.M. Rosenberg,“A Guide to MATLAB for Beginners and Experienced Users”, 2nd edition, Cambridge University press,2006.
  4. Knight, “Basics of MATLAB & Beyond”, Chapman and Hall, 2000.

 

 

   
 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1-L3

Introduction to the course , Starting MATLAB, Command window, Command history window, MATLAB Editor window, Current Directory window, Workspace window ,Variables, Variables naming  , Scalar arithmetic operations, Order of Precedence, Assignment Operator, Managing the Work Session, use of Tab and Arrow Keys, Predefined Constants, Complex Number Operations, file handling

L4-L6

One- and Two-Dimensional Numeric Arrays, Multidimensional Numeric Arrays, Element-by-Element Operations, Matrix Operations

L7-L8

Matrix Methods for Linear Equations, Polynomial Operations Using Arrays, Cell Arrays, Structure Arrays, Special Matrices

L9-L10

Exponential and Logarithmic Functions, Complex Number Functions, Numeric Functions, Trigonometric Functions.

L11-L13

User-Defined Functions, Some Simple Function Examples, Local and Global Variables, Function Handles, Methods for Calling Functions

L14-L16

Anonymous Functions, Primary Function, Sub-functions, Nested Functions, Relational Operators and Logical Variables, The logical Class, The logical Function

L17-L19

Accessing Arrays Using Logical Arrays, Logical Operators and Functions, Order of precedence for operator types, Short-Circuit Operators

L21-L23

Conditional Statements, The if, else, elseif  Statement, Strings and Conditional Statements, MATLAB program to solve linear equations and other practice examples

L24

MID EXAM

L25-L27

For Loops, Series Calculation with a for Loop, nested loops, The break and continue Statements, Using an Array as a Loop Index

L28-L30

While Loops, Series Calculation with a while Loop, The switch Structure, Practice examples.

L31-L32

Debugging MATLAB Programs, Cell Mode, The Debug Menu, Debugging Using Breakpoints

L33-L35

xy Plotting Functions, Saving Figures, Exporting Figures, Additional Commands and Plot Types, Interactive Plotting in MATLAB

L36-L38

Three-Dimensional Line Plots, Surface Mesh Plots, Contour Plots ,MATLAB ODE Solvers, ode45, ode15s

L39-L40

Symbolic Processing, Symbolic Expressions, Manipulating Expressions, Evaluating Expressions, Algebraic and Transcendental Equations solution

L41-L43

Sums, Limits, differentiation, integration and differential equations

L44-L46

Laplace Transforms, Symbolic Linear Algebra, Characteristic Polynomial and Roots

L47-L48

Solving Linear Algebraic Equations

 

Course Code: MATH325

Course Title:               Vector and Tensor Analysis

Credit Hours:              (3 0 3)

Pre-requisite(s):         MATH212

 

Course Objectives

The objectives of this course are:

  1. To learn about vector quantities and algebra of vector addition and multiplication.
  2. To understand differentiation and integration of vector valued functions and their applications.
  3. To learn about tensor quantities and algebra of tensor addition and multiplication.
  4. To understand differentiation of tensors fields.

 

Reading list:

  1. E. Bourne, P.C  Kendall , “Vector Analysis and Cartesian Tensors”, 3rd  edition,    Stanley Thornes, 1999.
  2. D. Smith, “Vector Analysis”, Oxford University Press, Oxford 1999.
  3. R. Spiegel, “Vector Analysis & Introduction to Tensor Analysis”, McGraw Hill, New York 2009.
  4. R. Spiegel, “Vector Analysis”, 2nd edition, McGraw Hill New York, 2009.
  5. G. Simmonds,  ”A Brief on Tensor Analysis”, Springer-Verlag, 2012.

 

 

   
 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1-L3

Introduction to the course ,Vectors , scalars ,Addition of vectors , Multiplication of a vector by a scalar , Algebra of vector addition and scalar multiplication ,Unit vector ,Components of a vector ,Scalar and Vector fields

L4-L6

Dot Product and Cross product of vectors, Properties and applications of dot and Cross product.

