BS Math 7th Semester
BS Mathematics (Semester Plan)
The BS Mathematics degree program is divided into eight semesters as follows.
Code 
Course Title 
CrHr 
PreReq 

Code 
Course Title 
CrHr 
PreReq 

Year 1 Semester 1 

Year 1 Semester II 

XXX*** 
URI 
4(3+1) 
None 

MATH102 
CalculusII 
3(3+0) 
MATH101 

MATH101 
CalculusI 
3(3+0) 
None 

MATH103 
Discrete Mathematics 
3(3+0) 
None 

XXX*** 
URII 
3(3+0) 
None 

XXX*** 
URIV 
3(3+0) 
None 

XXX*** 
URIII 
3(3+0) 
None 

XXX*** 
URV 
3(3+0) 
None 

XXX*** 
FRI 
3(3+1) 
None 

XXX*** 
FRII 
4(3+1) 
None 

Year 2 Semester III 

Year 2 Semester IV 

MATH201 
CalculusIII 
3(3+0) 
MATH102 

MATH211 
Group Theory 
3(3+0) 
None 

XXX*** 
FRIII 
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*** 
FRIV 
3(3+0) 
None 

XXX*** 
FRV 
3(3+0) 
None 

XXX*** 
URVI 
3(3+0) 
None 

XXX*** 
FRVI 
3(3+0) 
None 

XXX*** 
URVII 
3(3+0) 
None 

XXX*** 
FRVII 
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 AnalysisI 
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 
ElectiveII 
3(3+0)/4(3+1) 
MATHxxx 

MATH313 
Rings & Fields 
3(3+0) 
MATH211 

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

MATHxxx 
ElectiveI 
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 AnalysisII 
4(3+1) 
MATH331 

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

MATHXXX/ 
Elective/Report Writing & Presentation 
3(3+0)/4(3+1) 
MATHxxx 

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


Elective/Senior Design ProjectI 
3(0+9)/3(3+0)/4(3+1) 
MATHxxx 


Elective/Senior Design ProjectII 
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 
PreRequisite 
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 
PreRequisite 
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 
4(3+1) 
None 
PHY211 
Classical Mechanics 
3(3+0) 
None 
PHY212 
Quantum Mechanics 
3(3+0) 
None 
MATH241 
Mathematical StatisticsI 
3(3+0) 
None 
MATH242 
Mathematical StatisticsII 
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 
PreRequisite 
MATH101 
CalculusI 
3(3+0) 
None 
MATH101 
CalculusII 
3(3+0) 
MATH101 
MATH103 
Discrete Mathematics 
3(3+0) 
None 
MATH201 
CalculusIII 
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 AnalysisI 
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 AnalysisII 
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 fouryear 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 
PreRequisite 
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 
PreRequisite 
CS101 
Introduction to Computing 
4(2+1) 
None 
MATH101 
CalculusI 
3(3 + 0) 

MATH102 
CalculusII 
3(3 + 0) 
MATH101 
MATH103 
Discrete Mathematics 
3(3 + 0) 

MATH201 
CalculusIII 
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 
PreRequisite 
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 TechniquesI 
4(3 + 1) 
STAT211 
STAT331 
Regression Analysis 
4(3 + 1) 
STAT201 
STAT332 
Design and Analysis of ExperimentsI 
4(3 + 1) 
STAT201 
STAT313 
Statistical Packages 
3(3 + 0) 

