BS-4 YEAR(INTEGRATED)PROGRAMME WITH SPECIALIZATION IN MATHEMATICS

Scheme of Studies B.S (4 Year) Mathematics

(Semester System Program)

First Year

Semester-I Course Code Course Title Cr. Hours

MATH-301 Calculus-I 3(3 + 0)

MATH-302 Discrete Structures 3(3 + 0)

MATH-303 Introduction to Statistics-I 3(3 + 0)

PHY-301 Mechanics- I 3(3 + 0)

ENG-301 English Structure-I (Functional English) 3(3 + 0)

IS-311 Islamic Studies 2(2 + 0)

Total 17(17+0)

Semester-II Course Code Course Title Cr. Hours

MATH-351 Calculus-II 3(3 + 0)

MATH-352 Introduction to Statistics-II 3(3 + 0)

ENG-351 English-II (Communication Skills) 3(3 + 0)

PHY-351 Mechanics-II 3(3 + 0)

PS-351 Pakistan Studies 2(2 + 0)

CS-301 Introduction to Computers 2(2 +0)

Total 16(16+0)

Second Year Semester-III

Course Code Course Title Cr. Hours

MATH-401 Calculus-III 3(3 + 0)

MATH-402 Mathematical Spaces 3(3 + 0)

MATH-403 Number Theory 3(3 + 0)

PHY-401 Electricity & Magnetism-I 3(3 + 0)

PHY-352 Waves & Oscillation 3(3 + 0)

ENG-401 English-III (Technical Writing & Presentation Skills) 3(3 + 0)

Total 18(18+0)

Semester-IV

Course Code Course Title Cr. Hours

MATH-451 Operations Research-I 3(3 + 0)

MATH-452 Computing Tools for Mathematician 2(1 + 1)

MATH-453 Probability Theory 3(3 + 0)

PHY-451 Electricity & Magnetism-II 3(3 + 0)

PSY-101 Introduction to Psychology 3(2 + 1)

SOC-401 Introduction of Sociology 3(2 + 1)

Total 17(14+3)

Third Year

Semester-V

Course Code Course Title Cr. Hours

MATH-501 Algebra-I 3(3 + 0)

MATH-502 Vector & Tensor Analysis 3(3 + 0)

MATH-503 General Topology 3(3 + 0)

MATH-504 Complex Analysis-I 3(3 + 0)

MATH-505 Ordinary Differential Equations 3(3 + 0)

MATH-506 Real Analysis-I 3(3 + 0)

Total 18(18+0)

Semester-VI Course Code Course Title Cr. Hours

MATH-551 Algebra-II 3(3 + 0)

MATH-552 Classical Mechanics 3(3 + 0)

MATH-553 Complex Analysis-II 3(3 + 0)

MATH-554 Set Theory & Mathematical Logic 3(3 + 0)

MATH-555 Differential Geometry 3(3 + 0)

MATH-556 Real Analysis-II 3(3 + 0)

Total 18(18+0)

Fourth Year

Semester-VII Course Code Course Title Cr. Hours

MATH-xxx Elective-A I 3(3 + 0)

MATH-xxx Elective-A II 3(3 + 0)

MATH-601 Functional Analysis-I 3(3 + 0)

MATH-602 Partial Differential Equations 4(4 + 0)

MATH-603 Numerical Analysis-I 4(4 + 0)

Total 17(17+0)

Semester-VIII Course Code Course Title Cr. Hours

MATH-xxx Elective-A III 3(3 + 0)

MATH-xxx Project OR Elective-B 3(3 + 0)

MATH-651 Functional Analysis-II 3(3 + 0)

MATH-652 Optimization Theory 4(4 + 0)

MATH-653 Numerical Analysis-II 4(4 + 0)

Total 17(17+0)

The BS Course Contents Course contents for Communication Skills, English Structure, Islamic Studies, General A and General B (could be chosen from the list of subjectsoffered in the institution, e.g.chemistry, economics, geology, physics,statistics), Pakistan Studies, Social Sciences Course as well as TechnicalWriting, are to be drafted by the respective National-Curriculum-RevisionCommittees. Course contents for the mathematics courses are, alphabetically,given below:

Marks Distribution for each subject

Mid –term =20% Tests/assignment/quiz/attendance/presentation =20% Final examination =60%

MATH-301 CALCULUS I Prerequisite: 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

Antiderivative and integral; definite integral and applications; the fundamental theorem of calculus; inverse functions (Chapters 1-6 of the text)

Recommended Books:

Anton H, Bevens I, Davis S, Calculus: A New Horizon (8th edition), 2005, John Wiley, New York

Stewart J, Calculus (3rd edition), 1995, Brooks/Cole (suggested text)

Thomas GB, Finney AR, Calculus (11th edition), 2005, Addison-Wesley, Reading, Ma, USA

MATH-302 DISCRETE STRUCTURES Prerequisite: Mathematics at intermediate level Credit Hours:03+0 Specific Objectives of the Course: 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 Text:

Rosen KH, Discrete Mathematics and its Applications (12th edition), 1999, McGraw Hill, New York

Ross KA, Wright CRB, Discrete Mathematics, 2003, Prentice Hall, EnglewoodCliffs, NJ, USA

