bs scheme of studies

BS Scheme of Studies, Course Codes and Course Outlines (2013 onward)

MATHEMATICS DEPARTMENT ISLAMIA COLLEGE PESHAWAR

[PUBLIC SECTOR UNIVERSITY]

KHYBER PAKHTUNKHWA (PAKISTAN) Exchange: +92-091-9216514-15-16-17, Ext. 3043.

Subject: Maths Year Course Code Sub Code

Course Sub

Code

Subject Code

Foundations

Courses

1. Computing Tools for Mathematics 201

2. Discrete Structure 102

3. General Maths 103

4. Mathematical Statistics-I

(Probability Theory)

204

5. Mathematical Statistics-II 405

6. Programming Languages for

Mathematicians

206

7. Modeling and Simulation 407

8. Optimization Theory 408

Algebra

1. Linear Algebra 211

2. Algebra-I (Group Theory) 312

3. Algebra-II 313

4. (Algebra-III) Rings and Fields 414

5. Group Action 415

Analysis

1. Complex Analysis-I 421

2. Complex Analysis-II 422

3. Functional Analysis-I 423

4. Functional Analysis- II 424

5. Measure theory & Integration 325

6. Numerical Analysis- I 426

7. Numerical Analysis- II 427

8. Real Analysis- I 328

9. Real Analysis- II 329

S.No Course Code

1 FOUNDASATION COURSE 0

2 ALGEBRA 1

3 ANALYSIS 2

4 TECHNIQUES AND

CALCULUS

3

5 GEOMETRY 4

6 MECHANICS 5

7 NUMBER THEORY 6

8 TOPOLOGY 7


Techniques

Calculus

1. Calculus- I 131

2. Calculus- II 132

3. Calculus- III 233

4. Mathematical Space 434

5. Mathematical Methods 435

6. O. D. Equations 336

7. P. D. Equations 337

Mechanics

1. Classical Mechanics 451

2. Dynamics 452

3. Plasma Physics 453

4. Analytical Mechanics 454

5. Fluid Mechanics 455

Number Theory 1. Number Theory 161

Topology 1. General Topology 371

2. Advance Topology 372

Prof. Dr. Syed Inayat Ali Shah

Chairman,

Mathematics Department

Islamia College Peshawar

Geometry

1. Differential Geometry- I 341

2. Differential Geometry -II 342

3. Vector and Tensor Analysis 344


1st

Year

BS 1st

Semester BS 2nd

Semester Codes Course C H Code Course C H

102 Discrete Structure 3 132 Calculus-II 3

131 Calculus-I 3 161 Number Theory 3

Islamiyat 2 Pakistan Study 2

Physics-I 3 Introduction to Computer 3

Statistics-I 3 Statistics-II 3

Functional English-I 3 English-II 3

2nd

Year

BS 3rd

Semester BS 4th

Semester

Codes Course C H Code Course C H

233 Calculus-III 3 211 Linear Algebra 3

Electronics-I 3 Electronics-II 3

Mechanics- II 3 Introduction to Sociology 3

201 Computing Tools for

mathematicians

3 206 Programming Languages for

Mathematicians

3

Introduction to Psychology 3 204 Probability theory (Math. Statistics I) 3

English-III) Communication skills

for Mathematicians

3 (English- IV) Technical Writing 3

3rd

Year BS 5

th /M.Sc 1

st Semester BS 6

th / M.Sc 2

nd Semester

Code Course CH Code Course CH

312 Algebra I (Group Theory) 3 313 Algebra II 3

344 Vector & Tensor Analysis 3 325 Measure theory & Integration 3

371 General Topology. 3 372 Advance Topology 3

336 ODE’s 3 337 PDE’s 3

328 Real Analysis I 3 329 Real Analysis II 3

341 Differential Geometry I 3 342 Differential Geometry II 3

4th

Year

BS 7th

/ M.Sc 3rd

Semester BS 8th

/ M.Sc 4th

Semester

Codes Course C H Code Course C H

421 Complex Analysis I 3 422 Complex Analysis II 3

423 Functional Analysis I 3 424 Functional Analysis II 3

455 Fluid Mechanics 3 415 Group Action 3

426 Numerical Analysis I 3 427 Numerical Analysis II 3

435 Mathematical Methods 3 453 Plasma physics 3

414 Algebra III (Rings and Fields) 3 405 Mathematical Statistics II 3

452 Dynamics 3 434 Mathematical Space 3

408 Optimization theory 3 451 Classical Mechanics 3

454 Analytical Mechanics 3 407 Modeling and Simulation 3


BS 1st

Semester MATH-103: General Mathematics Course Contents

1. NUMBERS SYSTEMS:

Arithmetic of Signed Numbers, Concept of Real Numbers (Rational and Irrational Numbers)

Properties of Real Numbers (Equality and Inequality)

2. RATIO AND PROPORTION: Ratio, Proportion, Solving proportions, Applications of Proportions

3. PERCENT, Meaning of Percent, To convert a Percent to a Decimal, To convert a Decimal to a Percent,

To convert Fraction to a Decimal, Equations involving Percents, Applications of Percents (To calculate

Taxes, Zakat, commission etc), Percent Increase and Decrease (mark up and mark down)

4. AVERAGE:

5. LAWS OF EXPONENTS/INDICES: Algebraic Expressions, Polynomials, Apply the Laws of Exponents to

simplify Expressions with Real Exponents

6. LOGARITHMS: Scientific Notation (Express a number in Standard form of Scientific Notation and vice

versa), The Conversion of Exponential Numbers into Logarithms, Differentiate between Common and

Natural Logarithm, Laws of Logarithms, Application of Logarithms

7. INTRODUCTION TO ALGEBRA: Recall Base, Exponent Value, Linear Equations, Solution of linear

equations in one variable, Quadratic equations, To solve Quadratic Equations by Factorization and

Quadratic Formula, Simultaneous Equations, Solution of Simultaneous equations when, Both are linear

One is linear and the other is quadratic, both are quadratic

8. GRAPHING: The Cartesian Coordinate system, Graphing linear Equations in two variables, Line,

graphs

9. THE CONCEPT OF LIMIT: Define and Represent, Open Interval, Closed Interval, Half open and half

closed intervals on the number line, Explain the meaning of Phrase:

x tends to zero( 0→x ) , x tends to a( ax → ), x tends to infinity ( ∞→x )

State the Theorems on Limits of Sum, Difference, Product and Quotient of functions and demonstrate

through examples, Evaluate the Limits of functions of the type

( ) ( )1

2 1 01

1 0

...lim , lim , lim , lim

...

n nn nn n

n nx a x a x a xn n

a x a x ax abx c ax bx c

x a b x b x b

−−

−→ → → →∞−

+ + +−+ + +− + + +

10. DERAVITIVE OF A FUNCTION: Definition, Notations for Derivative, Differentiation of Algebraic

Expressions by using Power Rule, Sum Rule, Product, Quotient and Chain Rule, Differentiation of

Trigonometric functions, Differentiation of Exponential functions, Differentiation of logarithmic

functions

11. INTEGRATION: Introduction to Integration, Some Standard Integrals, Techniques of Integration

(Rules), To Evaluate Integrals of Simple Algebraic Functions, Trigonometric functions, Exponential

functions and Logarithmic functions

12. INTRODUCTION TO COORDINATE GEOMETRY: To derive Distance formula between two points given

in Cartesian plane, Application Distance Formula (Right, Rectangle, Collinear Points), To derive mid point

formula and to find the mid point if two points are given.