STAT322 
Probability and Probability Distribution II 
3(3 + 0) 
STAT321 
STAT312 
Sampling TechniquesII 
4(3 + 1) 
STAT311 
STAT334 
Econometrics 
4(3 + 1) 
STAT331 
STAT333 
Design and Analysis of ExperimentsII 
4(3 + 1) 
STAT332 
STAT341 
Population Studies 
3(3 + 0) 
STAT211 
STAT421 
Statistical InferenceI 
3(3 0 3) 
STAT221 
STAT431 
Applied Multivariate Analyses 
4(3 + 1) 
STAT222 
STAT411 
Survey and Research Methods 
3(3 + 0) 
STAT312 
STAT422 
Statistical InferenceII 
3(3 + 0) 
STAT421 
STAT412 
Nonparametric 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 
PreRequisite 
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: CalculusI
Credit Hours: (3 0 3)
Prerequisite(s): None
Course Objectives:
The objectives of this course are:
 To learn about limits and their central role in calculus.
 To learn about derivatives and their relationship to instantaneous rates of change.
 To understand practical applications of derivatives.
 To learn about integrals: their origin in the area problem and their relationship to derivatives.
Reading list:
 Anton, I. C. Bivens, S. Davis, “Calculus, Early Transcendental”, 11^{th} edition, John Wiley, New York, 2016.
 Stewart, “Calculus”, 8^{th} edition, Brooks/Cole, 2016.
 B. Thomas, A. R. Finney, “Calculus”, 14^{th} edition, Pearson, USA, 2017.
Lecturewise 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. 
L4L5 
Continuity. 
L7L7 
Limits Involving Infinity; Asymptotes. 
L8L9 
Tangents and Derivatives at a Point, The Derivative by first principle. 
L10 
Left/Right Derivatives. 
L11L12 
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 
L18L19 
Related rates of change 
L20L21 
Linearization and Differentials. 
L22 
MID EXAM 
L23L24 
Extreme Values of Functions. 
L25 
The Mean Value Theorem. 
L26L27 
Monotone Functions and the First Derivative Test. 
L28L29 
Concavity and Curve Sketching. 
L30L32 
Indeterminate Forms & L’Hopitals Rule. 
L33L35 
Applied Optimization. 
L36L38 
Antiderivatives and techniques. 
L39L41 
Area and Estimating with finite sums. 
L42L43 
Definite Integral. 
L44L45 
Fundamental Theorems of Calculus 
L46L48 
Improper integrals. 
Course Code: MATH102
Course Title: CalculusII
Credit Hours: (3 0 3)
Prerequisite(s): MATH101
Course Objectives:
The objectives of this course are:
 To develop greater depth of understanding of integration and its importance in scientific and engineering applications.
 To learn about series, including their convergence properties and their use in representing functions.
 To learn alternative coordinate systems which are natural for many problems and learn how calculus can be applied in these systems?
Reading list:
 Anton, I. C. Bivens, S. Davis, “Calculus, Early Transcendental”, 11^{th} edition, John Wiley, New York, 2016.
 Stewart, “Calculus”, 8^{th} edition, Brooks/Cole, 2016.
 B. Thomas, A. R. Finney, “Calculus”, 14^{th} edition, Pearson, USA, 2017.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Introduction to the course 
L2 
Review of Integration 
L3L4 
Area between curves 
L5L7 
Volumes Using Cross Sections 
L8L9 
Volumes Using Cylindrical Shells 
L10L11 
Arc Length 
L12L14 
Areas of Surfaces of Revolution 
L15L17 
Moments and center of mass 
L18L19 
Work done by the variable force 
L20 
MID EXAM 
L21L22 
Sequences and related theorems for the limits of sequences 
L23L24 
Infinite Series, sequence of partial sums 
L25L26 
Integral Test 
L27L28 
Comparison Tests 
L29L30 
Root and Ratio Tests 
L31L33 
Alternating Series, Absolute vs. Conditional Convergence 
L34L36 
Series Convergence, Power Series 
L37L39 
Conic Section in Cartesian coordinates 
L40L42 
Parameterization of Plane Curves 
L43L45 
Polar Coordinates 
L46L48 
Areas and Lengths in Polar Coordinates, Pedal equations 
Course Code: MATH103
Course Title: Discrete Mathematics
Credit Hours: (3 0 3)
Prerequisite(s): None
Course Objectives:
The objectives of this course are:
 To learn the basics of Discrete Mathematics.
 To help students in gaining the understanding of mathematical reasoning and to develop their
 To develop problem solving skills.
 To show students how discrete mathematics can be used in modern computer science
Reading list:
 H. Rosen, “Discrete Mathematics and its Applications”, 7^{th} edition, McGrawHill Education, 2011.
 S. Epp, Discrete Mathematics with Applications, 4^{th} edition, Cengage India, 2011.
 L. Biggs, “Discrete Mathematics”, 2^{nd} edition, Oxford University press, 2013.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1L2 
Introduction to the course, Set Theory(Set Operations, Venn Diagram, Algebra of sets, Counting Principle Classes of sets) 
L3L5 
Logic, Propositional Equivalences, Predicates, Quantifiers 
L6L8 
Nested Quantifiers, Methods of Proof 
L9L11 
Algorithms, The Growth of Algorithms, Complexity of Algorithms 
L12L14 
Proof Strategy, Sequences, Summations, Mathematical Induction 
L15L17 
Recursive, Structural Induction, Recursive Algorithms 
L18L21 
Program Correctness 
L22L23 
Basics of Counting, The Pigeonhole Principal 
L24 
MID EXAM 
L25L27 
Permutations, Combinations, Binomial Coefficients 
L28L29 
Recurrence Relations, Solving Recurrence Relations 
L30L32 
Generating Functions, InclusionExclusion 
L33L35 
Relations, nary Relations, Representing Relations ,Closures of Relations 
L36L38 
Equivalence Relations, Partial Orderings 
L39L41 
Introduction to Graphs, Graph Terminologies 
L42L43 
Graph Isomorphism 
L44L46 
Connectivity, Euler and Hamilton Paths 
L47L48 
Planar Graphs 
Course Code: MATH201
Course Title: CalculusIII
Credit Hours: (3 0 3)
Prerequisite(s): MATH102
Course Objectives:
The objectives of this course are:
 To apply previously developed skills learned in Calculus to learn Multivariable Calculus and Vectors
 To cover Vectors, Partial Derivatives, Multiple Integrals and Vector Fields in order to prepare students for further study in technological disciplines.
 To cover relevant applications in science and engineering to illustrate the utility of learning these topics
 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:
 Anton, I. C. Bivens, S. Davis, “Calculus, Early Transcendental”, 11^{th} edition, John Wiley, New York, 2016.
 Stewart, “Calculus”, 8^{th} edition, Brooks/Cole, 2016.
 B. Thomas, A. R. Finney, “Calculus”, 14^{th} edition, Pearson, USA, 2017.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Introduction to the course 
L2L3 
3Dimensional Coordinate Systems and Vectors. 
L4 
Dot Product. 
L5 
Cross Product. 
L6 
Lines and Planes in Space. 
L7L8 
Cylinders and Quadric Surfaces 
L9L10 
Curves and Tangents in Space. 
L11L12 
Integrals of Vector Functions: Projectile Motion. 
L13 
Arc Length of Space Curves. 
L14L15 
Functions of Several Variables. 
L16 
Partial Derivatives. 
L17 
The Chain Rule. 
L18L19 
Directional Derivatives and Gradients. 
L20L21 
Tangent Planes and Differentials. 
L22 
MID EXAM 
L22L23 
Extrema and Saddle Points. 
L24 
Lagrange Multipliers. 
L25L26 
Double and Iterated Integrals over Rectangles. 
L27 
Double Integrals over General Regions. 
L28 
Double Integrals in Polar Form. 
L29L30 
Triple Integrals in Rectangular Coordinates. 
L31L32 
Triple Integrals in Cylindrical Coordinates. 
L33L34 
Line Integrals. 
L35L37 
Vector Fields and Line Integrals: Work, Circulation and Flux. 
L38L40 
Path Independence, Conservative Fields and Potential Functions. 
L41L42 
Green’s Theorem in the plane. 
L43L44 
Surface Area and Surface Integrals. 
L45L46 
Divergence Theorem. 
L47L48 
Stoke’s Theorem. 
Course Code: MATH241
Course Title: Mathematical StatisticsI
Credit Hours: (3 0 3)
Prerequisite(s): MATH101
Course Objectives:
 To learn how to handle data numerically and graphically
 To understand the basic principles of random variables and random processes needed in applications
 To learn discrete random variables and their probability distributions
Reading List:
 S. Mann, “Introductory Statistics”, 9^{th} edition, John Wiley & Sons, 2016.
 T. Craig, J. McKean, R.V. Hogg, “Introduction to Mathematical Statistics”, 7^{th} edition, Pearson Education, 2013.
 R. Sheldo, “Introductory Statistics”, 3^{rd} edition, Oxford, 2010.
 R. Spiegel, J.L. Schiller, R.L. Sirinivasan, “Probability and Statistics”, 2^{nd} edition, McGraw Hill. NY, 2000.
 M. Clark, and D. Cooke, “A Basic Course in Statistics” 4^{th} edition, Arnold, London, 1998.
 T. Mclave, P.G. Benson,T. Snitch, “Statistics for Business & Economics” 9th edition, Prentice Hall, New Jersey, 2005.
 M. Chaudhry, S. Kamal, “Introduction to Statistical Theory” Parts I 6^{th} edition, Ilmi Kitab Khana, Lahore, Pakistan, 1996.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Review of Mathematical Statistics 
L2L4 
The nature and scope of the statistics, 
L5L7 
Organizing of data, classification of data and graphs, 
L8 L10 
Measures of central tendency: their properties, usage, limitations and comparison. 
L11L13 
Continued… 
L14L16 
Measures of dispersion: their properties, usage, limitations and comparison. 
L17L19 
Continued… 
L20L22 
Calculations for the ungrouped and grouped data, measures of skewness, Kurtosis and distribution shapes. 
L23 
Mid Exam 
L24L25 
Sets, probability concepts, permutation and combination. 
L26L28 
Addition and multiplication rules, bivariate frequency tables, joint and marginal probabilities. 
L29L31 
Conditional probability and independence, Bayes’ rule. 
L32L34 
Random variables, properties of random variables, discrete and continuous random variables 