MATH-303 INTRODUCTION TO STATISTICS-I

Pre-requisites: Intermediate with Math or A Level Math

Credit Hours: 03+0

Unit 1. What is Statistics? Definition of Statistics, Population, sample Descriptive and inferential Statistics, Observations, Data, Discrete and continuous variables, Errors of measurement, Significant digits, Rounding of a Number, Collection of primary and secondary data, Sources, Editing of Data. Exercises. Unit 2. Presentation of Data Introduction, basic principles of classification and Tabulation, Constructing of a frequency distribution, Relative and Cumulative frequency distribution, Diagrams, Graphs and their Construction, Bar charts, Pie chart, Histogram, Frequency polygon and Frequency curve, Cumulative Frequency Polygon or Ogive, Historigram, Ogive for Discrete Variable. Types of frequency curves. Exercises. Unit 3. Measures of Central Tendency Introduction, Different types of Averages, Quantiles, The Mode, Empirical Relation between Mean, Median and mode, Relative Merits and Demerits of various Averages. properties of Good Average, Box and Whisker Plot, Stem and Leaf Display, definition of outliers and their detection. Exercises. Unit 4.Measures of Dispersion Introduction, Absolute and relative measures, Range, The semi-Inter-quartile Range, The Mean Deviation, The Variance and standard deviation, Change of origin and scale, Interpretation of the standard Deviation, Coefficient of variation, Properties of variance and standard Deviation, Standardized variables, Moments and Moments ratios. Exercises. PHY-301 MECHANICS-I Pre-requisite A Level Physics and F.Sc. (Physics + Math) Credit Hours:3+0 Objectives: 1. To give concept of vector and their various properties. 2. To give basic understanding of laws of motion and their applications is daily life. 3. To give mathematical concept and expressions of various physical parameters usedin Mechanics.

Vector Analysis: Review of Vector in 3 dimensions and fundamental Operations, Direction, Cosines, Spherical polar coordinates, Cylindrical Coordinates. Vector and scalar triple products, gradient of a scalar, Divergence and curl of a vector, Physical significance of each type, Divergence of a vector, flux, curl and line integral (mutual relation). Vector identities, Divergence Theorem, Stoke‟s Theorem, their derivation, physical importance and applications to specific cases. Particle Dynamics: Dynamics of uniform, circular motion, the banked curve, Equations of motion, Deriving kinetic equations for x(t), v(t) via integration, Constant and variable forces, normal forces and contact forces, special examples, Time dependent forces, Obtaining x(t), v(t) for this case using integration method, Effect of drag forces on motion, Applying Newton‟s Laws to obtain v(t) for the case of motion with time dependent (Integration approach) drag (viscous) forces, terminal velocity, Projectile motion with and without air resistance, Non inertial frames and Pseudo forces, Qualitative discussion to develop understanding, Calculation of pseudo forces for simple cases (linearly accelerated reference frames), Centrifugal force as an example of pseudo force, Carioles force. Work, Power and Energy: Work done by a constant force, work done by a variable force (1-2dimension), (Essentially a review of grade-XII concepts via integration technique to calculate work done (e.g. in vibration of a spring obeying Hooke‟s Law), Obtaining general expression for work done (2-dimensional case) and applying to simple cases e.g. pulling a mass at the end of a fixed string against gravity, Work energy theorem, General 14proof of work energy theorem: Qualitative review of work energy theorem, Derivation using integral calculus, Basic formulae and applications, Power, Energy changes with respect to observers in different inertial frames, Conservation of Energy in 1, 2, and 3-dimensional conservative systems, Conservative and non-conservative forces: Conservation of energy in a system of particles, Law of conservation of total energy of an isolated system. Systems of Particles: Two particle systems and generalization to many particle systems, Centre of mass, Position, velocity and equation of motion, Centre of mass of solid objects, Calculation of Centre of Mass of solid objects using integral calculus, Calculating C.M. of Uniform Rod, Cylinder and Sphere, Momentum Changes in a system of variable mass, Derivation of basic equation, application to motion of a rocket (determination of its mass as a function of time). Collisions: Elastic Collisions, Conservation of momentum during collision in one and two dimensions, Inelastic collision, Collisions in centre of Mass reference frame (One and two dimensions), Simple applications, obtainingvelocities in C.M. frame. Recommended Books: 1. Halliday, D. Resnick, Krane, Physics, Vol. I & II, John Wiley, 5thed.1999. 2. D. Kleppner and R. Kolenkow, An Introduction to Mechanics, McGraw Hill,

1978. 3. M. R. Speigel, Vector Analysis and an Introduction to TensorAnalysis, Mc-Graw Hill, 1959.

ENG-301 English Structure-I (Functional English) Credit Hours: 03+0

Objectives: Enhance language skills and develop critical thinking. Course Contents

Basics of Grammar,

Parts of speech and use of articles

Sentence structure, active and passive voice

Practice in unified sentence

Analysis of phrase, clause and sentence structure

Transitive and intransitive verbs

Punctuation and spelling Comprehension

Answers to questions on a given text Discussion

General topics and every-day conversation (topics for discussion to be at the discretion of the teacher keeping in view the level of students)

Listening

To be improved by showing documentaries/films carefully selected by subject teachers

Translation skills

Urdu to English Paragraph writing

Topics to be chosen at the discretion of the teacher Presentation skills

Introduction Note: Extensive reading is required for vocabulary building Recommended books: 1. Functional English a) Grammar 1. Practical English Grammar by A.J. Thomson and A.V. Martinet. Exercises 1. Third edition. Oxford University Press. 1997. ISBN 0194313492 2. Practical English Grammar by A.J. Thomson and A.V. Martinet. Exercises 2. Third edition. Oxford University Press. 1997. ISBN 0194313506

b) Writing 1. Writing. Intermediate by Marie-Christine Boutin, Suzanne Brinand and Francoise Grellet. Oxford Supplementary Skills. Fourth Impression 1993. ISBN 0 19 435405 7 Pages 20-27 and 35-41. c) Reading/Comprehension 1. Reading. Upper Intermediate. Brain Tomlinson and Rod Ellis. Oxford Supplementary Skills. Third Impression 1992. ISBN 0 19 453402 2. d) Speaking

IS-311 ISLAMIC STUDIES (Compulsory)

Credit Hours: 02+0 Objectives:

This course is aimed at: 1 To provide Basic information about Islamic Studies 2 To enhance understanding of the students regarding Islamic Civilization 3 To improve Students skill to perform prayers and other worships 4 To enhance the skill of the students for understanding of issues related to faith and religious life.