References:

1. Calculus and Analytical Geometry George B. Thomas, JR

2. Basic Mathematics, (Rosanne Proga)

MATH- 131: CALCULUS I Prerequisite(s): Mathematics at intermediate level

Credit Hours: 3 + 0

Specific Objectives of the Course:

This is the first course of the basic sequence, Calculus I-III, serving as the foundation of advanced

subjects in all areas of mathematics. The sequence, equally, emphasizes basic concepts and skills needed

for mathematical manipulation. Calculus I & II focus on the study of functions of a single variable.


Course Outline:

Limits and continuity; derivative of a function and its applications; optimization problems; mean value

theorem (Taylor’s theorem and the infinite Taylor series with applications) and curve sketching; anti-

derivative and integral; definite integral and applications; the fundamental theorem of calculus; inverse

functions (Chapters 1-6 of the text)

Recommended Books:

1. Anton H, Bevens I, Davis S, Calculus: A New Horizon (8th

edition), 2005, John Wiley, New York

2. Stewart J, Calculus (3rd

edition), 1995, Brooks/Cole (suggested text)

3. Thomas GB, Finney AR, Calculus (11th

edition), 2005, Addison-Wesley, Reading, Ma, USA

MATH-102: DISCRETE STRUCTURES Prerequisite(s): Mathematics at intermediate level

Credit Hours: 3 + 0


This course shall assume background in number theory. It lays a strong emphasis on understanding and

utilizing various strategies for composing mathematical proofs.

Course Outline:

Set and Relations: Basic notions, set operations, Venn diagrams, extended-set operations, indexed

family of sets, countable and uncountable sets, relations, cardinality, equivalence relations, congruence,

partitions, partial order, representation of relations, mathematical induction.

Elementary Logic: Logics of order zero and one, Propositions and connectives, truth tables, conditionals

and biconditionals, quantifiers, methods of proof, proofs involving quantifiers.

Recommended Books:

1. Rosen KH, Discrete Mathematics and its Applications (12th

edition), 1999, McGraw Hill, New

York

2. Ross KA, Wright CRB, Discrete Mathematics, 2003, Prentice Hall, Englewood Cliffs, NJ, USA

BS 2nd

Semester

MATH-132: CALCULUS II Prerequisite(s): Calculus I

Credit Hours: 3 + 0


This is the second course of the basic sequence Calculus I-III serving as the foundation of advanced

subjects in all areas of mathematics. The sequence, equally, emphasizes basic concepts and skills needed

for mathematical manipulation. As continuation of Calculus I, it focuses on the study of functions of a

single variable.

Course Outline:

Continuation of Calculus I: Techniques of integration; further applications of integration; parametric

equations and polar coordinates; sequences and series; power series representation of functions

(Chapters 7-10 of the text)

Recommended Books:








MATH-161: NUMBER THEORY Prerequisite(s): Calculus I, Discrete Structures

Credit Hours: 3 + 0


This course shall assume no experience or background in number theory or theoretical mathematics.

The course introduces various strategies for composing mathematical proofs.

Course Outline:

Divisibility, Euclidean algorithm, GCD and LCM of 2 integers, properties of prime numbers, fundamental

theorem of arithmetic (UFT), congruence relation, residue system, Euler’s phi-function, solution of

system of linear congruences, congruences of higher degree, Chinese remainder theorem, Fermat’s little

theorem, Wilson’s theorem and applications, primitive roots and indices; integers belonging to a given

exponent (mod p), primitive roots of prime and composite moduli, indices, solutions of congruences

using indices., quadratic residues, composite moduli, quadratic residues of primes, the Legendre symbol, the Quadratic reciprocity law, the Jacobi symbol, Diophantine equations

Recommended Books:

1. Burton DM, Elementary Number Theory, Allyn and Bacon

2. Grosswald E, Topics from the Theory of Numbers, The Macmillan Company

3. LeVeque WJ, Topics in Number Theory, Vol.1, Addison-Wesley, Reading, Ma, USA

4. Niven I, Zuckerman HS, An Introduction to The Theory of Numbers, Wiley

5. Eastern Rosen KH, Elementary Number theory and its Applications (4th

edition), 2000, Addison-

Wesley, Reading, Ma, USA (suggested text)

BS 3rd

Semester

MATH-233: CALCULUS III Prerequisite(s): Calculus II Credit Hours: 3 + 0 Specific Objectives of the Course:

This is the third course of the basic sequence Calculus I-III serving as the foundation of advanced

subjects in all areas of mathematics.

Course Outline:

This course covers vectors and analytic geometry of 2 and 3 dimensional spaces; vector-valued functions

and space curves; functions of several variables; limits and continuity; partial derivatives; the chain rule;

double and triple integrals with applications; line integrals; the Green theorem; surface area and surface

integrals; the Green, the divergence and the Stokes theorems with applications (Chapters 11-14 of the

text)

Recommended Books:







MATH- 201: COMPUTING TOOLS FOR MATHEMATICIANS Prerequisite(s): Programming Languages for Mathematicians

Credit Hours: 1 + 1



The purpose of this course is to teach students the use of mathematical software like MATLAB, MAPLE,

MATHEMATICA for solving computationally-difficult problems in mathematics. The student shall become

well versed in using at least one mathematical software and shall learn a number of techniques that are

useful in calculus as well as in other areas of mathematics.

Course Outline:

The contents of the course are not fixed, however the following points should be kept in mind while

teaching the course. The course should be taught in a computer lab setting. Besides learning to use the

software, the students must be able to utilize the software to solve computationally difficult problems in

calculus and other areas of mathematics. At the end of the course, the students should have a good

command on at least two of the three programs mentioned above.

Recommended Books:

1. Etter DM, Kuncicky D, Hull D, Introduction to MATLAB 6, 2001, Prentice Hall, Englewood Cliffs,

NJ, USA

2. Garvan F, The Maple Book, 2002, Chapman & Hall/CRC

3. Kaufmann S, Mathematica as a Tool: An Introduction with Practical Examples,

4. 1994, Springer, New York

BS 4th

Semester MATH-211: LINEAR ALGEBRA Prerequisite(s): Mathematics at intermediate level

Credit Hours: 3 + 0


This is the first course in groups, matrices and linear algebra, which provides basic background needed

for all mathematics majors, a prerequisite for many courses. Many concepts presented in the course are

based on the familiar setting of plane and real three-space, and are developed with an awareness of

how linear algebra is applied.