L35L37 
Percentile for continuous random variables, multivariate distributions, marginal distributions, conditional distributions. 
L38L40 
Mathematical expectation, moments, Chebyshev’s theorem, moment generating Functions 
L41L43 
Relation between moments and cumulants, product moments, characteristic Function, conditional expectations. 
L44L46 
Probability distribution, discrete probability distributions, Bernoulli distribution, properties, binomial distribution, 
L47L48 
Poisson and hypergeometric distributions, negative binomial and geometric Distributions. 
Course Code: MATH202
Course Title: Number theory
Credit Hours: (3 0 3)
Prerequisite(s): None
Course Objectives:
The objectives of this course are:
 To learn about divisibility and its properties.
 To learn about Congruencies its properties and applications.
 To understand Primitive Roots and Quadratic Residue.
Reading list:
 W. Stein,” Elementary Number Theory”, SpringerVerlag,2017
 K.H Rosen, “Elementary Number Theory and its Applications”, 6th edition, Pearson, 2010.
 T. Koshy, “Elementary Number Theory with Applications”, 2^{nd} edition, Academic Press, 2007.
 D.M. Burton, “Elementary Number Theory”, 7^{th} edition, McGrawHill, 2010.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1L3 
Introduction to the course, Wellordering principle , Mathematical induction Divisibility ,Transitivity of divisibility ,Division of linear combination of integers and other related properties of divisibility 
L3L4 
The Division Algorithm, Proof and consequences of Division Algorithm 
L5L6 
Representation of integers in different base systems. Arithmetic in nondecimal systems. Negative bases 
L7L9 
Prime Numbers, Prime divisors, infinitude of primes, Upper bound for a prime factor of composite integers, Sieve of Eratosthenes , Prime number theorem(without proof) 
L10L12 
Greatest Common Divisor, Properties of GCD(theorems) ,GCD of more than two integers , The Euclidean Algorithm for finding GCD, 
L13L15 
Fibonacci numbers, GCD of successive Fibonacci numbers , Lame’s Theorem The Fundamental Theorem of Arithmetic, Least common multiple, Theorem relating GCD and LCM 
L16L18 
Fermat factorization, Fermat number, Prime divisors of Fermat number 
L19L21 
Linear Diophantine Equation, Solution in integers for Linear Diophantine Equation and its Applications. 
L22L24 
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 
L26L28 
Linear Congruencies, Criterion for existence of solution and method of solution for linear congruencies, System of linear congruencies in one variable , Chinese Reminder theorem . 
L29L31 
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 
L32L33 
Wilson's Theorem and Fermat's Little Theorem, Pseudoprim, Euler phifunction ,Euler Theorem 
L34L36 
Arithmetic function, Multiplicative functions, Euler phifunction as Multiplicative function , Mobius function, Mobius inversion formula 
L37L38 
Functions for The Sum and Number of Divisors, Perfect Numbers and Mersenne Primes 
L39L41 
Order of an Integer and primitive Roots, Incongruent Primitive roots of a positive integer ,Primitive roots for Primes 
L42L43 
Lagrange's Theorem, The Existence of Primitive Roots 
L44L45 
Index Arithmetic ,Quadratic Residue, Legendre symbol 
L46L48 
Euler's criterion, Gauss Lemma, law of quadratic reciprocity , Jacobi symbol 
Course Code: MATH211
Course Title: Group Theory
Credit Hours: (3 0 3)
Prerequisite(s): None
Course Objectives:
 To learn group structures
 To learn some fundamental results and techniques of group theory
 To classify groups (not all) up to isomorphism
Reading List:
 B. Bhattacharya, S.K. Jain and S.R. Nagpaul, “Basic Abstract Algebra”, 2^{nd} ed., Cambridge University Press, 1995.
 S. Dummit and R.M. Foote, “Abstract Algebra”, 3^{rd} ed., AddisonWesely, 2004.
 B. Fraleigh, “A First Course in Abstract Algebra”, 7^{th} ed., Pearson, 2002.
 A. Gallian, “Contemporary Abstract Algebra”, 7^{th} ed., Brooks/Cole, 2010.
 F. Humphreys, “A course in Group Theory”, Oxford University Press, 1996.
 Majeed, “Theory of Groups”, Ilmi Kitab Khana, 2012.
Lecturewise distribution of the course contents
Lecture # 
Topics 
L1 
Introduction to the course 
L2L3 
Operations, binary operations, usual and unusual operations (including modular arithmetic), semigroup, monoid 
L4L6 
Group, Caley’s table for finite groups, elementary properties of groups, order of a group, order of an element 
L7L9 
Group of symmetries ( e.g., equilateral triangle, square, rectangle etc) 
L10L12 
Subgroup with examples, subgroup test, finite subgroup test 
L13L15 
Subgroup lattice, product of subgroups, direct product of groups 
L16L18 
Generators and defining relations, cyclic groups and their properties 
L19L21 
Cosets, index of subgroup, Lagrange’s Theorem, its converse and consequences 
L22L23 
normalizer and centralizer of a subset of a group, center of a group 
L24 
MID EXAM 
L25L27 
Normal subgroups, factor groups 
L28L30 
Group homomorphism, kernel and image 
L31L33 
Properties elements/subgroups under homomrphism 
L34L36 
Injective and surjective homomorphism, endomorphism, isomorphism, 
L37L39 
The Fundamental Theorem of homomorphism and its applications 
L40L42 
2^{nd} and 3^{rd} isomorphism theorems 
L43L45 
Permutations, Cycles in group of permutations and their properties, even and odd permutations 
L46L48 
Symmetric and alternating subgroups, Caley’s Theorem 
Course Code: MATH271
Course Title: Ordinary Differential Equations
Credit Hours: (3 0 3)
Prerequisite(s): MATH101
Course Objectives:
The course objectives are:
 To learn elementary analytical solution techniques for the solution of ordinary differential equations (ODEs).
 To understand the solution structure of linear ODEs in terms of independent homogeneous solutions and nonhomogeneous solutions.
 To understand by exposure to examples how systems and phenomena from science and engineering can be modeled by ODEs.
 To learn how solution of different models can be used to analyze or predict a system’s behavior.
Reading list:
 W. E. Boyce, R. C. DiPrima, “Elementary Differential Equations and Boundary Value Problems, 10^{th} edition”, John Wiley & Sons, Inc., 2012.
 D. G. Zill, M. R. Cullen, “Differential Equations with BoundaryValue Problems”, 10^{th} edition, Brooks/Cole, 2013.
 R. L. Borrelli, C. S. Coleman, “Differential Equations: A Modeling Perspective”, 2^{nd} edition”, John Wiley & Sons Inc., 2004.
 R. K. Nagle, E. B. Saff, A. D. Snider, “Fundamentals of Differential Equations, 9^{th} edition”, Pearson Education Inc., 2017.
 E. Kreyszig, “Advanced Engineering Mathematics”, 10^{th} edition, John Wiley & Sons Inc., 2010.
 M. Colomb, M. Shanks, “Elements of Ordinary Differential Equations”, McGrawHill, 2001.
Lecturewise distribution of the course contents
Lecture # 
Topic 
L1 
Introduction to the course 
L2 
Differential Equations, Classification, Examples. 
L3L4 
Some Basic Models; Direction Fields. 
L5 
WellPosedness, Separable Differential Equations. 
L6 
Homogenous Differential Equations. 
L7L8 
Exact Differential Equations, Integrating factor in different cases. 
L9 
Linear Differential Equations. 
L10 
Differential equations reducible to linear DEs. 
L11L13 
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 NonLinear Equations: Logistic Population Model 
L16L17 
Higher Order Linear Differential Equations, IVP, BVPs with different possible solutions, Linear Dependence and Linear Independent functions, The Wronskian. 
L18L19 
Solutions of Linear Homogeneous Equations with constant coefficients:, Roots of the Characteristic equation. 
L20 
Reduction of Order method. 
L21 
MID EXAM 
L22L23 
Nonhomogeneous Equations; Method of Undetermined Coefficients. 
L24 
Method of Variation of Parameters. 
L25L26 
Cauchy Euler differential equation, Legendre differential equation. 
L27L29 
Modeling with 2^{nd} Order linear differential equations: Spring mass system (Free Undamped Motion, Free Damped Motion and Driven Motion). 
L30L31 
Review of Power series. 
L32L33 
Series Solutions of Second Order Linear ODEs with Nonconstant Coefficients; Solution Near an Ordinary Point. 
L34L36 
Singular points, Series Solutions Near a Regular Singular Point. 
L37L39 
Laplace Transform and Solution of Initial Value Problems. 
L40L42 
The Convolution Integral. 
L43L45 
System of Linear first order differential equations: Preliminary theory, Homogenous Linear system with constant coefficients. 
L46L48 
Qualitative approach for system of linear first order equations 
Course Code: MATH 242
Course Title: Mathematical StatisticsII
Credit Hours: (3 0 3)
Prerequisite(s): 241
Course Objectives:
 To learn continuous random variables and their probability distributions
 To learn sampling concepts and methods
 To help students in making predictions and decisions.
Reading List:
 T. Craig, J. McKean and R.V Hogg, “Introduction to Mathematical Statistics”, 7th edition, Pearson Education, 2013.
 A. Garaybill, A.M Mood & D.C. Boes, “Introduction to Theory of Statistics” 3rd edition, McGraw Hill Co, 1974.
 Miller and M. Miller, “John E. Freund's Mathematical statistics with Applications”, 8th edition, Pearson Education, 2014.
 B. Ash, “Basic Probability Theory, Dover”, Dover Books on Mathematics, 2008.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Introduction to Mathematical statistics II 
L2L4 
Continuous random variables, probability density function and its properties