L7-L8

Scalar and vector triple product ,Properties and applications of triple product

L9-L11

Derivatives of vector valued functions of scalar variable, Differentiation formulas Continuity and differentiability, Partial derivatives of vector functions

L12

Space curves ,unit tangent ,Principal normal , Bi-normal 

L13-L15

Gradient, Divergence ,Curl ,  Formulas involving gradient, Divergence and Curl

L16-L18

Integration of vector valued functions, Line integral, Work done by a variable force, conservative vector field, scalar potential, Path independence ,Work  done around a closed path    

L19-L21

Surface integrals, Volume integrals, Limit of sum definition and evaluation technique   

L22-L23

Transformation of coordinates, curvilinear coordinate , orthogonal curvilinear coordinate, Unit vectors in curvilinear systems

L24

MID EXAM

L25-L26

Contra-variant  and covariant components of a vector, Gradient, Divergence and Curl in curvilinear coordinate system

L27-L29

Special orthogonal coordinate systems , Cylindrical Coordinate, spherical Coordinates, Parabolic Cylindrical Coordinates, Paraboloidal Coordinates

L30-L32

Co-vector, Scalar product of vector and co-vector, Linear operators, Bilinear and quadratic forms, Dual Bilinear forms, Einstein summation convention ,General definition of tensors.

L33-L35

 Dot product and metric tensor, Tensors addition and multiplication by a scalar, Tensor product

L36-L38

Contraction, Kronecker symbol, Levi-Civita symbol, Tensor fields in Cartesian coordinates

L39-L41

Change of Cartesian coordinate system, Differentiation of tensor fields

L42-L44

Gradient, divergence, and curl, Laplace and d’Alambert operators

L45-L46

Tensor fields in curvilinear coordinate

L47-L48

Moving frame of curvilinear coordinates, Christoffel symbols

 

 

Course Code:     MATH322

Course Title:                   Real Analysis-II

Credit Hours:                  (3 0 3)

Pre-requisite(s):            MATH321

 

Course Objectives: 

 The course objectives are:

  1. To learn the differentiation and integration theory in
  2. To be able to construct proofs regarding sequences, series and their convergence
  3. To be able to construct proofs regarding the improper integrals
  4. To learn the Riemann–Stieltjes integrals

 

Reading list:           

  1. L. Brabenec, “Introduction to Real Analysis”, PWS Publishing Co., 1997.
  2. D. Gaughan, “Introduction to Analysis, 5th edition, Brooks/Cole, 1997.
  3. G. Bartle, D. R. Sherbert, “Introduction to Real Analysis” 4th edition, John Wiley & Sons, 2011.
  4. H. Protter, “Basic Elements of Real Analysis”, Springer-Verlag, New York, 1998.
  5. S.C Malik, S. Arora, “Mathematical Analysis”, Wiley Eastern Ltd. 2009.

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction to the course

L2

Functions of several variables

L3-L4

Limit and Continuity

L5-L7

Differentiability

L8-L10

Partial derivatives, Chain rule

L11-L12

Young’s theorem and Schwarz theorem

L13-L15

 Implicit functions, Implicit function theorem, Inverse function theorem

L16

Jacobian

L17

Functionally related functions

L18

Maxima and Minima for functions of two variables

L19-L20

Series of numbers and their convergence, Alternating Series, Leibnitz Test

L21-L23

Comparison test, Limit comparison test

L24

MID EXAM

L25

Cauchy integral test

L26-L28

D-Alembert Ratio test, Cauchy Root test   +   MIDTERM EXAMINATION

L29

Series of variable terms

L30

Uniform convergence

L31

 Weierstrass M theorem

L32-L34

Convergence and Divergence of improper integrals

L35

P-Test for convergence of improper integrals

L36-L38

Darboux upper and lower sums and integrals

L39

Definition and existence of the Riemann integral

L40- L42

Theorems on Riemann integration

L43-L45

Integration and differentiation Theorems

L46-L48

 Riemann-Steiltjes integration

 