Course Outline: Review of matrics and determinants , Linear Spaces, Bases and dimensions, subspaces, direct sums of

subspaces, factor spaces, linear forms, linear operators, matrix representation and sums and products of

linear operators, the range and null space of linear operators, invariant subspaces, eigen value and eigen

vector and linear transformation and matrices, transformation of matrix linear operator, diagonalization,

inner direct product spaces and linear functional, minimal polynomial of linear transformation.

Recommended Books

1. G.E, Shilov, Linear Algebra Dover Publications, Inc.,New York 1997.

2. D.G., Zill and M.R, Culle, Advanced Engineering Mathematics, PWS, Publishing company, Boston

1996.

3. Herstein, Topics in Algebra, John, Wiley, 1975.

4. A.M Trooper, Linear Algebra, Thomos Nelson and Sons, 1969.

Recommended Books:

1. Anton H, Linear Algebra with Applications (8th edition), John Wiley, New York

2. Herstein IN, Topics in Algebra (2nd edition), John Wiley, New York

3. Hill RO, Elementary Linear Algebra with Application (3 edition), 1995, rd Brooks/Cole

4. Leon SJ, Linear Algebra with Applications (6th edition), 2002, Prentice Hall, Englewood Cliffs, NJ,

USA

5. Nicholson WK, Elementary Linear Algebra with Applications (2nd edition),


MATH-204: MATHEMATICAL STATISTICS-I (PROBABILITY THEORY) Prerequisite(s): Calculus III

Credit Hours: 3 + 0


This course is designed to teach the students how to handle data numerically and graphically. If data are

influenced by chance effect, the concepts and rules of probability theory may be employed, being the

theoretical counterpart of the observable reality, whenever chance is at work.

Course Outline:

Introduction to probability theory; random variables; probability distributions; mean, standard

deviation, variance and expectation. Binomial, negative binomial, Poisson,, geometric, hypergeometric

and normal distributions; normal approximation to binomial distribution; distributions of 2 random

variables.

Recommended Books:

1. DeGroot MH, Schervish MJ, Probability and Statistics (3rd

edition), 2002,

2. Addison-Wesley, Reading, Ma, USA (suggested text)

3. Papoulis A, Probability, Random Variables, and Stochastic Processes, (3rd

edition), 1991, McGraw

Hill, New York

4. Sincich T, Statistics by Examples, 1990, Dellen Publishing Company

MATH-206: PROGRAMMING LANGUAGES FOR MATHEMATICIANS Prerequisite(s): Calculus II

Credit Hours: 3 + 1


The purpose of this course is to introduce students to operating systems and environments

Course Outline:

Introduction to operating systems, one Language (FORTRAN or C/C++), building blocks, variables,

input/output, loops (FOR, WHILE, DO), decisions (IF, IF ELSE, ELSE IF) construct switch statement,

conditional statement, function hat returns a value using argument to pass data to another function,

external variable, arrays and strings, pointers, structure, files and introduction to object-oriented

programming

Recommended Books:

1. Aho, AV, Ulman JD, Foundation of Computer Science, 1995, Computer Science Press, WH

Freeman, New York

2. Hein JL, Theory of Computation: An Introduction (1st

edition), Jones & Bartlett, Boston

3. Laffo R, Introduction to Object-Oriented Programming, McGraw Hill, New York

BS 5th

Semester

MATH-312: ALGEBRA I (Group Theory) Prerequisite(s): Mathematics at intermediate level Credit Hours: 3 + 0


This is the first course in groups, matrices and linear algebra, which provides basic background needed

for all mathematics majors, a prerequisite for many courses. Many concepts presented in the course are

based on the familiar setting of plane and real three-space, and are developed with an awareness of

how linear algebra is applied.


Course Outline:

Group Theory: Basic axioms of a group with examples, abelian groups, center of a group, derived

subgroup of a group, subgroups generated by subset of a group, system of generators, cyclic groups,

cosets and quotient sets, Lagrange’s theorem, introduction to permutations, even and odd

permutations, cycles, lengths of cycles, transpositions, symmetric group, alternating groups, rings, finite

and infinite fields (definition and examples), vector spaces, subspaces, linear span of a subset of a vector

space, bases and dimensions of a vector space

Algebra of Matrices: Determinants, matrix of a linear transformation. row and column operations, rank,

inverse of matrices, group of matrices and subgroups, orthogonal transformation, eigenvalue problem

with physical significance

Recommended Books:

1. Anton H, Linear Algebra with Applications (8th edition), John Wiley, New York

2. Herstein IN, Topics in Algebra (2nd edition), John Wiley, New York

3. Hill RO, Elementary Linear Algebra with Application (3 edition), 1995, rd Brooks/Cole

4. Leon SJ, Linear Algebra with Applications (6th edition), 2002, Prentice Hall, Englewood Cliffs, NJ,

USA

5. Nicholson WK, Elementary Linear Algebra with Applications (2nd edition),

MATH- 344: VECTOR AND TENSOR ANALYSIS Prerequisite(s): Calculus II

Credit Hours: 3 + 0


This course shall assume background in calculus. It covers basic principles of vector analysis, which are

used in mechanics

Course Outline:

3-D vectors, summation convention, kronecker delta, Levi-Civita symbol, vectors as quantities

transforming under rotations with ijk∈ notation, scalar- and vector-triple products, scalar- and vector-

point functions, differentiation and integration of vectors, line integrals, path independence, surface

integrals, volume integrals, gradient, divergence and curl with physical significance and applications,

vector identities, Green’s theorem in a plane, divergence theorem, Stokes’ theorem, coördinate systems

and their bases, the spherical-polar- and the cylindrical-coördinate meshes, tensors of first, second and

higher orders, algebra of tensors, contraction of tensor, quotient theorem, symmetric and skew-

symmetric tensors, invariance property, application of tensors in modeling anisotropic systems, study of

physical tensors (moment of inertia, index of refraction, etc.), diagonalization of inertia tensor as

aligning coördinate frame with natural symmetries of the system .

Recommended Books:

1. Bourne DE, Kendall PC, Vector Analysis and Cartesian Tensors (2nd

edition), Thomas Nelson

2. Shah NA, Vector and Tensor Analysis, 2005, A-One Publishers, Lahore

3. Smith GD, Vector Analysis, Oxford University Press, Oxford

4. Spiegel MR, Vector Analysis, 1974, McGraw Hill, New York

MATH-371: GENERAL TOPOLOGY Motivation and introduction, sets and their operations, countable and uncountable sets, cardinakl and

transfinite numbers, Topological spaces, open and closed sets, interior, closure and boundary of a set

,neighborhoods and neighborhood systems, isolated points ,some topological theorems, topology in

terms of closed sets, limit points, the derived and perfect sets, dense and separable spaces. Topology

bases, criteria for topological bases, local bases, first and second countable spaces, relationship between

sparability and second countability, relative or induced topologies, necessary and sufficient condition

for a subset of a subspace to be open in the original space ,induced bases. Metric spaces , topology

induced by a metric, equivalent topology, formulation with closed sets, Cauchy sequence , complete

metric spaces, characterization of completeness, Cantor’s intersection theorem, the completion of

metric space, metrizable spaces, Continuous functions, various characterizations of continuous

functions, geometric meaning, homeomorphisms, open and closed continuous functions, topological

properties and homeomorphisms. Separation axioms, T1 and T2 spaces and their characterization,


regular and normal spaces and their characterizations, Urysohn’s lemma, Urysohn’s metrizablity

theorem (without proof).Compact spaces their characterization and some theorems, construction

of compact spaces, characterization and some properties of connected spaces.