L5L7 
Continuous distributions: rectangular distribution, exponential distribution 
L8 L10 
Gamma distribution 
L11L13 
Beta distribution, relation between gamma and beta distribution with normal Distribution 
L14L16 
Bivariate discrete random variables 
L17L19 
Bivariate continuous random variables, conditional distributions, independence of random variables 
L20L22 
Numerical examples on continuous random variables, conditional distributions, independence of random variables 
L23L24 
Covariance of bivariate random variables, variance of the linear combination of random variables 
L25 
MID EXAM 
L26L28 
Correlation and independence, moment generating functions 
L29L31 
Differences between partial and multiple correlation. 
L32L34 
The chi square table, introduction of F distribution 
L35L37 
Some properties of F distribution, the F tables of area, tests based on F Distribution 
L38L40 
Introduction to sampling theory. Sampling distribution and properties. limiting 
L41L43 
Some examples on sampling distribution for illustration, 
L44L46 
Transformation of variables reasons and importance, transforming to uniform Distribution 
L47L48 
Statistical hypothesis and tests 
Course Code: MATH212
Course Title: Linear Algebra
Credit Hours: (3 0 3)
Prerequisite(s): None
Course Objectives:
The objectives of this course are:
 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.
 To improve their ability to reason abstractly and understand proofs and know the geometric approaches which help them in visualizing an idea.
 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:
 C. Lay, S. R. Lay, J. J. McDonald, “Linear Algebra and Its Applications”, 5^{th} edition, Pearson Education, 2015.
 A. Beezer, “A First Course in Linear Algebra”, 3^{rd} edition, Congruent Press, 2012.
 Strang, “Introduction to Linear Algebra”, 4^{th} edition, Wellesley Press, 2009.
 K. Nicholson, “Linear Algebra and Applications, 7^{th} edition, McGrawHill, 2013.
 Anton, “Elementary Linear Algebra”, 9^{th} edition”, John Wiley & Sons Inc., USA, 2005.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Introduction to the course 
L2L3 
System of Linear Equations, Row Reduction and Echelon Forms, Vector Equations,The Matrix Equation Ax=b. 
L4L6 
Solution sets of Linear systems, Applications of Linear systems. 
L7 
Linear Independence. 
L8L10 
Introduction to Linear Transformations, The Matrix of a Linear Transformations. 
L11L13 
Linear Models in Business, Science and Engineering. 
L14L16 
Matrix Operations, The Inverse of a Matrix, Characterizations of Invertible matrices, Matrix Factorizations. 
L17L18 
Applications to Computer Graphics. 
L19L20 
Subspaces of . 
L21 
Dimension and Rank. 
L22 
MID EXAM 
L23L25 
Introduction to Determinants, properties, Cramer’s Rule, Voume, and Linear Transformations. 
L26L28 
Vector spaces and subspaces, Null spaces, Column spaces, and Linear Transformations 
L29L31 
Linear Independent Sets, Bases, Coordinate system, The Dimension of a vector space, Rank 
L32L33 
Change of Basis. 
L34L36 
Applications to Difference equations and Markov Chain. 
L37L39 
Eigenvalues and Eigenvectors, The Characteristic Equation, Diagonalization. 
L40L41 
Discrete Dynamical Systems. 
L42L43 
Applications to Differential Equations. 
L44L45 
Inner product, Length and Orthogonality, Orthogonal sets, Orthogonal Projections 
L46 
The GramSchmidth Process 
L47L48 
LeastSquares Problems 
Course Code: MATH321
Course Title: Real AnalysisI
Credit Hours: (3 0 3)
Prerequisite(s): MATH101
Course Objectives:
The objectives of this course are:
 To understand and deal with the real number system.
 To be able to construct proofs regarding limit and continuity.
 To learn basic properties of functions on R.
 To learn the elementary theory of differentiation.
Reading list:
 L. Brabenec, “Introduction to Real Analysis”, PWS Publishing Co, USA 1997
 D. Gaughan, “Introduction to Analysis”, 5th edition, Brooks/Cole, 1997.
 G. Bartle, D. R. Sherbert, “Introduction to Real Analysis”, 4^{th} edition, John Wiley & Sons Inc, 2011.
 H. Protter, “Basic Elements of Real Analysis”, Springer Verlag, New York, 1998
 C Malik, S. Arora, “Mathematical Analysis”, Wiley Eastern Ltd., 2009.
Lecturewise 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 
Subsequences 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)
Prerequisite(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”, 2^{nd }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”, McGrawHill, 1965. 
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Introduction to the course 
L2L4 
Topology of the Line and Plane 
L5 
Topological spaces. 
L6L7 
Closet sets, closure of a set and related results. 
L8L10 
Interior, exterior and boundary of a set and related results. 
L11L13 
Neighborhoods and neighborhood systems, Accumulation points , Coarser and finer topologies. 
L14 
Subspaces, relative topologies. 
L15L17 
Base for a topology, Subbases, Local bases and related theorems. 
L18L20 
Continuous functions, Continuity at a point,open and closed functions. 
L21 
MID EXAM 
L22L23 
Homeomorphism and related theorems. 
L24L26 
Metrics, Distance between sets, diameters, open spheres. 
L27L29 
Metric topologies, metric spaces, Properties of metric topologies, Metrization problem, Isometric metric spaces, Euclidean mspace. 
L30L32 
Convergence and Continuity in metric spaces. 
L33L35 
T1spaces, Hausdorff spaces, Regular spaces, Normal spaces and related theorems. 
L36L38 
Urysohn’s lemma and metrization theorem, Completely regular spaces. 
L39L41 
Covers, Compact sets, subset of compact sets, Finite intersection property. 
L42L43 
Locally compact spaces, Compactness in metric spaces. 
L44L46 
Product topology, Base for a finite product topology. 
L47L48 
Separated sets, Connected sets, Connected spaces, Connectedness on the real line , Components, Locally connected spaces, Paths, Arcwise connected sets. 
Course Code: MATH 371
Course Title: Partial Differential Equations
Credit Hours: (3 0 3)
Prerequisite(s): MATH271
Course Objectives:
The objectives of this course are:
 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.
 To apply analytical methods to solve these PDEs and be able to interpret the results.
Reading list:
 Richard Haberman, “Applied Partial Differential Equations with Fourier Series and Boundary Value Problems”, 5^{th} edition, Pearson Prentice Hall, 2012.
 Amaranath , “An Elementary Course in Partial Differential Equations”, 4^{th} edition”,
Jones and Barlett learning Massachusetts, 2009.
 Sankarra Rao, “Introduction to Partial Differential Equations” 3^{rd} edition, Prentice Hall, 2010.
 Walter A. Strauss, “Partial Differential Equations: An Introduction”, 2^{nd} edition, Willey, 2007.
 David Logan, “Applied Partial Differential Equations”, 3^{rd} edition”, Springer, 2014.
 Donald Greenspan, “Introduction to Partial Differential Equations”, 1^{st} edition, Dover Publications, 2000.
Lecturewise Distribution of the Contents
Lecture # 
Topics 
L1L3 
Introduction to the course, Differential equations, PDEs; Order, linearity of PDEs, IVPs and BVPs, Wellposedness, Mathematical Models, 
L4L6 
Formation of PDEs, Elimination of arbitrary constants and arbitrary functions 
L7L9 
PDEs solvable like ODEs, Linear 2nd Order PDEs, Classification of PDEs, 
L10L12 
Heat Equation: 1D Derivation & Boundary Conditions, Steady State Diffusion equation(Laplace equation), Wave Equation: 1D Derivation 
L13L15