Detail of Courses Introduction to Quranic Studies

1) Basic Concepts of Quran 2) History of Quran 3) Uloom-ul -Quran

Study of Selected Text of Holly Quran

1) Verses of Surah Al-Baqra Related to Faith(Verse No-284-286) 2) Verses of Surah Al-Hujrat Related to Adab Al-Nabi

(Verse No-1-18) 3) Verses of Surah Al-Mumanoon Related to Characteristics of faithful (Verse No1-11) 4) Verses of Surah al-Furqan Related to Social Ethics (Verse No.63-77) 5) Verses of Surah Al-Inam Related to Ihkam(Verse No-152-154)

Study of Sellected Text of Holly Quran

1) Verses of Surah Al-Ihzab Related to Adab al-Nabi (Verse No.6,21,40,56,57,58.) 2) Verses of Surah Al-Hashar (18,19,20) Related to thinking, Day of Judgment 3) Verses of Surah Al-Saf Related to Tafakar,Tadabar (Verse No-1,14)

Seerat of Holy Prophet (S.A.W) I

1) Life of Muhammad Bin Abdullah ( Before Prophet Hood) 2) Life of Holy Prophet (S.A.W) in Makkah 3) Important Lessons Derived from the life of Holy Prophet in Makkah

Seerat of Holy Prophet (S.A.W) II 1) Life of Holy Prophet (S.A.W) in Madina

2) Important Events of Life Holy Prophet in Madina 3) Important Lessons Derived from the life of Holy Prophet in Madina

Introduction To Sunnah

1) Basic Concepts of Hadith 2) History of Hadith 3) Kinds of Hadith 4) Uloom –ul-Hadith 5) Sunnah & Hadith 6) Legal Position of Sunnah

Selected Study from Text of Hadith Introduction To Islamic Law & Jurisprudence

1) Basic Concepts of Islamic Law & Jurisprudence 2) History & Importance of Islamic Law & Jurisprudence 3) Sources of Islamic Law & Jurisprudence 4) Nature of Differences in Islamic Law 5) Islam and Sectarianism

Islamic Culture & Civilization 1) Basic Concepts of Islamic Culture & Civilization

2) Historical Development of Islamic Culture & Civilization 3) Characteristics of Islamic Culture & Civilization 4) Islamic Culture & Civilization and Contemporary Issues

Islam & Science

1) Basic Concepts of Islam & Science 2) Contributions of Muslims in the Development of Science 3) Quranic & Science

Islamic Economic System

1) Basic Concepts of Islamic Economic System 2) Means of Distribution of wealth in Islamic Economics 3) Islamic Concept of Riba 4) Islamic Ways of Trade & Commerce

Political System of Islam

1) Basic Concepts of Islamic Political System 2) Islamic Concept of Sovereignty 3) Basic Institutions of Govt. in Islam

Islamic History 1) Period of Khlaft-E-Rashida 2) Period of Ummayyads 3) Period of Abbasids Social System of Islam 1) Basic Concepts Of Social System Of Islam 2) Elements Of Family 3) Ethical Values Of Islam Reference Books: 1) Hameed ullah Muhammad, “Emergence of Islam” , IRI, Islamabad 2) Hameed ullah Muhammad, “Muslim Conduct of State” 3) Hameed ullah Muhammad, „Introduction to Islam 4) Mulana Muhammad Yousaf Islahi,” 5) Hussain Hamid Hassan, “An Introduction to the Study of Islamic Law” leaf Publication Islamabad, Pakistan. 6) Ahmad Hasan, “Principles of Islamic Jurisprudence” Islamic Research Institute, International Islamic University, Islamabad (1993) 7) Mir Waliullah, “Muslim Jrisprudence and the Quranic Law of Crimes” Islamic Book Service (1982) 8) H.S. Bhatia, “Studies in Islamic Law, Religion and Society” Deep & Deep Publications New Delhi (1989) 9) Dr. Muhammad Zia-ul-Haq, “Introduction to Al Sharia Al Islamia” Allama Iqbal Open University, Islamabad (2001)

MATH-351 CALCULUS II Prerequisite: Calculus I Credit Hours: 03+0 Specific Objectives of the Course: 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:

Anton H, Bevens I, Davis S, Calculus: A New Horizon (8th edition), 2005, John Wiley, New York

Stewart J, Calculus (3rd edition), 1995, Brooks/Cole (suggested text)

Thomas GB, Finney AR, Calculus (11th edition), 2005, Addison-Wesley, Reading, Ma, USA

MATH-352 Introduction of Statistics-II Pre-requisites: Intermediate with Math and A-Level Math Credit Hours: 03+0 Unit 5. Sampling and Sampling Distributions Introduction, sample design and sampling frame, bias, sampling and non-sampling errors, sampling with and without replacement, probability and non-probability sampling, Sampling distributions for single mean and proportion, Difference of means and proportions. Exercises

Unit6. Hypothesis Testing Introduction, Statistical problem, null and alternative hypothesis, Type-I and Type-II errors, level of significance, Test statistics, acceptance and rejection regions, general procedure for testing of hypothesis. Exercises. Unit 7. Testing of Hypothesis- Single Population Introduction, Testing of hypothesis and confidence interval about the population mean and proportion for small and large samples, Exercises Unit 8. Testing of Hypotheses-Two or more Populations

Introduction, Testing of hypothesis and confidence intervals about the difference of population means and proportions for small and large samples, Analysis of Variance and ANOVA Table. Exercises Unit 9. Testing of Hypothesis-Independece of Attributes

Introduction, Contingency Tables, Testing of hypothesis about the Independence of attributes. Exercises. Unit 11. Regression and Correlation

Introduction, cause and effect relationships, examples, simple linear regression,

estimation of parameters and their interpretation. r and R2

. Correlation. Coefficient of linear correlation, its estimation and interpretation. Multiple regression and interpretation of its parameters. Examples Recommended Books

1. Walpole, R. E. 1982. “Introduction to Statistics”, 3rd

Ed., Macmillan Publishing Co., Inc. New York.