RECOMMENDED BOOKS:

1. Munkres, J.R.,Topology A First Course, Prentice-Hall, Inc. London, 1975.

2. Simon, G. F., Introduction to Topology and Modern Analysis McGraw –Hill, NewYork, 1963.

3. Pervin, W.J., Foundation of General Topology, Academic Press, London, 2nd

, ed., 1965.

MATH-328: REAL ANALYSIS-I Prerequisite(s): Calculus III

Credit Hours: 2 + 0


This is the first rigorous course in analysis and has a theoretical emphasis. It rigorously develops the

fundamental ideas of calculus and is aimed to develop the students’ ability to deal with abstract

mathematics and mathematical proofs.

Course Outline:

Ordered sets, supremum and infimum, completeness properties of the real numbers, limits of numerical

sequences; limits and continuity, properties of continuous functions on closed bounded intervals;

derivatives in one variable; the mean value theorem; Sequences of functions, power series, point-wise

and uniform convergence. Functions of several variables: open and closed sets and convergence of

sequences in Rn

; limits and continuity in several variables, properties of continuous functions on

compact sets; differentiation in n-space; the Taylor series in Rn

with applications; the inverse and implicit

function theorems.

Recommended Books:

1. Bartle RG, Sherbert DR, Introduction to Real Analysis (3rd

edition), 1999, JohnWiley, New York

2. Brabenec RL, Introduction to Real Analysis, 1997, PWS Publishing Company

3. Gaughan ED, Introduction to Analysis (5th

edition), 1997, Brooks/Cole

4. Rudin W, Principles of Mathematical Analysis (3rd

edition), 1976, McGraw Hill, New York

MATH-341: DIFFERENTIAL GEOMETRY-I Historical background; Motivation and applications. Index notion and summation convention; Space

curves; The tangent vector field; Reparametrization; Arc length; Curvature Principal normal; Binormal;

Torsion; The osculating, the normal and the rectifying planes; The frenet-serral Theorem; Spherical

images; Sphere curves; Spherical contects; Fundamental theorem of space curves ; Line integrals and

Green s theorem; Local surface theory; Coordinate transformations; The tangent and the normal planes;

Parametric curves ; The first fundamental form and the metric tensor; Normal and geodesic

curvatures; Gauss s formulae; Christoffel symbols of first and second kinds; Parallel vector fields along

a curve and parallelism; The second fundamental form and the Weingarten map; Principal, Gaussian,

Mean and normal curvatures; Dupin indicatrices; Conjugate and asymptotic directions; Isometries

and the fundamental theorem of surfaces.

RECOMMENDED BOOKS:

1. Millman, R.S and Parker., G.D. Elements of Differential Geometry, Prentice-Hall Inc., New Jersey,

1977.

2. Struik, D.J., Lectures on classical differential Geometry, Addison–Wesley, publishing Company,

Inc, Massachusetts, 1977.

3. Do Carmo, M.P, Differential Geometry of Curves and surfaces, Prentice-Hall, Inc., Englewood,

New Jersey, 1985.

4. Neil, B, O., Elementary Differential Geometry, Academic Press.1966.

5. Goetz, A., Introduction to Differential Geometry, Addison-Wesley, 1970.

6. Charlton, F., Vector and Tensor Methods, Ellis Horwood. 1976


MATH-336: ORDINARY-DIFFERENTIAL EQUATIONS Prerequisite(s): Calculus III, Computing Tools for Mathematicians Credit Hours: 3 + 0


This course provides the foundation of all advanced subjects in Mathematics. Strong foundation and

applications of Ordinary Differential Equations is the goal of the course.

Course Outline:

Introduction; formation, solution and applications of first-order-differential equations; formation and

solution of higher-order-linear-differential equations; differential equations with variable coefficients;

Sturm-Liouville (S-L) system and boundary-value problems; series solution and its limitations; the

Frobenius method, solution of the Bessel, the hypergeometric, the Legendre and the Hermite equations,

properties of the Bessel function.

Recommended Text:

1. Zill DG, Cullen MR, Differential Equations with Boundary-Value Problems, (3rd

Edition), 1997,

PWS Publishing Co.

BS 6th

Semester

MATH- 313: ALGEBRA II Prerequisite(s): Algebra I

Credit Hours: 3 + 0


This is a course in advanced abstract algebra, which builds on the concepts learnt in Algebra I.

Course Outline:

Group Theory: Normalizers and centralizers of a subset of a group, congruency classes of a group,

normal subgroup, quotient groups, conjugacy relation between elements and subgroups,

homomorphism and isomorphism between groups, Homomorphism and isomorphism theorems, group

of automorphisms, finite p-groups, internal and external direct products, group action on sets, isotropy

subgroups, orbits, 1st

, 2nd

and 3rd

Sylow theorems.

Ring Theory: Types of rings, matrix rings, rings of endomorphisms, polynomial rings, integral domain,

characteristic of a ring, ideal, types of ideals, quotient rings, homomorphism of rings, fundamental

theorem of homomorphism of rings.

Recommended Books:

1. Allenby RBJT, Rings, Fields and Groups: An Introduction to Abstract Algebra, 1983,Edward Arnold

2. Farleigh J.B, A First Course in Abstract Algebra (7th

edition), Addison-Wesley, Reading, Ma., USA

3. Macdonald I.D., The Theory of Groups, 1975, Oxford Clarendon Press, Ma., USA

MATH-325: MEASURE THEORY & INTEGRATION

Foundation of Analysis, A development of integral rational, real and complex numbers system from the

Peano axioms, Denumerable and non-denumerable sets, cardinal and ordinal numbers, partially ordered

sets and totally ordered sets, well-ordered sets, transfinite induction, axiom of choice and well-ordering

theorem.

Theory of set of points, covering theorems, theory of measure, Measurable functions, the lebasgue

integral convergence theorem, the fundamental theorem of the integral calculus, derivative, non

differentiable functions, the lebesgue set, the lebesgue classes. Strong convergence, simple treatment

of Riemann-Stieltjes and Lebesgue Stieltjes integral.