Course Code:  MATH331

Course Title:               Numerical Analysis I

Credit Hours:              (3 3 4)

Pre-requisite(s):         MATH321

 

Course Objectives:

The objectives of this course are:

  1. To demonstrate understanding of common numerical methods and how they are used to obtain approximate solutions to otherwise intractable mathematical problems. Root-finding iterative methods will be discussed both in respect of their derivations and convergence performance.
  2. To demonstrate numerical methods to obtain solutions of system of linear and nonlinear algebraic equations.
  3. To perform an error analysis for various numerical methods
  4. To implement such numerical methods in MATLAB or any programming language.

Reading list:

   
 
  1. L. Burden and J.D. Faires, “Numerical analysis 10th edition”,Brooks Cole, 2015.
  2. F. Gerald, P.O. Wheatley, “Applied Numerical Analysis 7th edition”, Pearson , 2003.
  3. Atkinsonan , W. Han, " Elementary Numerical Analysis 3rd edition”,Wiley,2003.
  4. K. Jain, S.R.K. Iyengar, R.K. Jain “Numerical Methods for Scientific and Engineering computation 6th edition”, New Age International Pvt Ltd, 2010.
  5. W. Hamming, ”Numerical Methods for Scientists and Engineer 2nd revised edition”, Dover, 1987.
  6. B. Hildebrand , “Introduction to Numerical Analysis 2nd edition”, Dover,1987.
  7. Bradie , “A Friendly Introduction to Numerical Analysis 1st edition”, Pearson, 2005.

 

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction and overview  to the course

L2-L3

Calculus Review:  Continuity, differentiability and related theorems, convergence of sequences, Taylor’s theorem

L4-L5

Error, types, sources and propagation, computer arithmetic’s

L6-L7

Algorithms and convergence, stability analysis of algorithms, error growth, rate and order of convergence. Big-O and Little-O notations.

L8-L10

Solutions of equations in one variable: The Bisection method, algorithm (or pseudo code) and Implementation in Matlab. Error Analysis

L11-L13

Fixed point, existence and uniqueness of fixed point, fixed point iteration, cobwebbing diagram.

L14-L16

Newton’s Method, Derivation, Algorithm and Implementation in MATLAB, Error Analysis. Modified Newton’s Method for roots with multiplicity.

L17-L19

Secant Method, Derivation, Algorithm and Implementation, Error Analysis

L20

Method of false position, Algorithm

L21

MID EXAM

L22-L23

Linear system of equations, Pivoting strategies, Linear algebra and matrix inversions.

L24-L26

 Elimination methods: Gauss Elimination and Gauss-Jordan, operations analysis, Algorithms.

L27-L29

Matrix Factorizations: Doolittle’s method, Crout’s method, Cholesky method with algorithms.

L30-L32

Norms of vectors and matrices, convergence and perturbation theorems

L33-L35

Eigen values and eigen vectors, power and inverse power method.

L36-L37

Spectral radius of a matrix, Greshgorin Circle theorem for bounds of eigen values

L38-L40

Iterative methods for solving linear systems: Jacobi Iterative method, Gauss Seidel Iterative method and SOR method with algorithms and implementation in MATLAB.

L41

Condition number of a matrix

L42-L44

Solving Sparse systems: Gradient vectors, quadratic forms, Residuals, Krylov subspace, Steepest descent method

L45-L48

Numerical solution of Nonlinear system using Newton’s method

 

 

Course Code:  MATH323

Course Title:               Complex Analysis

Credit Hours:              (3 0 3)

Pre-requisite(s):         MATH321

 

Course Objectives

The objectives of this course are:

  1. To understand basic theory of algebraic and geometric structures of the complex numbers.
  2. To understand the concepts of analyticity, Cauchy-Riemann relations and harmonic functions are then introduced with some applications in fluid dynamics.
  3. To learn Complex integration and complex power series.
  4. To learn the classification of isolated singularities and examine the theory and illustrate the applications of the calculus of residues in the evaluation of integrals.