Characteristics of PDEs, Canonical forms of different types of PDEs, 
L16L18 
working examples, General solutions of PDEs using method of characteristics 
L19L21 
Heat equation/Wave equation and Laplace equation in cylindrical and spherical coordinates, Fourier Series and termbyterm operations 
L22L24 
Method of Separation of Variables, Working Examples with Heat Equation, Wave equation with homogeneous boundary conditions 
L25 
MID EXAM 
L25L27 
Working Examples with the Heat Equation, Laplace equation, Wave equation with nonhomogeneous boundary conditions 
L28L30 
PDE’s in 2D: Vibration of a Rectangular Membrane, Vibration of a Circular Membrane, Bessel Functions, 
L31L33 
Nonhomogeneous Problems: Heat flow with sources and Nonhomogenous BCs 
L34L36 
Integral transform methods, Fourier Transform and its properties, Infinite Domain Problems 
L37L39 
Fourier Transform Solutions of PDEs, Working Examples 
L40L42 
Laplace transform, its properties, SemiInfinite Domain Problems: 
L43L45 
Laplace transform solutions of PDEs, working examples 
4648 
Hankel transform, its properties, Hankel transform solutions of PDEs, working examples 
Course Code: MATH313
Course Title: Rings and Fields
Credit Hours: (3 0 3)
Prerequisite(s): MATH 211
Course Objectives:
 To learn construction of rings
 To study structural properties of rings
 To understand familiar concepts, such as fractions, factorization, primness, divisibility etc., in general setting of rings.
Reading list:
 P.B. Bhattacharya, S.K. Jain and S.R. Nagpaul, “Basic Abstract Algebra”, 2^{nd} ed., Cambridge University Press, 1995.
 D.S. Dummit and R.M. Foote, “Abstract Algebra”, 3^{rd} ed., AddisonWesely, 2004.
 J.B. Fraleigh, “A First Course in Abstract Algebra”, 7^{th} ed., Pearson, 2002.
 J. A. Gallian, “Contemporary Abstract Algebra”, 7^{th} ed., Brooks/Cole, 2010.
 S. Lang, “Algebra”, 3^{rd} ed., Springer, 2005
 C. Musili, “Introduction to rings and Modules”, 2^{nd} ed., Narosa Publishing House, 2009.
Lecturewise distribution of the course contents
Lecture # 
Topics 
L1 
Introduction to the course 
L2L3 
Rings: examples and basic notions 
L4L6 
Units and their properties, idempotent and nilpotent elements in a ring 
L7L9 
Some important rings: rings of continuous functions, matrix rings, polynomial rings, power series rings, Laurent rings, Boolean rings, endomorphism ring, group ring 
L10L12 
Opposite rings, direct product of rings, characteristic of a ring 
L13L15 
Division ring, field, zero divisors, integral domains and their properties 
L16L18 
Subrings, subring test, examples and properties, center of a ring 
L19L21 
Motivation for an ideal, left, right and two sided ideals, their examples and properties 
L22L23 
Ideals in a field, annihilators, radical of an ideal 
L24 
MID EXAM 
L25L27 
Ideal generated by a subset, finitely generated ideal, ideals in a commutative ring with unity, principal ideal 
L28L30 
Factor rings, constructing examples, further structural properties, ideals in factor rings 
L31L33 
Prime ideals, maximal ideals, examples and important characterizations, local rings 
L34L36 
Ring homomorphism, kernel and image, monic and kernel, fundamental theorem of homomorphism and its consequences 
L37L39 
Field of fractions, examples and construction 
L40L42 
Factorization of polynomials over a field, irreducible polynomials 
L43L45 
Division in domains, Euclidean domains 
L46L48 
Principal ideal domains , factorization domains, unique factorization domains 
Course Code: MATH131
Course Title: Computing Tools
Credit Hours: (2 3 3)
Prerequisite(s): None
Course Objectives:
The objectives of this course are:
 To learn about arrays, cell and structure.
 To learn how to plot 2 and 3D functions with in Computing Tools (e.g, Matlab/ Maple/ Mathematica).
 To learn about symbolical processing Tools (e.g, Matlab/ Maple/ Mathematica)
 To learn how to construct user define functions and do simple programming in Computing Tools (e.g, Matlab/ Maple/Mathematica)
Reading list:
 J. Palm III, “A Concise Introduction to MATLAB”, McGrawHill, 2008.
 Attaway, “A Practical Introduction to Programming and Problem solving”, 2^{nd} edition, Elsevier Inc, 2011.
 R. Hunt, R.L. Lipsman and J.M. Rosenberg,“A Guide to MATLAB for Beginners and Experienced Users”, 2^{nd} edition, Cambridge University press,2006.
 Knight, “Basics of MATLAB & Beyond”, Chapman and Hall, 2000.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1L3 
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 
L4L6 
One and TwoDimensional Numeric Arrays, Multidimensional Numeric Arrays, ElementbyElement Operations, Matrix Operations 
L7L8 
Matrix Methods for Linear Equations, Polynomial Operations Using Arrays, Cell Arrays, Structure Arrays, Special Matrices 
L9L10 
Exponential and Logarithmic Functions, Complex Number Functions, Numeric Functions, Trigonometric Functions. 
L11L13 
UserDefined Functions, Some Simple Function Examples, Local and Global Variables, Function Handles, Methods for Calling Functions 
L14L16 
Anonymous Functions, Primary Function, Subfunctions, Nested Functions, Relational Operators and Logical Variables, The logical Class, The logical Function 
L17L19 
Accessing Arrays Using Logical Arrays, Logical Operators and Functions, Order of precedence for operator types, ShortCircuit Operators 
L21L23 
Conditional Statements, The if, else, elseif Statement, Strings and Conditional Statements, MATLAB program to solve linear equations and other practice examples 
L24 
MID EXAM 
L25L27 
For Loops, Series Calculation with a for Loop, nested loops, The break and continue Statements, Using an Array as a Loop Index 
L28L30 
While Loops, Series Calculation with a while Loop, The switch Structure, Practice examples. 
L31L32 
Debugging MATLAB Programs, Cell Mode, The Debug Menu, Debugging Using Breakpoints 
L33L35 
xy Plotting Functions, Saving Figures, Exporting Figures, Additional Commands and Plot Types, Interactive Plotting in MATLAB 
L36L38 
ThreeDimensional Line Plots, Surface Mesh Plots, Contour Plots ,MATLAB ODE Solvers, ode45, ode15s 
L39L40 
Symbolic Processing, Symbolic Expressions, Manipulating Expressions, Evaluating Expressions, Algebraic and Transcendental Equations solution 
L41L43 
Sums, Limits, differentiation, integration and differential equations 
L44L46 
Laplace Transforms, Symbolic Linear Algebra, Characteristic Polynomial and Roots 
L47L48 
Solving Linear Algebraic Equations 
Course Code: MATH325
Course Title: Vector and Tensor Analysis
Credit Hours: (3 0 3)
Prerequisite(s): MATH212
Course Objectives:
The objectives of this course are:
 To learn about vector quantities and algebra of vector addition and multiplication.
 To understand differentiation and integration of vector valued functions and their applications.
 To learn about tensor quantities and algebra of tensor addition and multiplication.
 To understand differentiation of tensors fields.
Reading list:
 E. Bourne, P.C Kendall , “Vector Analysis and Cartesian Tensors”, 3rd edition, Stanley Thornes, 1999.
 D. Smith, “Vector Analysis”, Oxford University Press, Oxford 1999.
 R. Spiegel, “Vector Analysis & Introduction to Tensor Analysis”, McGraw Hill, New York 2009.
 R. Spiegel, “Vector Analysis”, 2nd edition, McGraw Hill New York, 2009.
 G. Simmonds, ”A Brief on Tensor Analysis”, SpringerVerlag, 2012.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1L3 
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 
L4L6 
Dot Product and Cross product of vectors, Properties and applications of dot and Cross product. 
L7L8 
Scalar and vector triple product ,Properties and applications of triple product 