RECOMMENDED BOOKS:

1. Natonson, Theory of Functions of Real Variables.

2. Burkill, Lebesgue Integral.

3. Titchmarsh, Theory of Functions.

4. Edmond Landau, Foundation of the Analysis.

5. Seymour Lipschutz, Set Theory and Related Topics.

MATH-372: ADVANCE TOPOLOGY Directed sets and nets, subnets and cluster points, sequences and subsequences, quotient spaces, the

Tychonoff theorem, completely regular space, the Stone-Eeih compactification, meterization theorem

and paracompactness, function spaces.

RECOMMENDED BOOKS:

1. J. L. Kelly, 1975. General Topology, Springer Verlag.

2. S. Willard, 1970. General Topology, Addison Wesley Pub. Co.

3. J. R. Munkers, 1975. Topology (a first course), Prentice Hill Inc.

MATH-337: PARTIAL-DIFFERENTIAL EQUATIONS Prerequisite(s): Real Analysis I, Ordinary-Differential Equations

Credit Hours: 3 + 0


The course provides a foundation to solve Partial Differential Equations with special emphasis on wave,

heat and Laplace equations. Formulation and some theory of these equations are also intended.

Course Outline:

First-order-partial-differential equations; classification of second-order PDE; canonical form for second-

order equations; wave, heat and the Laplace equation in Cartesian, cylindrical and spherical-polar

coördinates; solution of partial differential equation by the methods of: separation of variables; the

Fourier, the Laplace and the Hankel transforms, non-homogeneous-partial-differential equations .

Recommended Books:

1. Myint UT, Partial Differential Equations for Scientists

2. and Engineers (3rd

edition), 1987, North Holland, Amsterdam

MATH-329: REAL ANALYSIS II Prerequisite(s): Real Analysis I

Credit Hours: 2 + 0


A continuation of Real Analysis I, this course rigorously develops integration theory. Like Real Analysis I,

Real Analysis II emphasizes proofs.

Course Outline:

Series of numbers and their convergence, Series of functions and their convergence, Dabroux upper and

lower sums and integrals; Dabroux integrability. Riemann sums and the Riemann integral. Riemann

integration in R2

, change of order of variables of integration, Riemann integration in R3

, and Rn

.

Riemann-Steiltjes integration, Functions of bounded variation, The length of a curve in Rn.

Recommended Books:

1. Bartle RG, Sherbert DR, Introduction to Real Analysis (3rd


2. Brabenec RL, Introduction to Real Analysis, 1997, PWS Publishing Company

3. Fulks W, Advanced Calculus, John Wiley, New York (suggested text)

4. Gaughan ED, Introduction to Analysis (5th

edition), 1997, Brooks/Cole

5. Rudin W, Principles of Mathematical Analysis (3rd



MATH-342: DIFFERENTIAL GEOMETRY-II Definition and examples of manifolds; Differential maps; Submanifolds; Tangents; Coordinate vector

fields; Tangent spaces; Dual spaces; Multilinear functions; Algebra of tensors; Vector fields; Tensor

fields; Integral curves; Flows; Lie derivatives; Brackets; Differential forms; Introduction to integration

theory on manifolds; Riemannian and semi Riemannian metrics; Flat spaces; Affine connextions; Parallel

translations; Covariant differentiation of tensor fields; Curvature and torsion tensors; Connexion of a

semi-Riemannian tensor; Killing equations and killing vector fields; Geodesics; Sectional curvature.

RECOMMENDED BOOKS:

1. Bishop, R. L. and Goldberg, S.I., Tensor Analysis on Manifolds, Dover Publications, Inc. N.Y.,

1980.

2. do carom, M.P., Riemannian Geometry, Birkhauser, Boston, 1992.

3. Lovelock, D, and Rund, H. Tensors., Differential Forms and variational principles, John-Wiley,

1975.

4. Langwitz, D. Differential and Riemannian geometry, Academic Press, 1970.

5. Abraham, R., Marsden, J.E. and Ratiu, T., Manifolds, Tensor Analysis and Applications, Addison-

Wesley, 1983.

BS 7th

Semester

MATH- 421: COMPLEX ANALYSIS I Prerequisite(s): Real Analysis I

Credit Hours: 3 + 0


This is an introductory course in complex analysis, giving the basics of the theory along with applications,

with an emphasis on applications of complex analysis and especially conformal mappings. Students

should have a background in real analysis (as in the course Real Analysis I), including the ability to write

a simple proof in an analysis context.

Course Outline:

Complex no, complex plane, complex function, topological aspects of complex no, limit point, continuity,

types of function, limits involving points at infinity, analytical functions, entire function, Cauchy Riemann

equations, Cauchy Riemann in polar form, Laplace equation, harmonic function, elementary functions

and their inverses, complex line integral, properties of complex line integral, greens theorem, Cauchy

theorem, Cauchy goursat theorem, Cauchy goursat theorem for multiply connected region, Cauchy

integral formula, Generalization of Cauchy integral formula, liouville’s theorem, fundamental theorem of

algebra statement, gauss mean value theorem.

Recommended Text:

1. Churchill RV, Brown JW: Complex Variables and

2. Applications (5th


MATH-414: ALGEBRA III (RING AND FIELDS) Definitions and basic concepts, homomorphism, homomorphism theorems, polynomial rings, unique

factorization domain, factorization theory, Euclidean domains, arithmetic in Euclidean domain,

extension fields, algebraic and transcendental elements, simple extension, introduction to Galois theory.

RECOMMENDED BOOKS:

1. Fraleigh, J.A., A first course in Abstract Algebra, Addision Wesley Publishing Company, 1987.

2. Herstein, I.N., Topies in Algebra, John Wiley and Sons 1975.

3. Lang, S., Algebra, Addison Wesley 1965.

4. Hartley, B., and Hawkes, T.O., Ring, Modules and Linear Algebra, Chapman and Hall, 1980.


MATH-423: FUNCTIONAL ANALYSIS- I Prerequisite(s): Complex Analysis

Credit Hours: 3 + 0


This course extends methods of linear algebra and analysis to spaces of functions, in which the

interaction between algebra and analysis allows powerful methods to be developed. The course will be

mathematically sophisticated and will use ideas both from linear algebra and analysis.

Course Outline:

Metric Spaces: A quick review, completeness and convergence, completion.

Normed Spaces: Linear spaces, Normed spaces, Difference between a metric and a normed space,

Banach spaces, Bounded and continuous linear operators and functionals, Dual spaces, Finite

dimensional spaces, F. Riesz Lemma, The Hahn-Banach Theorem, The HB theorem for complex spaces,

The HB theorem for normed spaces, The open mapping theorem, The closed graph theorem, Uniform

boundedness principle and its applications

Banach-Fixed-Point Theorem: Applications in Differential and Integral equa-tions

Recommended Books:

1. Curtain RF, Pritchard AJ, Functional Analysis in Modern Applied Mathematics, Aademic Press,

New York

2. Friedman A, Foundations of Modern Analysis, 1982, Dover Kreyszig E, Introductory Functional

Analysis with Applications, John Wiley, New York Rudin W, Functional Analysis, 1973, McGraw

Hill, New York

3.