2. Muhammad, F. 2005. “Statistical Methods and Data Analysis”, Kitab Markaz, Bhawana Bazar Faisalabad

PHY-351 MECHANICS-II

Pre-requisite: A Level Physics and F.Sc. (Physics + Math)

Credit Hours: 03+0 Objectives: 1. To give the basic concept of rotational motion, law of gravitation, Physical properties of matter and relativistic mechanics 2. Uses of above concepts in daily life in a scientific way Rotational Dynamics: Relationships between linear & angular variables, scalar and vector form. Kinetic energy of rotation, Moment of Inertia, Parallel axis and Perpendicular axis theorems, Proof and Illustration, application to simple cases, Determination of moment of inertia of various shapes i.e. for disc, bar and solid sphere, Rotational dynamics of rigid bodies, Equations of rotational motion and effects of application of torques, Combined rotational and translational motion, Rolling without slipping. Angular Momentum: Angular Velocity, Conservation of angular momentum, effects of Torque and its relation with angular momentum, Stability of spinning objects, Discussion with examples, The spinning Top, Effects of torque on the angular momentum, processional motion. Gravitation: Gravitational effect of a spherical mass distribution, Its mathematical treatment, Gravitational Potential Energy (develop using integration techniques), calculation of escape velocity, Gravitational field & Potential, Universal Gravitational Law. Radial and transversal velocity and acceleration, Motion of Planets and Keplers' Laws (Derivation & explanation) Motion of Satellites, Energy considerations in planetary and satellite motion, Qualitative discussion on application of gravitational law to the Galaxy. Bulk Properties of Matters. Elastic Properties of Matter, Physical basis of elasticity, Tension, Compression & shearing, Elastic Modulus, Elastic limit. Poisson‟s ratio, Relation between three types of elasticity, Fluid Statics, Variation of Pressure in fluid at rest and with height in the atmosphere, Surface Tension, Physical basis; role in formation of drops and bubbles, Viscosity, Physical basis, obtaining the Coefficient of viscosity, practical example of viscosity; fluid flow through a cylindrical pipe (Poiseulle's law).

Special Theory of Relativity: Inertial and non-inertial frame, Postulates of Relativity, The Lorentz Transformation,

Derivation, Assumptions on which inverse transformation is derived, Consequences of

Lorentz transformation, Relativity of time, Relativity of length, Relativity of mass,

Transformation of velocity, variation of mass with velocity, mass energy relation and its

importance, relativistic momentum and Relativistic energy, (Lorentz invariants) E2=c2

p2+m2oc

CS-301 Introduction to Computer

Credit Hours: 03(2+1)

Objectives:

This is an introductory course on information and communication Technologies. Topics include

ICT terminologies, hardware and software components, the internet and world wide web, and

ICT based applications.

After completing this course, a student will be able to:

Understand different terms associated with ICT

Identify various components of a computer system

Identify the various categories of software and their usage

Define the basic terms associated with communications and networking

Understand different terms associated with the Internet and World Wide Web.

Use various web tools including Web Browsers, E-mail clients and search utilities

Use text processing, spreadsheets and presentation tools

Understand the enabling/pervasive features of ICT

Course Contents: Basic Definitions & concepts, Hardware, Computer System & Components, Storage Devices, Number Systems, Software: Operating Systems, Programming and Application Software, Introduction to Programming, Databases and Information Systems, Networks, Data Communication, The Internet, Browsers and Search Engines, The Internet: E-mail, Collaborative Computing and Social Networking, The Internet, E-Commerce, IT Security and other issues, Project Week, Review Week.

Books Recommended: 1 Introduction to Computers by Peter Norton, 6th International Edition (McGraw

HILL)

2 A Practical Introduction to Computer & Communications by Williams Sawyer, 6th Edition

(McGraw HILL)

ENG-351 English II (Communication Skills)

Credit Hours: 03+0 Objectives: Enable the students to meet their real life communication needs. Course Outline: Paragraph writing

Practice in writing a good, unified and coherent paragraph Essay writing

Introduction CV and job application

Translation skills

Urdu to English

Study skills

Skimming and scanning, intensive and extensive, and speed reading, summary and précis writing and comprehension

Academic skills

Letter/memo writing, minutes of meetings, use of library and internet

Presentation skills

Personality development (emphasis on content, style and pronunciation) Recommended books: Communication Skills a) Grammar 1. Practical English Grammar by A.J. Thomson and A.V. Martinet. Exercises 2. Third edition. Oxford University Press 1986. ISBN 0 19 431350 6. b) Writing 1. Writing. Intermediate by Marie-Chrisitine Boutin, Suzanne Brinand and Francoise Grellet. Oxford Supplementary Skills. Fourth Impression 1993.ISBN 019 4354057 Pages 45-53 (note taking). 2. Writing. Upper-Intermediate by Rob Nolasco. Oxford Supplementary Skills. Fourth Impression 1992. ISBN 0 19 435406 5 (particularly good for writing memos, introduction to presentations, descriptive and argumentative writing). c) Reading 1. Reading. Advanced. Brian Tomlinson and Rod Ellis. Oxford Supplementary Skills. Third Impression 1991. ISBN 0 19 453403 0.