 

Reading list:

  1. G. Zill, P. D. Shanahan, “A First Course in Complex Analysis with Applications”, 3rd edition, Jones and Bartlett Publishers, 2013
  2. W. Brown, R. V. Churchill, “Complex Variables and Applications”, 9th edition, McGraw-Hill, 2013.
  3. A. Silverman, “Complex Analysis with Applications’’, 1st edition, Dover, 2010.
  4. B. Saff, A. D. Snider, “Fundamentals of Complex Analysis with Applications to Science and Engineering”, 3rd edition”, Pearson Education, 2003.
  5. K. Jain, S. R. K. Iyengar, “Advanced Engineering Mathematics” 10th edition, John Wiley & Sons Inc., 2011.

 

   
 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1-L3

Introduction to the course, Complex Numbers and Their Properties, Complex Plane, Polar Form of Complex Numbers, Powers and Roots, Sets of Points in the Complex Plane,

L4-L6

Complex Functions, Complex Functions as Mappings, Linear Mappings, Special Power Functions, Reciprocal Function, differences between real and complex functions

L7-L9

Limits and Continuity, Complex functions as vector fields

L10-L12

Differentiability and Analyticity

L13-L14

Cauchy-Riemann Equations

L15-L16

Harmonic Functions, Applications: Orthogonal families, Gradient fields, Complex potentials and ideal fluids, Heat flow

L17-L19

Elementary functions: Exponential and Logarithmic Functions, Complex Powers, Trigonometric and Hyperbolic Functions, Inverse Trigonometric and Hyperbolic

Functions

L20-L22

Real Integrals, Complex Integrals

L23

MID EXAM

L24-L26

Cauchy-Goursat Theorem

L27

Independence of Path

L28-L30

Cauchy’s  Integral Formula, Cauchy’s  Integral Formula for derivatives,

L31-L33

Consequences of Cauchy’s  Integral Formula: Cauchy’s Inequality,Liouville’s theorem, Fundamental theorem of algebra,Morera’s theorem,Maximum Modulus theorem.

L34-L36

Sequences and series, Taylor series, Laurent series, zeros and poles

L37-L38

Residues and Residue theorem

L39

Evaluation of Real Trigonometric integrals

L40

Evaluation of Real improper integrals

L41

Integration along a Branch cut

L42-L43

Miscellaneous integrals

L44-L45

The Argument Principle and Roche’s theorem

L46-L48

Summing infinite series

 

Course Code: MATH471

Course Title:               Mathematical Modeling

Credit Hours:              (3 0 3)

Pre-requisite(s):        MATH271/MATH212

 

Course Objectives:

The objectives of this course are:

  1. To learn the basics of deterministic modeling.
  2. To learn how to apply balance laws, conservative laws and constitutive laws to construct a mathematical model.
  3. To analyze the derived model with dynamical system point of view.
  4. To interpret the qualitative behavior of the model.

Reading list:

  1. R. Adler, “Modeling The Dynamics of Life: Calculus and Probability for Life Scientists”, 3rd Brooks/Cole, 2013.
  2. H. Strogatz, “Nonlinear dynamics and chaos: With applications in Physics, Biology,

Chemistry and Engineering”, 2nd, This email address is being protected from spambots. You need JavaScript enabled to view it. Press, 2014.

  1. Edelstein-Keshet, “Mathematical Models in Biology”, Leah SIAM, 2005.
  2. H. Taubes, “Modeling Differential Equations in Biology”, 2nd edition, Cambridge University Press, 2008.

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction to the course

L2-L3

Introduction: Deterministic Vs Stochastic modeling, Modeling Components, Modeling laws.