MATH- 426: NUMERICAL ANALYSIS I Prerequisite(s): Computing Tools for Mathematicians

Credit Hours: 3 + 0


This course is designed to teach the students about numerical methods and their theoretical bases. The

students are expected to know computer programming to be able to write program for each numerical

method. Knowledge of calculus and linear algebra would help in learning these methods.

Course Outline:

Computer arithmetic, approximations and errors; methods for the solution of nonlinear equations and

their convergence: bisection method, regula falsi method, fixed point iteration method, Newton-

Raphson method, secant method; error analysis for iterative methods. Interpolation and polynomial

approximation: Lagrange interpolation, Newton’s divided difference, forward-difference and backward-

difference formulae, Hermite interpolation. Numerical integration and error estimates: rectangular rule,

trapezoidal rule, Simpson’s one-three and three-eight rules. Numerical solution of systems of algebraic

linear equations: Gauss-elimination method, Gauss-Jordan method; matrix inversion; LU-factorization;

Doolittle’s, Crount’s, Cholesky’s methods; Gauss-Seidel and Jacobi methods.

Recommended Books:

1. Atkinson KE, An Introduction to Numerical Analysis (2nd


(suggested text)

2. Burden RL, Faires JD, Numerical Analysis (5th

edition), 1993, PWS Publishing Company Chapra SC,

Canale RP, Numerical Methods for Engineers, 1988, McGraw Hill, New York

MATH-452: DYNAMICS (a)Dynamics of a Rigid Body

Moments and product of ineruia, D’ Alembert’s principles, Motion about a fixed axis, Linear momentum

and Kinetic energy of a rigid body, Compound pendulum, Motion in two dimension, Finite forces,

impulsive forces, Lagrange’s equations in generalized coordinates.

(b) Dynamics of a Particle

uniplanar motion, acceleration parallel to fixed axes, polar coordinates, moving axes, central forces,

stability of orbits, acceleration varying as the inverse square of the distance, Kapler’s laws, Planetary


motion, Tangential and normal accelerations, Motion in a resisting medium, Angular momentum and

rate of change of angular momentum for a system of particles.

RECOMMENDED BOOKS:

1. Dynamics of a particle and Rigid Body by S. L. Loney

2. A Text Book of Dynamics by F. Charlton.

MATH-408: OPTIMIZATION THEORY Prerequisite(s): Algebra I, Real Analysis I

Credit Hours: 3 + 0


The main objective is to teach the basic notions and results of mathematical programming and

optimization. The focus will be to understand the concept of optimality conditions and the construction

of solutions. Students should have a good background in analysis, linear algebra and differential

equations.

Course Outline:

Linear programming: simplex method, duality theory, dual and primal-dual simplex methods.

Unconstrained optimization: optimality con-ditions, one-dimensional problems, multi-dimensional

problems and the method of steepest descent. Constrained optimization with equality cons-traints:

optimality conditions, Lagrange multipliers, Hessians and bordered Hessians. Inequality constraints and

the Kuhn-Tucker Theorem, The calculus of variations, the Euler-Lagrange equations, functionals

depending on several variables, variational problems in parametric form, transportation models and

networks.

Recommended Books:

1. Elsgolts L, Differential Equations and the Calculus of Variations, 1970, Mir Publishers, Moscow

2. Gotfried BS, Weisman J, Introduction to Optimization Theory, 1973, Prentice Hall, Englewood

Cliffs, NJ, USA

3. Luenberger DG, Introduction to Linear and Non-Linear Programming, 1973, Addision-Wesley,

Reading, Ma, USA

BS 8th

Semester MATH- 422: COMPLEX ANALYSIS II Prerequisite(s): Real Analysis II

Credit Hours: 3 + 0


This is an introductory course in complex analysis, giving the basics of the theory along with applications,

with an emphasis on applications of complex analysis and especially conformal mappings. Students

should have a background in real analysis (as in the course Real Analysis II), including the ability to write

a simple proof in an analysis context.

Course Outline:

Sequences, absolutely convergence, conditionally convergence, some tests on convergence series,

power series, divergence tests, radius of convergence, Taylor’s series, Laurent expansion, poles and

residue, roche’s theorem, singular point, Zero’s of a functions, removable singular point, essential

singular point, proper and improper integrals, special functions, gamma functions, Euler’s formula for

gamma functions, beta functions, hyper geometric functions, integral representation of hyper

geometric.

Recommended Text:

1. Churchill RV, Brown JW: Complex Variables and

2. Applications (5th



MATH-424: FUNCTIONAL ANALYSIS- II Prerequisite(s): Complex Analysis

Credit Hours: 3 + 0


This course extends methods of linear algebra and analysis to spaces of functions, in which the

interaction between algebra and analysis allows powerful methods to be developed. The course will be

mathematically sophisticated and will use ideas both from linear algebra and analysis.

Course Outline:

Inner-Product Spaces: Inner-product space, Schwarz’s Inequality, Polarization identity, Parallelogram

Law, Pythagorean Theorem.

Hilbert space: Projection theorem, orthogonal and orthonormal sets, orthogonal comple-ments, Bessel’s

inequality, Gram-Schmidt orthogonalization process, representation of functionals, Reiz-representation

theorem, weak and weak* Convergence.

Operators on Hilbert Spaces: Self Adjoint Operator, Normal Operator, Unitary Operator, Projections.

Finite Dimensional Spectral Theory: Spectral properties of Self Adjoint linear operator, Spectral Radius of

an operator, The Spectral Theorem.

Recommended Books:

4. Curtain RF, Pritchard AJ, Functional Analysis in Modern Applied Mathematics, Aademic Press,

New York

5. Friedman A, Foundations of Modern Analysis, 1982, Dover Kreyszig E, Introductory Functional

Analysis with Applications, John Wiley, New York Rudin W, Functional Analysis, 1973, McGraw

Hill, New York

MATH-405: MATHEMATICAL STATISTICS-II Statistics inference, Maximum likelihood estimators, Properties of maximum likelihood estimators,

Sufficient statistics, Jointly sufficient statistics, Minimal sufficient statistics, The sampling distribution of

a statistics, The Chi square distribution, Joint distribution of the sample mean and sample variance, The t

distribution, Confidence intervals, Unbiased estimators, Fisher information, Testing simple hypotheses,

Uniformly most powerful tests, the t test, The f distribution, Comparing the means of two normal

distributions, Tests of goodness of fit, Contingency tables, Equivalence of confidence sets and tests.

Kolmogorov-smirnov tests, The Wilcoxon signed ranks test, the Wilcoxon mann whitney rank test.

RECOMMENDED BOOKS:

1. Mood, A.M., Graybill, F.A., Boes, D.C., Introduction to the theory of statistics, 2nd

edition,

McGraw-Hill Book Company New York 1986.

2. Degroot, M.H., Probability and statistics, Addison – Wesley Publising Company, USA, 1986.

MATH-415: Group Action Action of a group on a set G-spaces, G-morphisms, The symmetric and alternating groups, orbits,

transitivity, linear groups and their types, Graphical representation of group.