2. Reading and Study Skills by John Langan 3. Study Skills by Riachard Yorky.

PS-351 Pakistan Studies (Compulsory) Credit Hours: 02+0 Introduction/Objectives

• Develop vision of historical perspective, government, politics, contemporary Pakistan, ideological background of Pakistan.

• Study the process of governance, national development, issues arising in the modern age and posing challenges to Pakistan.

Course Outline 1. Historical Perspective

a. Ideological rationale with special reference to Sir Syed Ahmed Khan, Allama Muhammad Iqbal and Quaid-i-Azam Muhammad Ali Jinnah.

b. Factors leading to Muslim separatism c. People and Land

i. Indus Civilization ii. Muslim advent iii. Location and geo-physical features.

2. Government and Politics in Pakistan Political and constitutional phases:

a. 1947-58 b. 1958-71 c. 1971-77 d. 1977-88 e. 1988-99 f. 1999 onward

3. Contemporary Pakistan a. Economic institutions and issues b. Society and social structure c. Ethnicity d. Foreign policy of Pakistan and challenges e. Futuristic outlook of Pakistan

Books Recommended 1. Burki, Shahid Javed. State & Society in Pakistan, The Macmillan Press Ltd 1980.

2. Akbar, S. Zaidi. Issue in Pakistan’s Economy. Karachi: Oxford University Press, 2000.

3. S.M. Burke and Lawrence Ziring. Pakistan‟s Foreign policy: An Historical analysis. Karachi: Oxford University Press, 1993.

4. Mehmood, Safdar. Pakistan Political Roots & Development. Lahore, 1994. 5. Wilcox, Wayne.The Emergence of Banglades., Washington: American Enterprise,

Institute of Public Policy Research, 1972. 6. Mehmood, Safdar. Pakistan Kayyun Toota, Lahore: Idara-e-Saqafat-e-Islamia, Club

Road, nd. 7. Amin, Tahir. Ethno - National Movement in Pakistan, Islamabad: Institute of Policy

Studies, Islamabad. 8. Ziring, Lawrence. Enigma of Political Development. Kent England: WmDawson &

sons Ltd, 1980. 9. Zahid, Ansar. History & Culture of Sindh. Karachi: Royal Book Company, 1980. 10. Afzal, M. Rafique. Political Parties in Pakistan, Vol. I, II & III. Islamabad: National

Institute of Historical and cultural Research, 1998. 11. Sayeed, Khalid Bin. The Political System of Pakistan. Boston: Houghton Mifflin,

1967. 12. Aziz, K.K. Party, Politics in Pakistan, Islamabad: National Commission on Historical

and Cultural Research, 1976. 13. Muhammad Waseem, Pakistan Under Martial Law, Lahore: Vanguard, 1987. 14. Haq, Noor ul. Making of Pakistan: The Military Perspective. Islamabad: National

Commission on Historical and Cultural Research, 1993.

MATH-401 CALCULUS III Prerequisite: Calculus II Credit Hours: 03+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 3dimensional 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:

1. Anton H, Bevens I, Davis S, Calculus: A New Horizon (8th edition), 2005, JohnWiley, 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-402 MATHEMATICAL SPACES Prerequisite: Discrete Structures, Real Analysis I Credit Hours: 03+0 Specific Objectives of the Course: 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, neighbourhoods, 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 sfields (Holmos) Recommended Books:

1. Friedmann A, Foundations of Modern Analysis, 1982, Dover 2. Holmos PR, Measure Theory, van Nostrand, New York 3. Munkres JR, Topology: A First Course, Prentice Hall, Englewood Cliffs,

NJ,USA 4. Simmon GF, Introduction to Topology and Modern Analysis, 1963,

McGrawHill, New York MATH-403 NUMBER THEORY Prerequisite(s): Calculus I, Discrete Structures Credit Hours: 03+0

Specific Objectives of the Course: This course shall assume no experienceor background in number theory or theoretical mathematics. The courseintroduces 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 systemof linear congruence, congruence of higher degree, Chinese remaindertheorem, Fermat‟s little theorem, Wilson‟s theorem and applications, primitiveroots and indices; integers belonging to a given exponent (mod p), primitiveroots of prime and composite moduli, indices, solutions of congruence usingindices., 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. Gross wald 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

Eastern 5. Rosen KH, Elementary Number theory and its Applications (4th edition), 2000,

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

PHY-401 ELECTRICITY AND MAGNETISM-I Credit Hours: 03+0 Pre-requisites: Intermediate with Physics and Math or A level Physics Objectives 1. To give the concept of electric field, electrical potential and dielectrics 2. To understand the DC circuits 3. To know the effect of magnetic field and basic magnetic properties of materials Electric Field: Field due to a point charge: due to several point charges. Electric dipole. Electric field of continuous charge distribution e.g. Ring of charge, disc of charge, infinite line of charge. Point charge in an electric field. Dipole in an electric field, Torque and energy of a dipole in uniform field. Electric flux: Gauss's law; (Integral and differential forms) and its application. Charge in isolated conductors, conductor with a cavity, field near a charged conducting sheet. Field of infinite line of charge, field of infinite sheet of charge, field of spherical shell and field of spherical charge distribution.