L4

Review of some physical and biochemical laws.

L5-L6

Units and Dimensions, Dimensional analysis and Scaling.

L7-L8

Buckingham Pi theorem and its Importance, examples.

L9-L11

One-dimensional flows: geometric approach, fixed points and stability, Potentials.

L12-L13

Linear stability analysis for 1D system.

L14-L16

1D Bifurcations:  Saddle Node Bifurcation, Transcritical Bifurcation, Pitchfork Bifurcation, examples.

L17-L19

Linear systems: Definitions and examples, Classifications.

L20-L22

Phase Plane: Phase portrait, Existence, Uniqueness and Topological consequences.

L23

MID EXAM

L24-L25

Nullclines, Fixed points and linearization.

L26-L27

Conservative systems, Reversible systems, Nonlinear Pendulum.

L28-L29

Biological Models Using Difference equations: Cell division, An insect population.

L30-L32

Propagation of Annual plants: problem statement, assumptions, equations, condensing the equations, validation.

L33-L34

System of linear difference equations.

L35-L37

Nonlinear difference equations:  steady states, stability and critical parameters, system of non-linear difference equations.

L38-L39

Applications of non-linear difference equations to population Biology.

L40-L41

Continuous time models: Formulating a model, dimensional analysis, steady states, stability and linearization, examples.

L42-L44

Applications of continuous Models to Population dynamics: Malthus model, logistic model, Allee effect, Gomoertz growth in tumors,predator-prey systems and Lotka-Volterra equations, populations in competition.

L45-L48

Models for molecular events: Chemical reactions and law of mass action, Michaelis-Menten kinetics, The Quasi-Steady state assumptions, Sigmoidal kinetics.

 

Course Code: MATH424

Course Title:               Functional Analysis

Credit Hours:              (3 0 3)

Pre-requisite(s):         MATH212

 

Course Objectives:

  1. To understand the notion of norm and inner product in an arbitrary linear space
  2. To learn general theory of operators
  3. To study Hilbert spaces
  4. To learn the basic techniques and methods of functional analysis.

Reading List:

  1. J.B. Conway, “A Course in Functional Analysis”, 2nd ed., Springer-Verlag, 1997.
  2. E. Kreyszig, “Introductory Functional Analysis with Applications”, John Wiley & Sons, 2004.
  3. P.D. Lax, “Functional Analysis”, John Wiley & Sons, Inc., 2002.
  4. A. Majeed, “Elements of Topology and Functional Analysis”, Ilmi Kitab Khana, Lahore, 1997.
  5. W. Rudin, “Functional Analysis”, 2nd Edition, McGraw Hill, Inc., 1991.
  6. K. Saxe, “Beginning Functional Analysis”, Springer-Verlag, 2001.
  7. A.E. Taylor and David C. Lay, “Introduction to Functional Analysis”, John Wiley & Sons, 1980.
  8. K. Yosida, “Introduction to Functional Analysis with applications”, 5th ed., Springer-Verlag, 1995.

Lecture-wise distribution of the course contents

Lecture #

Topics

L1

Introduction and overview  to the course

L2-L3

Review of metric spaces (e.g., convergence, completeness etc. ) and linear spaces,

L4-L6

Young’s inequality, Hilbert’s inequality, Cauchy-Schwarz inequality, Minkowski’s inequality

L7-L9

Normed spaces, complete normed spaces (Banach),  examples of incomplete normed space, Completeness and finite dimension

L10-L12

Equivalent norms, finite dimension and equivalent norm

L13-L15

Linear operator,  bounded and continuous linear operator

L16-L18

 Bounded linear extension, normed space of bounded linear operators and its completeness

L19-L21

Linear functional (LF), bounded and continuous LF

L22-L23

Dual spaces, reflexivity

L24

MID EXAM

L25-L27

Hahn-Banach  Theorem for real (without proof) complex and normed spaces with some  important consequences