Recommended Books:

1. Rose, J.S., A course on group theory, Cambridge University Press, 1978.

2. Magnus, W. Karras., A. and Solitar, Conbinatorial group theory, Dover Publications 1966.

3. Husain, Taqdir introduction to topological groups. W.B. Saunder’s company Philadelpna and

London 1966.

MATH-407: MODELING AND SIMULATION Prerequisite(s): Partial-Differential Equations

Credit Hours: 2 + 1



Mathematics is used in many areas such as engineering, ecological systems, biological systems, financial

systems, economics, etc. In all such applications one approximates the actual situation by an idealized

model. This is an introductory course of modeling, consisting of three parts: modeling with ordinary

differential equations and their systems; partial differential equations; and integral equations. The

course will not be concerned with the techniques for solving the equations but with setting up the

equations in specific applications. Whereas the first two types of equations have already been dealt

with, the third type has not. Consequently, solutions of the former will be discussed but of the latter will

barely be touched upon.

Course Outline:

Concepts of model, modeling and simulation, functions, linear equations, linear-differential equations,

nonlinear-differential equations and integral equations as models, introduction to simulation techniques

Ordinary-Differential Equations: Modeling with first order differential equations: Newton’s law of

cooling; radioactive decay; motion in a gravitational field; population growth; mixing problem;

Newtonian mechanics. Modeling with second order differential equations: vibrations; application to

biological systems; modeling with periodic or impulse forcing functions, Modeling with systems of first

order differential equations; competitive hunter model; predator-prey model.

Partial-Differential Equations: Methodology of mathematical modeling; objective, background,

approximation and idealization, model validation, compounding. Modeling wave phenomena (wave

equation); shallow water waves, uniform transmission line, traffic flow, RC circuits, Modeling the heat

equation and some application to heat conduction problems in rods, lamina, cylinders etc. Modeling the

potential equation (Laplace equation), applications in fluid mechanics, gravitational problems, Equation

of continuity.

Simulation: Techniques of simulation (students are required to simulate at least one system)

Recommended Books:

1. Giordano FR, Weir MD, Differential Equations: A Modeling Approach, 1994, Addison-Wesley,

Reading, Ma, USA (suggested text)

2. Jerri AJ, Introduction to Integral Equations with Applications, 1985, Marcel Dekker, New York

3. Myint UT, Debnath L, Partial Differential Equations for Scientists and Engineers (3rd

edition),

1987, North Holland, Amsterdam

MATH-451: CLASSICAL MECHANICS Prerequisite(s): Vector and Tensor Analysis

Credit Hours: 3 + 0


This course builds grounding in principles of classical mechanics, which are to be used while studying

quantum mechanics, statistical mechanics, electromagnetism, fluid dynamics, space-flight dynamics,

astrodynamics and continuum mechanics.

Course Outline:

Particle kinematics, radial and transverse components of velocity and acceleration, circular motion,

motion with a uniform acceleration, the Newton laws of motion (the inertial law, the force law and the

reaction law), newtonian mechanics, the newtonian model of gravitation, simple-harmonic motion,

damped oscillations, conservative and dissipative systems, driven oscillations, nonlinear oscillations,

calculus of variations, Hamilton’s principle, lagrangian and hamiltonian dynamics, symmetry and

conservation laws, Noether’s theorem, central-force motion, two-body problem, orbit theory, Kepler’s

laws of motion (the law of ellipses, the law of equal areas, the harmonic law), satellite motion,

geostationary and polar satellites, kinematics of two-particle collisions, motion in non-inertial reference

frame, rigid-body dynamics (3-D-rigid bodies and mechanical equivalence, motion of a rigid body,

inverted pendulum and stability, gyroscope).

Recommended Books:

1. Bedford A, Fowler W, Dynamics: Engineering Mechanics, Addision-Wesley, Reading, Ma, USA

2. Chow TL, Classical Mechanics, 1995, John Wiley, New York

3. Goldstein H, Classical Mechanics (2nd

edition), 1980, Addison-Wesley, Reading,Ma, USA


4. Marion JB, Classical Dynamics of Particles and Fields (2nd

edition), 1970, Academic Press, New

York (suggested text)

MATH- 453: PLASMA PHYSICS Introduction to plasma/ what is plasma state. Occurrence of plasma in nature, definition of plasma,

concept of temperature, debye shielding, the plasma parameter, criteria for plasma, application of

plasma physics, single particle motions, introductions, uniform E and B fields. Non uniform B field, Non

uniform E field, Time varying E field. Time varying B field, Summery of guiding centre drifts, Adiabatic

invariants.

Plasma as fluids, Introductions, Relation of plasma physics to ordinary Electromagnetic, the fluid

equations of motion, Fluid drifts perpendicular, Fluids drifts parallel to B the plasma approximation.

Waves in plasma: representation of waves, Group velocity, Plasma oscillation, Electro plasma waves,

sound waves, Ion waves, Validity of the plasma approximation, Comparison of Ion and Electron waves,

Electro-Static Electron oscillation perpendicular to B Electro static Ion waves perpendicular to B, the

lower hybrid frequency, Electromagnetic waves with 0=oB , Electromagnetic waves perpendicular to

oB , Hydro-magnetic waves, Magneto sonic waves.

Magnetically confined fusion, Progress in Tokmaks, Motivation for fusion research, Basic reactions,

Charged particle energy, Energy Losses, Ideal break even Temperature, Lawson criterion, Supplementary

(or Auxiliary) Heating, progress in Tokamaks, Next generation experiments.

RECOMMENDED BOOKS:

1. F.F. Chen, Introduction to plasma physics and controlled fusion. (Plasma, NY, 1983) 2nd

Ed.

2. Bettencourt (Pergamon, Oxford/ NY, 1986).

MATH-427. NUMERICAL ANALYSIS-II Osculating polynomials, Differentiation and integration in multidimension, Ordinary differential

equations; predictor methods, Modified Eulers method, Truncation error and stability, The Taylor series

method, Runge-Kutta methods, Differential equations of high order, System of differential equations;

Runge-Kutta methods, shooting methods, finite difference methods.

Partial differential equations, Elliptic hyperbolic and parabolic equations; Explicit and implicit finite

difference methods, stability, convergence and consistency analysis, the method of characteristic.

Eigen value problems; Estimation of eigen values and corresponding error bounds, Gerschgorin’s

theorem and its applications Schur’s theorem, Power method, Shift of origin, Deflation method for the

subdominant eigen values.

RECOMMENDED BOOKS:

1. Conte, S.D., and De Boor., Elementary Numerical Analysis, McGraw-Hill 1972.

2. Gerald, C.F., Applied Numerical Analysis, Addison Wesely, 1984.

3. Froberg, C.E., Introduction to Numerical Analysis, Addison Wesely, 1972.

4. Gourlay, A.R., and Watson, G.A., Computational Methods for Matrix Eigene Problems. John

Wiley and Sons 1973.