Electric Potential: Potential due to point charge, potential due to collection of point charges, potential due to dipole. Electric potential of continuous charge distribution. Poisson‟s and Laplace equation without solution. Field as the gradient or derivative of potential. Potential and field inside and outside an isolated conductor. Capacitors and dielectrics: Capacitance, calculating the electric field in a capacitor. Capacitors of various shapes, cylindrical, spherical etc. and calculation of their capacitance. Energy stored in an electric field. Energy per unit volume.Capacitor with dielectric, Electric field of dielectric. An atomic view. Application of Gauss's Law to capacitor with dielectric. D C Circuits: Electric Current, current density J, resistance, resistivity, ρ, and conductivity, σ, Ohm‟s Law, energy transfer in an electric circuit.Equation of continuity. Calculating the current in a single loop, multiple loops, voltages at various elements of a loop. Use of Kirchhoff's Ist& 2nd law, Thevenin theorem, Norton theorem and Superposition theorem, Growth and Decay of current in an RC circuit and their analytical treatment. Magnetic Field Effects and Magnetic Properties of Matter: Magnetic force on a charged particle, magnetic force on a current, Recall the previous results. Do not derive. Torque on a current loop. Magnetic dipole: Energy of magnetic dipole in field. Discuss quantitatively, Lorentz Force with its applications in CRO. Biot-Savart Law: Analytical treatment and applications to a current loop, force on two parallel current changing conductors. Ampere's Law, Integral and differential forms, applications to solenoids and toroids. (Integral form), Gauss's Law for Magnetism: Discuss and develop the concepts of conservation of magnetic flux, Differential form of Gauss‟s Law. Origin of Atomic and Nuclear magnetism, Basic ideas.Bohr Magneton. Magnetization, Defining M, B, μ. Magnetic Materials, Para magnetism, Diamagnetism, Ferromagnetism - Discussion. Hysteresis in Ferromagnetic materials.

Recommended Books:

1. F. J. Keller, W. E. Gettys, M. J. SkovePhysics Classical and Modern (2nd

edition), McGraw-Hill, Inc., 1993.

2. A. F. Kip Fundamentals of Electricity and Magnetism (2nd

Ed.), McGraw-Hill Book Co., 1969.

3. D. Halliday, R. Resnick, K. S. KranePhysics (Vol-II), John Willey & sons, Inc.,1992. 4. D. N. VasudevaMagnetism and Electricity, S. Chand & Co., 1959. 5. J. A. EdministerSchaum’s Outline Series; Theory and Problems of

Electromagnetism, McGraw-Hill Book Co., 1986. PHY-352 WAVES & OSCILLATIONS

Credit Hours: 03+0

Pre-requisite: A Level Physics and F.Sc. (Physics + Math)

Objective: 1. To understand the basics of waves, mechanism of wave production, propagation and interaction with other waves 2. use of basic concept of waves in their application in daily life Harmonic Oscillations: Simple harmonic motion (SHM), Obtaining and solving the basic equations of motion x(t), v(t), a(t), Longitudinal and transverse Oscillations, Energy considerations in SHM. Application of SHM, Torsional oscillator, Physical pendulum, simple pendulum, SHM and uniform circular motion, Combinations of harmonic motions, Lissajous patterns, Damped harmonic motion, Equation of damped harmonic motion, Quality factor, discussion of its solution, Forced oscillations and resonances, Equation of forced oscillation, Discussion of its solution, Natural frequency, Resonance, Examples of resonance. Waves in Physical Media: Mechanical waves, Travelling waves, Phase velocity of traveling waves, Sinusoidal waves, Group speed and dispersion, Waves speed, Mechanical analysis, Wave equation, Discussion of solution, Power and intensity in wave motion, Derivation & discussion, Principle of superposition (basic ideas), Interference of waves, Standing waves. Phase changes on reflection. Sound: Beats Phenomenon, Analytical treatment. Coupled Oscillators and Normal modes: Two coupled pendulums, General methods of finding normal modes, Beats in coupled oscillations, Two coupled masses, Two coupled LC circuits, Energy relations in coupled oscillations, Forced oscillations of two coupled oscillators, Many coupled oscillator. Normal Modes of Continuous systems: Transverse vibration of a string, Longitudinal vibrations of a rod, Vibrations of air columns, Normal modes, Fourier methods of analyzing general motion of a continuous system, Atomic vibrations. Recommended Books: 1. Halliday, D. Resnick, Krane, Physics, Vol. I & II, John Wiley, 5th ed. 1999. 2. N.K. Bajaj, The Physics of Waves & Oscillations, Tata McGraw-Hill Publishing company Limited, 1986. 3. H. J. Pain, The Physics of Vibrations and Waves, 5th Edition 1999

ENG-401 English III (Technical Writing and Presentation Skills) Credit Hours: 03+0 Objectives: Enhance language skills and develop critical thinking Course Contents Presentation skills Essay writing Descriptive, narrative, discursive, argumentative Academic writing How to write a proposal for research paper/term paper How to write a research paper/term paper (emphasis on style, content, language, form, clarity, consistency) Technical Report writing Progress report writing Recommended Books: Technical Writing and Presentation Skills a) Essay Writing and Academic Writing 1. Writing. Advanced by Ron White. Oxford Supplementary Skills. Third Impression 1992. ISBN 0 19 435407 3 (particularly suitable for discursive, descriptive, argumentative and report writing). 2. College Writing Skills by John Langan. Mc=Graw-Hill Higher Education. 2004. 3. Patterns of College Writing (4th edition) by Laurie G. Kirszner and Stephen R. Mandell. St. Martin‟s Press. b) Presentation Skills c) Reading The Mercury Reader. A Custom Publication. Compiled by norther Illinois University. General Editiors: Janice Neulib; Kathleen Shine Cain; Stephen Ruffus and Maurice Scharton. (A reader which will give students exposure to the best of twentieth century literature, without taxing the taste of engineering students). MATH-451 Operations Research-I Credit Hours. 03+0

Specific Objectives of the Course: 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: Introduction, Formulation and graphical solution of two variables linear programs, Simplex method, Method of Penalty, the two-phase technique, Sensitivity analysis Dual Simplex method, Duality, Sensitivity and parametric Analysis, Transportation and Assignment Models. Linear Programming: Advanced Topics. RECOMMENDED BOOKS: 1. H.A.Taha; 1982. Introduction to Operations Research , 3rd Ed. McMillan Pub.