L28-L30

Inner product, inner product space, Hilbert Space and properties

L31-L33

Orthogonality , Orthogonal complements and direct sums, annihilators

L34-L36

Orthonormal sets and sequences with properties, Bessel’s inequality

L37-L39

Gram-Schmidt process of orthonormalization

L40-L42

Total Orthonormal sets and sequences, Parseval relation

L43-L45

 Representation of functionals on Hilbert spaces,  Riesz’s Theorem,  Sesquilinear forms and their representation

L46-L48

Hilbert-adjoint operator, self-adjoint, unitary and normal operators

Course Code:  MATH431

Course Title:               Numerical Analysis-II

Credit Hours:              (3 3 4)

Pre-requisite(s):         MATH331

Course Objectives:

The objectives of this course are:

  1. To understand the basic problems of interpolation and approximation both theoretically and computationally.
  2. To understand quadrature rules and numerical rules for the solution of ODEs both theoretically and computationally.
  3. To perform an error analysis for the methods discussed in this course.

 

Reading list:

  1. L. Burden, J. D. Faires, “Numerical analysis” 8th edition, Brooks Cole, 2004.
  2. F. Gerald, “Applied Numerical Analysis, 8th edition”, Pearson Education, 2008.
  3. Atkinson, “Elementary Numerical Analysis”, 3rd edition, John Wiley & Sons Inc., 2003.
  4. K. Jain, S. R. K. Iyengar, R. K. Jain, “Numerical Methods for Scientific and Engineering Computation”, New Age International Pvt. Ltd., 2007.
  5. W. Hamming, “Numerical Methods for Scientists and Engineer”, 2nd revised edition”, 1987.
  6. B. Hildebrand , “Introduction to Numerical Analysis”, 2nd edition, Dover, 1987.
  7. Bradie, “A Friendly Introduction to Numerical Analysis”, 1st edition”, 2005.

 

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1

Introduction to the course

L2-L3

Introduction to Interpolation Problem, Weirstrass approximation theorem, Interpolation with Taylor polynomials and limitations.

L4-L6

Lagrange basis, Lagrange polynomial Interpolation, Algorithm, Error Analysis with Lagrange Interpolating Polynomial.

L7-L9

Operators, Divided Differences, Newton’s Interpolating divided difference formula, Newton-forward, backward and centered difference formulas. Divided difference Algorithm.

L10-L12

Equally spaced interpolation drawbacks, Runge’s phenomenon.

L13-L14

Interpolation with Chebyshev nodes.

L15-L17

Splines, Linear, quadratic and Cubic Spline Interpolation with algorithms. Error bounds.

L18-L20

Approximation Vs Interpolation, The Minimax approximation problem, The least squares approximation problem.

L21-L22

Orthogonal polynomials and least squares approximation revisited.

L23

MID EXAM

L24-L25

Numerical Differentiation, Error Analysis.

L26-L27

 Newton-cotes quadrature formulas: Trapezoidal rule, derivation, error term and algorithms.

L28-L29

Simpson’s rule with various forms, derivation, error term, algorithm.

L30-L31

Midpoint rule, derivation, error term, algorithm.

L32

Degree of precision of a Quadrature rule.

L33-L35

Drawbacks with Newton-cotes quadrature, Gaussian Quadrature with different weights of classes of orthogonal polynomials.

L36

Romberg Integration.

L37-L39

Numerical Solution of IVP and BVP for ordinary differential equations: Existence, Uniqueness and stability theory.

L40-L41

Euler’s method, derivation, error analysis, algorithm.

L42

Modified Euler’s method, error analysis, algorithm.

L43

Runge-Kutta method of general order.

L44-L45

Multistep method, derivations, convergence and stability.

L46-L47

Stiff differential equations, Boundary value problems, Shooting method.

L48

A tour guide of MATLAB built-in ODE solvers.