L9L11 
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 , Binormal 
L13L15 
Gradient, Divergence ,Curl , Formulas involving gradient, Divergence and Curl 
L16L18 
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 
L19L21 
Surface integrals, Volume integrals, Limit of sum definition and evaluation technique 
L22L23 
Transformation of coordinates, curvilinear coordinate , orthogonal curvilinear coordinate, Unit vectors in curvilinear systems 
L24 
MID EXAM 
L25L26 
Contravariant and covariant components of a vector, Gradient, Divergence and Curl in curvilinear coordinate system 
L27L29 
Special orthogonal coordinate systems , Cylindrical Coordinate, spherical Coordinates, Parabolic Cylindrical Coordinates, Paraboloidal Coordinates 
L30L32 
Covector, Scalar product of vector and covector, Linear operators, Bilinear and quadratic forms, Dual Bilinear forms, Einstein summation convention ,General definition of tensors. 
L33L35 
Dot product and metric tensor, Tensors addition and multiplication by a scalar, Tensor product 
L36L38 
Contraction, Kronecker symbol, LeviCivita symbol, Tensor fields in Cartesian coordinates 
L39L41 
Change of Cartesian coordinate system, Differentiation of tensor fields 
L42L44 
Gradient, divergence, and curl, Laplace and d’Alambert operators 
L45L46 
Tensor fields in curvilinear coordinate 
L47L48 
Moving frame of curvilinear coordinates, Christoffel symbols 
Course Code: MATH322
Course Title: Real AnalysisII
Credit Hours: (3 0 3)
Prerequisite(s): MATH321
Course Objectives:
The course objectives are:
 To learn the differentiation and integration theory in
 To be able to construct proofs regarding sequences, series and their convergence
 To be able to construct proofs regarding the improper integrals
 To learn the Riemann–Stieltjes integrals
Reading list:
 L. Brabenec, “Introduction to Real Analysis”, PWS Publishing Co., 1997.
 D. Gaughan, “Introduction to Analysis, 5^{th} edition, Brooks/Cole, 1997.
 G. Bartle, D. R. Sherbert, “Introduction to Real Analysis” 4^{th} edition, John Wiley & Sons, 2011.
 H. Protter, “Basic Elements of Real Analysis”, SpringerVerlag, New York, 1998.
 S.C Malik, S. Arora, “Mathematical Analysis”, Wiley Eastern Ltd. 2009.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Introduction to the course 
L2 
Functions of several variables 
L3L4 
Limit and Continuity 
L5L7 
Differentiability 
L8L10 
Partial derivatives, Chain rule 
L11L12 
Young’s theorem and Schwarz theorem 
L13L15 
Implicit functions, Implicit function theorem, Inverse function theorem 
L16 
Jacobian 
L17 
Functionally related functions 
L18 
Maxima and Minima for functions of two variables 
L19L20 
Series of numbers and their convergence, Alternating Series, Leibnitz Test 
L21L23 
Comparison test, Limit comparison test 
L24 
MID EXAM 
L25 
Cauchy integral test 
L26L28 
DAlembert Ratio test, Cauchy Root test + MIDTERM EXAMINATION 
L29 
Series of variable terms 
L30 
Uniform convergence 
L31 
Weierstrass M theorem 
L32L34 
Convergence and Divergence of improper integrals 
L35 
PTest for convergence of improper integrals 
L36L38 
Darboux upper and lower sums and integrals 
L39 
Definition and existence of the Riemann integral 
L40 L42 
Theorems on Riemann integration 
L43L45 
Integration and differentiation Theorems 
L46L48 
RiemannSteiltjes integration 
Course Code: MATH331
Course Title: Numerical Analysis I
Credit Hours: (3 3 4)
Prerequisite(s): MATH321
Course Objectives:
The objectives of this course are:
 To demonstrate understanding of common numerical methods and how they are used to obtain approximate solutions to otherwise intractable mathematical problems. Rootfinding iterative methods will be discussed both in respect of their derivations and convergence performance.
 To demonstrate numerical methods to obtain solutions of system of linear and nonlinear algebraic equations.
 To perform an error analysis for various numerical methods
 To implement such numerical methods in MATLAB or any programming language.
Reading list:
 L. Burden and J.D. Faires, “Numerical analysis 10^{th} edition”,Brooks Cole, 2015.
 F. Gerald, P.O. Wheatley, “Applied Numerical Analysis 7^{th} edition”, Pearson , 2003.
 Atkinsonan , W. Han, " Elementary Numerical Analysis 3^{rd} edition”,Wiley,2003.
 K. Jain, S.R.K. Iyengar, R.K. Jain “Numerical Methods for Scientific and Engineering computation 6^{th} edition”, New Age International Pvt Ltd, 2010.
 W. Hamming, ”Numerical Methods for Scientists and Engineer 2^{nd} revised edition”, Dover, 1987.
 B. Hildebrand , “Introduction to Numerical Analysis 2^{nd} edition”, Dover,1987.
 Bradie , “A Friendly Introduction to Numerical Analysis 1^{st} edition”, Pearson, 2005.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Introduction and overview to the course 
L2L3 
Calculus Review: Continuity, differentiability and related theorems, convergence of sequences, Taylor’s theorem 
L4L5 
Error, types, sources and propagation, computer arithmetic’s 
L6L7 
Algorithms and convergence, stability analysis of algorithms, error growth, rate and order of convergence. BigO and LittleO notations. 
L8L10 
Solutions of equations in one variable: The Bisection method, algorithm (or pseudo code) and Implementation in Matlab. Error Analysis 
L11L13 
Fixed point, existence and uniqueness of fixed point, fixed point iteration, cobwebbing diagram. 
L14L16 
Newton’s Method, Derivation, Algorithm and Implementation in MATLAB, Error Analysis. Modified Newton’s Method for roots with multiplicity. 
L17L19 
Secant Method, Derivation, Algorithm and Implementation, Error Analysis 
L20 
Method of false position, Algorithm 
L21 
MID EXAM 
L22L23 
Linear system of equations, Pivoting strategies, Linear algebra and matrix inversions. 
L24L26 
Elimination methods: Gauss Elimination and GaussJordan, operations analysis, Algorithms. 
L27L29 
Matrix Factorizations: Doolittle’s method, Crout’s method, Cholesky method with algorithms. 
L30L32 
Norms of vectors and matrices, convergence and perturbation theorems 
L33L35 
Eigen values and eigen vectors, power and inverse power method. 
L36L37 
Spectral radius of a matrix, Greshgorin Circle theorem for bounds of eigen values 
L38L40 
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 
L42L44 
Solving Sparse systems: Gradient vectors, quadratic forms, Residuals, Krylov subspace, Steepest descent method 
L45L48 
Numerical solution of Nonlinear system using Newton’s method 
Course Code: MATH323
Course Title: Complex Analysis
Credit Hours: (3 0 3)
Prerequisite(s): MATH321
Course Objectives:
The objectives of this course are:
 To understand basic theory of algebraic and geometric structures of the complex numbers.
 To understand the concepts of analyticity, CauchyRiemann relations and harmonic functions are then introduced with some applications in fluid dynamics.
 To learn Complex integration and complex power series.
 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:
 G. Zill, P. D. Shanahan, “A First Course in Complex Analysis with Applications”, 3^{rd} edition, Jones and Bartlett Publishers, 2013
 W. Brown, R. V. Churchill, “Complex Variables and Applications”, 9^{th} edition, McGrawHill, 2013.
 A. Silverman, “Complex Analysis with Applications’’, 1^{st} edition, Dover, 2010.
 B. Saff, A. D. Snider, “Fundamentals of Complex Analysis with Applications to Science and Engineering”, 3^{rd} edition”, Pearson Education, 2003.
 K. Jain, S. R. K. Iyengar, “Advanced Engineering Mathematics” 10^{th} edition, John Wiley & Sons Inc., 2011.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1L3 