Company N.Y. 2. G.Hadley; Introduction to Linear Programming. MATH-452 COMPUTING TOOLS FOR MATHEMATICIANS Credit Hours: 2(01+01) Specific Objectives of the Course: The purpose of this course is to teachstudents the use of mathematical software like MATLAB, MATHEMATICA for solving computationally-difficult problems in mathematics.The student shall become well versed in using at least one mathematicalsoftware and shall learn a number of techniques that are useful in calculus aswell as in other areas of mathematics. Course Outline: The contents of the course are not fixed, however thefollowing points should be kept in mind while teaching the course. The courseshould be taught in a computer lab setting. Besides learning to use thesoftware, the students must be able to utilize the software to solvecomputationally 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. Atkinson KE, An Introduction to Numerical Analysis (2nd edition), 1989, John Wiley, New York (suggested text)

2. Burden RL, Faires JD, Numerical Analysis (5th edition), 1993, PWS Publishing Company

3. Chapra SC, Canale RP, Numerical Methods for Engineers, 1988, McGraw Hill, New York

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

5. Garvan F, The Maple Book, 2002, Chapman & Hall/CRC 6. Kaufmann S, MathematicaAs a Tool: An Introduction with Practical Examples,

1994, Springer, New York MATH-453 PROBABILITY THEORY Prerequisite: Calculus III Credit Hours: 03+0 Specific Objectives of the Course: 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, hyper geometric 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, Addison-Wesley, Reading, Ma, USA (suggested text)

2. Papoulis A, Probability, Random Variables, and Stochastic Processes,(3rd edition), 1991, McGraw Hill, New York

3. Sincich T, Statistics by Examples, 1990, Dellen Publishing Company PHY-451 ELECTRICITY AND MAGNETISM-II Pre-requisites: FSC level Physics and Electricity and Magnetism I Credit Hours: 03+0 Objectives: 1. To understand the laws of electromagnetic induction 2. To understand the AC circuits 3. To know the generation and propagation of Electromagnetic waves Course Outline: Inductance:

Faraday‟s Law of Electromagnetic Induction, Review of emf, Faraday Law and Lenz‟s Law, Induced electric fields, Calculation and application using differential and integral form, Inductance, “Basic definition”. Inductance of a Solenoid; Toroid. LR Circuits, Growth and Decay ofcurrent, analytical treatment. Energy stored in a magnetic field, Derive. Energy density and the magnetic field. Electromagnetic Oscillation, Qualitative discussion. Quantitative analysis using differential equatins. Forced electromagnetic oscillations and resonance. Alternating Current Circuits: Alternating current, AC current in resistive, inductive and capacitative elements. Single loop RLC circuit, Series and parallel circuits i.e. acceptor and rejecter, Analytical expression for time dependent solution. Graphical analysis, phase angles. Power in A.C circuits: phase angles, RMS values, power factor. Electro-Magnetic Waves (Maxwell's Equations): Summarizing the electro- magnetic equations, (Gauss's law for electromagnetism, Faraday Law, Ampere's Law). Induced magnetic fields & displacement current. Development of concepts, applications. Maxwell's equations, (Integral & Differential forms) Discussion and implications. Generating an electro- magnetic wave. Travelling waves and Maxwell's equations. Analytical treatment; obtaining differential form of Maxwell's equations, obtaining the velocity of light from Maxwell's equations. Energy transport and the Poynting Vector. Analytical treatment and discussion of physical concepts. Recommended Books:

1. F. J. Keller, W. E. Gettys, M. J. Skove Physics Classical and Modern (2nd

edition), McGraw-Hill, Inc., 1993.

2. A. F. Kip Fundamentals of Electricity and Magnetism (2nd

Ed.), McGraw-Hill Book Co., 1969.

3. D. Halliday, R. Resnick, K. S. Krane Physics (Vol-II), John Willey & sons, Inc., 1992. 4. D. N. Vasudeva Magnetism and Electricity, S. Chand & Co., 1959. 5. J. A. Edminister Schaum’s Outline Series; Theory and Problems of

Electromagnetism, McGraw-Hill Book Co., 1986.

PSY-101 INTRODUCTION TO PSYCHOLOGY

Credit Hours: 03(2+1)

Course Objectives Describe psychology with major areas in the field, and identify the parameters of this discipline. Distinguish between the major perspectives on human thought and behavior. Appreciate the variety of ways psychological data are gathered and evaluated. Gain insight into human behavior and into one's own personality or personal relationships.

Explore the ways that psychological theories are used to describe, understand, predict, and control or modify behavior.

Course Contents

1. Introduction to Psychology

a. Nature and Application of Psychology with special reference toPakistan.

b. Historical Background and Schools of Psychology (A BriefSurvey)

2. MethodsofPsychology

a. Observation

b. Case History Method Experimental Method

c. Survey Method

d. Interviewing Techniques

3. Biological Basis of Behavior

a. Neuron: Structure and Functions

b. Central Nervous Systemand Peripheral Nervous System

c. EndocrineGlands

4. Sensation, Perception and Attention

a. Sensation

i. Characteristics and Major Functions of Different Sensations ii. Vision: Structureand functions of the Eye. iii. Audition: Structure and functions of the Ear.

b. Perception

i. Nature of Perception ii. Factors of Perception: Subjective, Objective and Social iii. Kinds of Perception: iv. Spatial Perception (Perception of Depth and Distance) v. Temporal Perception; Auditory Perception.

c. Attention

i. Factors, Subjective and Objective ii. Span of Attention iii. Fluctuation of Attention iv. Distraction of Attention (Causes and Control)

5. Motives

a. Definition and Nature

b. Classification

Primary(Biogenic) Motives: Hunger, Thirst, Defection and Urination, Fatigue, Sleep,

Pain, Temperature, Regulation, Maternal Behavior, Sex

Secondary (Sociogenic) Motives:Play and Manipulation, Exploration and Curiosity,

Affiliation, Achievement and Power, Competition, Cooperation, Social Approval and

Self Actualization.