 

 

 

 

 

Course Code: MATH472

Course Title:               Integral Equations

Credit Hours:              (3 0 3)

Pre-requisite(s):         MATH271

 

Course Objectives:

The objectives of this course are:

  1. To learn the theory of linear and nonlinear integral equations.
  2. To learn the connection of integral equations with ordinary differential equations.
  3. To learn different classes of integral equations.

Reading list:

  1. Moiseiwitsch, “Integral Equations”, Longman London and New York, 1977.
  2. P. Kanwal, “Linear Integral Equations Theory and Technique”, Academic Press, 1971.
  3. M. Wazwaz, “Linear and Nonlinear Integral Equations: Methods and Applications”,

Springer,  2011.

  1. Hochstadt, “Integral equations”, John Wiley and Sons Inc., 1973.

 

   
 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1-L2

Classification of integral equations , Historical introduction , Linear integral equations , Special types of kernel , Symmetric kernels

L3

Kernels producing convolution integrals, Separable kernels

L4-L5

Square integrable functions and kernels , Singular integral equations

L6

Non-linear integral equations

L7-L9

Linear differential equations, Green's function, Influence function

L10

Integral transforms

L11-L13

Fredholm equation of the first kind, Stieltjes integral equation, Volterra equation of the first kind

L14-L16

Fredholm equation of the second kind, Volterra equation of the second kind

L17-L19

Method of successive approximations: Neumann series, Iterates and the resolvent kernel

L20-L22

Generalization to higher dimensions, Green's functions in two and three dimensions

L23

MID EXAM

L24-L26

Dirichlet's problem, Poisson's formula for the unit disc

L27-L29

Poisson's formula for the half plane

L30

Hilbert kernel, Hilbert transforms

L31-L32

Singular integral equation of Hilbert type

L33-L34

Resolvent equation, Uniqueness theorem, Characteristic values and functions

L35-L37

Neumann seres, Volterra integral equation of the second kind, Bacher's example, Fredholm equation in abstract Hilbert space , Degenerate kernels , Approximation by degenerate kernels

L38-L40

Fredholm';, theorems, Fredholm theorems for completely continuous, Operators, Fredholm formulae for continuous kernels

L41-L43

Hermitian kernels, Spectrum of a Hilbert-Schmidt kernel

L44-L46

Expansion theorems, Hilbert-Schmidt theorem, Hilbert's formula, Expansion theorem for iterated kernels, Solution of Fredholm equation of second kind.

L47-L48

Bounds on characteristic values, Positive kernels, Mercer's theorem, Variational principles, Rayleigh-Ritz variational method.

 

 

Course Code: MATH451

Course Title:               Differential Geometry

Credit Hours:              (3 0 3)

Pre-requisite(s):         MATH351

Course Objectives

The objectives of this course are:

  1. To learn about space curves their curvature and torsion.
  2. To understand intrinsic and non-intrinsic properties of surfaces.
  3. To understand the application of vector calculus to explore geometry of curves and

  surfaces.

                                         

Reading list:

  1. B.E. Weather, “Differential Geometry of Three Dimensions”, Cambridge University Press, 1961.
  2. S. Millman, G. D. Parker “Elements of Differential Geometry”, 1st edition, Prentice Hall, 1977.
  3. . D.J. Struik, “Lectures on Classical Differential Geometry”, Addison Wesley, 1962.

 

 

Lecture-wise distribution of the Contents

Lecture #

Topic

L1-L3

Introduction to the course, Vectors and scalars, Addition of vectors, scalar multiplication, Direction angels, cosine and ratios, dot product, cross product, Scalar triple product, Vector triple product

L4-L6

Differential geometry, Space curves, parametric equations, Tangent at a point, Equation of tangent line

L7-L9

Arc length as a parameter, conversion of parameters

L10-L12

Unit tangent, Principal normal, Bi-normal, Moving tetrahedron,

L13-L15

Fundamental planes associated with space curve, Equation of osculating plane in different forms, curvature