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, 
L4L6 
Complex Functions, Complex Functions as Mappings, Linear Mappings, Special Power Functions, Reciprocal Function, differences between real and complex functions 
L7L9 
Limits and Continuity, Complex functions as vector fields 
L10L12 
Differentiability and Analyticity 
L13L14 
CauchyRiemann Equations 
L15L16 
Harmonic Functions, Applications: Orthogonal families, Gradient fields, Complex potentials and ideal fluids, Heat flow 
L17L19 
Elementary functions: Exponential and Logarithmic Functions, Complex Powers, Trigonometric and Hyperbolic Functions, Inverse Trigonometric and Hyperbolic Functions 
L20L22 
Real Integrals, Complex Integrals 
L23 
MID EXAM 
L24L26 
CauchyGoursat Theorem 
L27 
Independence of Path 
L28L30 
Cauchy’s Integral Formula, Cauchy’s Integral Formula for derivatives, 
L31L33 
Consequences of Cauchy’s Integral Formula: Cauchy’s Inequality,Liouville’s theorem, Fundamental theorem of algebra,Morera’s theorem,Maximum Modulus theorem. 
L34L36 
Sequences and series, Taylor series, Laurent series, zeros and poles 
L37L38 
Residues and Residue theorem 
L39 
Evaluation of Real Trigonometric integrals 
L40 
Evaluation of Real improper integrals 
L41 
Integration along a Branch cut 
L42L43 
Miscellaneous integrals 
L44L45 
The Argument Principle and Roche’s theorem 
L46L48 
Summing infinite series 
Course Code: MATH471
Course Title: Mathematical Modeling
Credit Hours: (3 0 3)
Prerequisite(s): MATH271/MATH212
Course Objectives:
The objectives of this course are:
 To learn the basics of deterministic modeling.
 To learn how to apply balance laws, conservative laws and constitutive laws to construct a mathematical model.
 To analyze the derived model with dynamical system point of view.
 To interpret the qualitative behavior of the model.
Reading list:
 R. Adler, “Modeling The Dynamics of Life: Calculus and Probability for Life Scientists”, 3^{rd} Brooks/Cole, 2013.
 H. Strogatz, “Nonlinear dynamics and chaos: With applications in Physics, Biology,
Chemistry and Engineering”, 2^{nd}, This email address is being protected from spambots. You need JavaScript enabled to view it. Press, 2014.
 EdelsteinKeshet, “Mathematical Models in Biology”, Leah SIAM, 2005.
 H. Taubes, “Modeling Differential Equations in Biology”, 2^{nd} edition, Cambridge University Press, 2008.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Introduction to the course 
L2L3 
Introduction: Deterministic Vs Stochastic modeling, Modeling Components, Modeling laws. 
L4 
Review of some physical and biochemical laws. 
L5L6 
Units and Dimensions, Dimensional analysis and Scaling. 
L7L8 
Buckingham Pi theorem and its Importance, examples. 
L9L11 
Onedimensional flows: geometric approach, fixed points and stability, Potentials. 
L12L13 
Linear stability analysis for 1D system. 
L14L16 
1D Bifurcations: Saddle Node Bifurcation, Transcritical Bifurcation, Pitchfork Bifurcation, examples. 
L17L19 
Linear systems: Definitions and examples, Classifications. 
L20L22 
Phase Plane: Phase portrait, Existence, Uniqueness and Topological consequences. 
L23 
MID EXAM 
L24L25 
Nullclines, Fixed points and linearization. 
L26L27 
Conservative systems, Reversible systems, Nonlinear Pendulum. 
L28L29 
Biological Models Using Difference equations: Cell division, An insect population. 
L30L32 
Propagation of Annual plants: problem statement, assumptions, equations, condensing the equations, validation. 
L33L34 
System of linear difference equations. 
L35L37 
Nonlinear difference equations: steady states, stability and critical parameters, system of nonlinear difference equations. 
L38L39 
Applications of nonlinear difference equations to population Biology. 
L40L41 
Continuous time models: Formulating a model, dimensional analysis, steady states, stability and linearization, examples. 
L42L44 
Applications of continuous Models to Population dynamics: Malthus model, logistic model, Allee effect, Gomoertz growth in tumors,predatorprey systems and LotkaVolterra equations, populations in competition. 
L45L48 
Models for molecular events: Chemical reactions and law of mass action, MichaelisMenten kinetics, The QuasiSteady state assumptions, Sigmoidal kinetics. 
Course Code: MATH424
Course Title: Functional Analysis
Credit Hours: (3 0 3)
Prerequisite(s): MATH212
Course Objectives:
 To understand the notion of norm and inner product in an arbitrary linear space
 To learn general theory of operators
 To study Hilbert spaces
 To learn the basic techniques and methods of functional analysis.
Reading List:
 J.B. Conway, “A Course in Functional Analysis”, 2^{nd} ed., SpringerVerlag, 1997.
 E. Kreyszig, “Introductory Functional Analysis with Applications”, John Wiley & Sons, 2004.
 P.D. Lax, “Functional Analysis”, John Wiley & Sons, Inc., 2002.
 A. Majeed, “Elements of Topology and Functional Analysis”, Ilmi Kitab Khana, Lahore, 1997.
 W. Rudin, “Functional Analysis”, 2^{nd} Edition, McGraw Hill, Inc., 1991.
 K. Saxe, “Beginning Functional Analysis”, SpringerVerlag, 2001.
 A.E. Taylor and David C. Lay, “Introduction to Functional Analysis”, John Wiley & Sons, 1980.
 K. Yosida, “Introduction to Functional Analysis with applications”, 5^{th} ed., SpringerVerlag, 1995.
Lecturewise distribution of the course contents
Lecture # 
Topics 
L1 
Introduction and overview to the course 
L2L3 
Review of metric spaces (e.g., convergence, completeness etc. ) and linear spaces, 
L4L6 
Young’s inequality, Hilbert’s inequality, CauchySchwarz inequality, Minkowski’s inequality 
L7L9 
Normed spaces, complete normed spaces (Banach), examples of incomplete normed space, Completeness and finite dimension 
L10L12 
Equivalent norms, finite dimension and equivalent norm 
L13L15 
Linear operator, bounded and continuous linear operator 
L16L18 
Bounded linear extension, normed space of bounded linear operators and its completeness 
L19L21 
Linear functional (LF), bounded and continuous LF 
L22L23 
Dual spaces, reflexivity 
L24 
MID EXAM 
L25L27 
HahnBanach Theorem for real (without proof) complex and normed spaces with some important consequences 
L28L30 
Inner product, inner product space, Hilbert Space and properties 
L31L33 
Orthogonality , Orthogonal complements and direct sums, annihilators 
L34L36 
Orthonormal sets and sequences with properties, Bessel’s inequality 
L37L39 
GramSchmidt process of orthonormalization 
L40L42 
Total Orthonormal sets and sequences, Parseval relation 
L43L45 
Representation of functionals on Hilbert spaces, Riesz’s Theorem, Sesquilinear forms and their representation 
L46L48 
Hilbertadjoint operator, selfadjoint, unitary and normal operators 
Course Code: MATH431
Course Title: Numerical AnalysisII
Credit Hours: (3 3 4)
Prerequisite(s): MATH331
Course Objectives:
The objectives of this course are:
 To understand the basic problems of interpolation and approximation both theoretically and computationally.
 To understand quadrature rules and numerical rules for the solution of ODEs both theoretically and computationally.
 To perform an error analysis for the methods discussed in this course.
Reading list:
 L. Burden, J. D. Faires, “Numerical analysis” 8^{th} edition, Brooks Cole, 2004.
 F. Gerald, “Applied Numerical Analysis, 8^{th} edition”, Pearson Education, 2008.
 Atkinson, “Elementary Numerical Analysis”, 3^{rd} edition, John Wiley & Sons Inc., 2003.
 K. Jain, S. R. K. Iyengar, R. K. Jain, “Numerical Methods for Scientific and Engineering Computation”, New Age International Pvt. Ltd., 2007.
 W. Hamming, “Numerical Methods for Scientists and Engineer”, 2^{nd} revised edition”, 1987.