5. Smith, G.D., Numerical Solution of Partial Differential Equations, Oxford University Press.

6. Mitchel A.R. and Griffiths D.F., The Finite Difference Methods in Partial Differential Equations,

John Wiley and Sons 1973.

MATH- 454: ANALYTICAL MECHANICS Review of basic principles: kinematics of particle and rigid body in three dimensions; Euler’s theorem,

Work, power, Energy, Conservative field of force, Motion in a resisting medium, Variables mass

problem, Moving coordinate system, Rate of change of a vector, Motion relative to the rotating earth,

The motion of a system of marticles, Conservation laws, Generalized coordinates. Lagrange’s equations,

Simple applications, Motion of a rigid body, Moments and products of inertia. Angular momentum.

Kinetic energy about a fixed point, Principles axes, Momental ellipsoid, Equimomental systems,

Gyroscopic motion, Euler’s dynamical equations, Properties of a rigid body motion under forces. Review

of material.

RECOMMENDED BOOKS:

1. Chorlton, F., Principles of mechanics, McGraw-Hill, N.Y. 1983.

2. Symon, K.R., Mechanics Addison Wesley, 1964.


3. Goldstein, H., Classical Mechanics, Addison Wesley, 2nd

Edition, 1980.

4. Synge, J.I. and Griffith, B.A., Principles of Mechanics, McGraw-Hill, N.Y. 1986.

5. Beer, F.P., and Johnson, E.R., Mechanics for Engineers, Vols. I and II, McGraw-Hill.

MATH-455: FLUID MECHANICS Real fluids and ideal fluids, velocity of a fluid at a point, streamlines and path lines, steady and unsteady

flows, velocity potential, vorticity vector, local and particle rates of change, equation of continuity,

Acceleration of a fluid, conditions at a rigid boundary, general analysis of fluid motion. Euler’s equations

of motion, Bernoulli’s equations, steady motion under conservative body forces, some potential

theorems, impulsive motion, Sources, Sinks and doublets, images in rigid infinite plane and solid

spheres, axi-symmetric flows, stokes stream function, Stream functions, Complex potential for two

dimensional, irrotational, incompressible flow, complex velocity potential for uniform stream, line

sources and line sinks, line doublets and line vortices, image systems, Miline-Thomson circle theorem,

Blasius’s theorem, the use of conformal transformation and the Schwarz-Christoffel transformation in

solving problems, vortex rows, Kelvin’s minimum energy theorem, Uniqueness theorem, fluid streaming

past a circular cylinder, irrotational motion produced by a vortex filament, The Helmholtz vorticity

equations, Karman’s vortex sterrt.

RECOMMENDED BOOK:

1. Charlton, F., Textbook of fluid Dynamics, D. Van Nostrand Co. Ltd. 1967.

2. Thomson, M., Theoretical Hydrodynamics, Macmillan Press, 1979.

3. Jaunzemics, W., Continum Mechanic, Machmillan Company 1967.

4. Landau, L.D., and Lifshitz, E.M., Fluid Mechanics, Pergoman Press, 1966.

5. Batchlor, G.K., An Introduction to fluid dynamics, Cambridge University Press, 1969.

MATH-434: MATHEMATICAL SPACES Prerequisite(s): Discrete Structures, Real Analysis I

Credit Hours: 2 + 0


This course is designed primarily to develop pure mathematical skills of students. Students will need

some background in writing proofs. They will lean notions of spaces, metric, measure and topology

Course Outline:

Notion of Spaces: Example of set, group, field, ring, affine space, Banach space, normed space, Hilbert

space (Simmon)

a) Notion of Topology: Calculus on manifolds, continuity of functions on spaces, neighborhoods,

topological spaces, finer and weaker topologies, homomorphism, homomorphic spaces, compactness,

connectedness, normal spaces, Urysohn’s lemma (Munkres)

b) Notion of Metric: Metric space, complete metric space, Baire category theorem, metrization of spaces

(Friedmann)

c) Notion of Measure: Spaces with measure, measurable function, idea of –σ fields (Holmos)

Recommended Books:

1. Friedmann A, Foundations of Modern Analysis, 1982, Dover Holmos PR, Measure Theory, van

Nostrand, New York

2. Munkres JR, Topology: A First Course, Prentice Hall, Englewood Cliffs, NJ, USA

3. Simmon GF, Introduction to Topology and Modern Analysis, 1963, McGraw Hill, New York

MATH-435: MATHEMATICAL METHODS Fourier series, Generalized Fourier series, Fourier Cosine series, Fourier Sine series, Fourier integrals,

Fourier transform, Laplace transform, Z-transform, Hankel transform, Mellin transform. Solution of

differential equation by Laplace and Fourier transform methods. General solution of Bessel equation,

Recurrence relations, Orthogonal sets of Bessel functions, Modified Bessel functions, Applications,

General solution of Legendre equation, Legendre polynomials, Associated Legendre polynomials,

Rodrigues formula, Orthogonality of Legendre polynomials.

Recommended Books :

1. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944.


2. G. F. Roach, Green’s Functions, Cambridge University Press, 1995.

3. D. Poularikas, The Transforms and Applications – Handbook, CRC Press, 1996.

4. J. W. Brown and R. Churchill, Fourier Series and Boundary Value Problems, McGraw Hill, 1993.

5. Allen Jeffery, Advanced Engineering Mathematics, Academic Press, 2002.

MEMBER BOARD OF STUDIES.

Name and Institution Signature

Under Section 6(2)(i) Prof. Dr. Syed Inayat Ali Shah Chairman Mathematics Department

Islamia College Peshawar.

Under Section 6(2)(ii)

i. Dr. Sareer Badshah, Associate Professor, Department of

Statistics Islamia College Peshawar.

ii. Dr. Shazia Naeem, Associate Professor, Department of

Physics, Islamia College Peshawar.

Under Section 6(2)(iii)

i. Mr. Arbab Safeer Assistant Professor, Department Physics

Islamia College Peshawar.

ii. Mr. Atta Ullah Assistant Professor, Department of

Computer Science Islamia College Peshawar.

Under Section 6(2)(iv)

i. Dr. Saeed Islam, Associate Professor, Mathematics

Department, Abdul Wali Khan University Mardan.

ii. Dr. Noor Badshah Assistant Professer, Department of

Basic Sciences and Islamyat University of Engineering and

Technology Peshawar.

iii. Dr. Abdul Samad, Assistant Professor, Mathematics

Department of Peshawar.

Under Section 6(2)(v)

i. Dr. Haider Zaman, Assistant Professor Mathematics

Department Islamia College Peshawar.


ii. Mr. Murad Ullah Assistant Professor Mathematics

Department Islamia College Peshawar.

Convener Board of Studies Chairman

Dean of Physical and Numerical Sciences Prof. Dr. Syed Inayat Ali Shah

Islamia College Peshawar (Public Sector University) Islamia College Peshawar (Public Sector University)

bs scheme of studies

Documents