L16-L18

Torsion for space curves, Serret-Frenet formulae, Different type of relations for Curvature and Torsion calculation

L19-L21

Plane and Skewed curves, Different criterion for a curve to be planar, Osculating Circle at a point of a curve

L22-L23

Involutes and Evolutes, Derivation of equation for the curves

L24

MID EXAM

L25-L27

properties of evolutes, Order of contact between curves and surfaces, Osculating sphere of a curve at a point

L29-L30

equation for the locus of center of osculating sphere and its properties, Spherical and Cylindrical helices

L31-L33

Spherical indicatrices and their properties, Theorems on different properties of the curves, Surfaces

L34-L36

tangent plane, Normal vector, family of surfaces, Envelope and characteristics of a family of surfaces, Edge of regression

L37-L39

Developable surfaces, Developable surfaces associated with a space curve,

L40-L42

First fundamental form of a surface, Geometrical meaning, First fundamental magnitudes Properties, Applications, Second fundamental form of a surface,

L43-L45

Geometrical meaning of Second fundamental form of a surface, Second fundamental magnitudes, Properties, Applications, Normal section, Normal curvature

L46-L48

Principal directions and principal curvatures. Gaussian and mean curvature, Euler’s Theorem,

 

 Technical Electives

An important aspect of the Mathematics curriculum is the Technical Elective courses. Students are given choices towards the end of the program to choose specialization of their own interest. Students are required to complete 15-20 CrHr from the following list of courses as per the guidance of the Institute.

 

Code

Title

Cr Hrs

Pre-Requisite

General Electives Courses

MATH434

Numerical Solution of PDEs

4(3+1)

MATH331

MATH461

Analytical Dynamics

3(3+0)

None

MATH462

Introduction to Special Relativity

3(3+0)

None

MATH435

Cryptography

3(3+0)

MATH202

MATH474

Discrete Dynamical Systems

3(3+0)

MATH271

MATH332

Operations Research

4(3+1)

None

MATH361

Fluid Mechanics

  3(3+0)

MATH371

MATH423

Measure Theory & Integration

3(3+0)

MATH321

MATH475

 Introduction to Mathematical Biology

3(3+0)

MATH271

MATH413

 Theory of Modules

3(3+0)

MATH313

MATH441

Stochastic Processes

3(3+0)

MATH242

MATH312

Advanced Group Theory

3(3+0)

MATH211

MATH432

Optimization Theory-I

4(3+1)

MATH131

MATH433

Optimization Theory-II

4(3+1)

MATH432

MATH412

Galois Theory

3(3+0)

MATH313

MATH302

Set Theory & Mathematical Logic

3(3+0)

None

 

Senior Design Project: (6 CrHr)

The project is spread over two semesters. Students are required to work on a real-world problem under the supervision of a senior faculty member. Students can complete this segment in group form as well.

Code

Title

CrHr

Pre-Requisite

MATH499

Senior Design Project – I

3(0+9)

None

MATH499

Senior Design Project – II

3(0+9)

MATH499 Senior Design Project – I

 

Master of Mathematics (M.Sc) - 16 Years.

The Institute of Numerical Sciences offers two years undergraduate degree program under the title ‘Master of Mathematics (MSc)’. This program is offered for students who have obtained two years conventional BSc degree with relevant courses, i.e., Maths A and Maths B. This degree program is spread over four semesters. The year and semester wise details of MSc degree program are given below.   

Sr #

Category

Description

Credit Hours

1

General Education

Courses under this category are compulsory which are completed by all bachelor students at KUST.

6

2

Core Course

Core courses related to the degree program as per the HEC and KUST guidelines.

53

3

Technical Elective

In the senior year students are provided opportunity to choose area of specializations of their interest. The Institute offers various important latest courses for students, so they can choose market-oriented courses for their electives.

12-16

4

Senior Design Project/Electives

In order to train students to tackle a project related real world problems, senior design project is offered in the last semester. 

3

 

Total

 

74-78

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