 B. Hildebrand , “Introduction to Numerical Analysis”, 2^{nd} edition, Dover, 1987.
 Bradie, “A Friendly Introduction to Numerical Analysis”, 1^{st} edition”, 2005.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1 
Introduction to the course 
L2L3 
Introduction to Interpolation Problem, Weirstrass approximation theorem, Interpolation with Taylor polynomials and limitations. 
L4L6 
Lagrange basis, Lagrange polynomial Interpolation, Algorithm, Error Analysis with Lagrange Interpolating Polynomial. 
L7L9 
Operators, Divided Differences, Newton’s Interpolating divided difference formula, Newtonforward, backward and centered difference formulas. Divided difference Algorithm. 
L10L12 
Equally spaced interpolation drawbacks, Runge’s phenomenon. 
L13L14 
Interpolation with Chebyshev nodes. 
L15L17 
Splines, Linear, quadratic and Cubic Spline Interpolation with algorithms. Error bounds. 
L18L20 
Approximation Vs Interpolation, The Minimax approximation problem, The least squares approximation problem. 
L21L22 
Orthogonal polynomials and least squares approximation revisited. 
L23 
MID EXAM 
L24L25 
Numerical Differentiation, Error Analysis. 
L26L27 
Newtoncotes quadrature formulas: Trapezoidal rule, derivation, error term and algorithms. 
L28L29 
Simpson’s rule with various forms, derivation, error term, algorithm. 
L30L31 
Midpoint rule, derivation, error term, algorithm. 
L32 
Degree of precision of a Quadrature rule. 
L33L35 
Drawbacks with Newtoncotes quadrature, Gaussian Quadrature with different weights of classes of orthogonal polynomials. 
L36 
Romberg Integration. 
L37L39 
Numerical Solution of IVP and BVP for ordinary differential equations: Existence, Uniqueness and stability theory. 
L40L41 
Euler’s method, derivation, error analysis, algorithm. 
L42 
Modified Euler’s method, error analysis, algorithm. 
L43 
RungeKutta method of general order. 
L44L45 
Multistep method, derivations, convergence and stability. 
L46L47 
Stiff differential equations, Boundary value problems, Shooting method. 
L48 
A tour guide of MATLAB builtin ODE solvers. 
Course Code: MATH472
Course Title: Integral Equations
Credit Hours: (3 0 3)
Prerequisite(s): MATH271
Course Objectives:
The objectives of this course are:
 To learn the theory of linear and nonlinear integral equations.
 To learn the connection of integral equations with ordinary differential equations.
 To learn different classes of integral equations.
Reading list:
 Moiseiwitsch, “Integral Equations”, Longman London and New York, 1977.
 P. Kanwal, “Linear Integral Equations Theory and Technique”, Academic Press, 1971.
 M. Wazwaz, “Linear and Nonlinear Integral Equations: Methods and Applications”,
Springer, 2011.
 Hochstadt, “Integral equations”, John Wiley and Sons Inc., 1973.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1L2 
Classification of integral equations , Historical introduction , Linear integral equations , Special types of kernel , Symmetric kernels 
L3 
Kernels producing convolution integrals, Separable kernels 
L4L5 
Square integrable functions and kernels , Singular integral equations 
L6 
Nonlinear integral equations 
L7L9 
Linear differential equations, Green's function, Influence function 
L10 
Integral transforms 
L11L13 
Fredholm equation of the first kind, Stieltjes integral equation, Volterra equation of the first kind 
L14L16 
Fredholm equation of the second kind, Volterra equation of the second kind 
L17L19 
Method of successive approximations: Neumann series, Iterates and the resolvent kernel 
L20L22 
Generalization to higher dimensions, Green's functions in two and three dimensions 
L23 
MID EXAM 
L24L26 
Dirichlet's problem, Poisson's formula for the unit disc 
L27L29 
Poisson's formula for the half plane 
L30 
Hilbert kernel, Hilbert transforms 
L31L32 
Singular integral equation of Hilbert type 
L33L34 
Resolvent equation, Uniqueness theorem, Characteristic values and functions 
L35L37 
Neumann seres, Volterra integral equation of the second kind, Bacher's example, Fredholm equation in abstract Hilbert space , Degenerate kernels , Approximation by degenerate kernels 
L38L40 
Fredholm';, theorems, Fredholm theorems for completely continuous, Operators, Fredholm formulae for continuous kernels 
L41L43 
Hermitian kernels, Spectrum of a HilbertSchmidt kernel 
L44L46 
Expansion theorems, HilbertSchmidt theorem, Hilbert's formula, Expansion theorem for iterated kernels, Solution of Fredholm equation of second kind. 
L47L48 
Bounds on characteristic values, Positive kernels, Mercer's theorem, Variational principles, RayleighRitz variational method. 
Course Code: MATH451
Course Title: Differential Geometry
Credit Hours: (3 0 3)
Prerequisite(s): MATH351
Course Objectives:
The objectives of this course are:
 To learn about space curves their curvature and torsion.
 To understand intrinsic and nonintrinsic properties of surfaces.
 To understand the application of vector calculus to explore geometry of curves and
surfaces.
Reading list:
 B.E. Weather, “Differential Geometry of Three Dimensions”, Cambridge University Press, 1961.
 S. Millman, G. D. Parker “Elements of Differential Geometry”, 1^{st} edition, Prentice Hall, 1977.
 . D.J. Struik, “Lectures on Classical Differential Geometry”, Addison Wesley, 1962.
Lecturewise distribution of the Contents
Lecture # 
Topic 
L1L3 
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 
L4L6 
Differential geometry, Space curves, parametric equations, Tangent at a point, Equation of tangent line 
L7L9 
Arc length as a parameter, conversion of parameters 
L10L12 
Unit tangent, Principal normal, Binormal, Moving tetrahedron, 
L13L15 
Fundamental planes associated with space curve, Equation of osculating plane in different forms, curvature 
L16L18 
Torsion for space curves, SerretFrenet formulae, Different type of relations for Curvature and Torsion calculation 
L19L21 
Plane and Skewed curves, Different criterion for a curve to be planar, Osculating Circle at a point of a curve 
L22L23 
Involutes and Evolutes, Derivation of equation for the curves 
L24 
MID EXAM 
L25L27 
properties of evolutes, Order of contact between curves and surfaces, Osculating sphere of a curve at a point 
L29L30 
equation for the locus of center of osculating sphere and its properties, Spherical and Cylindrical helices 
L31L33 
Spherical indicatrices and their properties, Theorems on different properties of the curves, Surfaces 
L34L36 
tangent plane, Normal vector, family of surfaces, Envelope and characteristics of a family of surfaces, Edge of regression 
L37L39 
Developable surfaces, Developable surfaces associated with a space curve, 
L40L42 
First fundamental form of a surface, Geometrical meaning, First fundamental magnitudes Properties, Applications, Second fundamental form of a surface, 
L43L45 
Geometrical meaning of Second fundamental form of a surface, Second fundamental magnitudes, Properties, Applications, Normal section, Normal curvature 
L46L48 
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 1520 CrHr from the following list of courses as per the guidance of the Institute.
Code 
Title 
Cr Hrs 
PreRequisite 
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 TheoryI 
4(3+1) 
MATH131 
MATH433 
Optimization TheoryII 
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 realworld problem under the supervision of a senior faculty member. Students can complete this segment in group form as well.
Code 
Title 
CrHr 
PreRequisite 
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 marketoriented courses for their electives. 
1216 
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 

7478 