6. Emotions

a. Definition and Nature

b. Physiological changes during Emotions (Neural, Cardial, Visceral, Glandular), Galvanic Skin Response; Pupilliometrics

c. Theories of Emotion

d. James Lange Theory; Cannon-Bard Theory

e. Schachter –Singer Theory

7. Learning

a. Definition of Learning

b. Types of Learning: Classical and Operant Conditioning Methods of Learning: Trial and Error; Learning by Insight; Observational Learning

8. Memory

a. Definition and Nature

b. Memory Processes: Retention, Recall and Recognition

c. Forgetting: Nature and Causes

9. Thinking

a. Definition and Nature

b. Tools of Thinking: Imagery; Language; Concepts

c. Kinds of Thinking

d. Problem Solving; Decision Making; Reasoning

10. Individual differences

a. Definition concepts of;

b. Intelligence, personality, aptitude, achievement RECOMMENDED BOOKS

1. Atkinson R. C., & Smith E. E. (2000). Introduction to psychology(13thed.). Harcourt

Brace College Publishers.

2. Fernald,L.D.,&Fernald,P.S.(2005). Introduction to psychology. USA:

WMCBrownPublishers.

3. Glassman, W. E. (2000). Approaches to psychology. Open University Press. Hayes, N.

(2000). Foundation of psychology (3rd ed.). Thomson Learning. Lahey, B. B. (2004).

Psychology: An introduction (8th ed.). McGraw-HillCompanies, Inc.

4. Leahey, T. H. (1992). A history of psychology: Main currents in psychological thought.

New Jersey: Prentice-Hall International, Inc.

5. Myers, D. G. (1992).Psychology. (3rd ed.). New York: WadsworthPublishers.

6. Ormord, J. E. (1995). Educational psychology: Developing learners. Prentice- Hall, Inc.

SOC-401 INTRODUCTION OF SOCIOLOGY CREDIT HOURS: 03(02+01) Course Aims and Objective: The course is designed to introduce the students with sociological concepts and the discipline. The focus of the course shall be on major concepts like social systems and structures, socio-economic changes and social processes. The course will provide due foundation for further studies in the field of sociology. Course Outline 1. Introduction

Definition, Scope, and Subject Matter

Sociological imagination

Nature of Sociology

Historical background of Sociology

Importance of studying sociology 2. Perspectives in Sociology

Structural Functionalist perspective

Conflict perspective

Symbolic Integrationist perspective

Global perspective 3. Culture

Definition, aspects and characteristics of Culture

Material and non-material culture/culture and civilization

Ideal and real culture

Elements of culture

Norms and social sanctions

Cultural Relativism

Sub Cultures

Ethnocentrism and Xenocentrism

Cultural change and related concepts 4. Socialization & Personality

Socialization, Agencies of Socialization

Self

Personality, Factors in Personality Formation

Theories of socialization and personality development

Role & Status

5. Social Processes

Social interaction and forms of social interaction

Cooperation

Competition

Conflict

Assimilation and acculturation

Accommodation 6. Social Groups

Definition

Types of social groups

In and out groups

Primary and Secondary group

Reference groups

Bureaucracy

Pressure groups

7. Social Inequality and Social Stratification

Social Class

Caste

Gender

Race

Social Mobility and types of social mobility

Income inequality Recommended Books:

1. Ballantine, Jeanne H. and Roberts, Keith A. (Condensed Version) 2010. Our Social World. California: Pine Forge Press/Sage Publication.

2. Brown, Ken 2004. Sociology. United Kingdom: Polity Press 3. Brym, Robert J. and Lie, John. Sociology: Your compass for a new world (Brief Edition) 2007

Belmont: Thomson Wadsworth.

4. Colander, David C. and Hunt, Elgin F. (Thirteenth Edition) (2010) Social Sciences: An introduction to the study of Society. India: Pearson Education/Dorling Dindersley.

5. Gidden, Anthony 2002. Introduction to Sociology. UK: Polity Press. 6. Rao, C. N. Shankar (2008) „Sociology: Principles of Sociology with an Introduction to Social

Throughts‟ New Delhi: S. Chand & Company.

7. James M. Henslin. (2004). Sociology: A Down to Earth Approach. Toronto: Allen and Bacon. 8. Macionis, John J. (2006). 10th Edition Sociology New Jersey: Prentice-Hall 9. Montuschi, Eleonora. (2006). The Objects of Social Sciences New York: Continuum.

Hortun, Paul B. and Hunt, Chester L. 1984. Sociology. New York: McGraw-Hill MATH-501 ALGEBRA I Prerequisite: Mathematics at intermediate level Credit Hours: 3 + 0

Specific Objectives of the Course: This is the first course in groups, matrices and linear algebra, which provides basic background needed for allmathematics majors, a prerequisite for many courses. Many conceptspresented in the course are based on the familiar setting of plane and realthree-space, and are developed with an awareness of how linear algebra isapplied. Course Outline: Group Theory: Basic axioms of a group with examples, Abelian groups, centre of a group, derived subgroup of a group, subgroups generated by subset of a group, system of generators, cyclic groups, co-sets 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 (3rd edition),

1995,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),

1994, PWS Publishing Co. MATH-502 VECTOR AND TENSOR ANALYSIS Prerequisite: Calculus II Credit Hours: 03+0 Specific Objectives of the Course: This course shall assume background incalculus. It covers basic principles of vector analysis, which are used inmechanics

Course Outline: 3-D vectors, summation convention, kronecker delta, Levi-Civita symbol, vectors as quantities transforming under rotations with ijknotation, 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 modelling anisotropic systems, study of physical tensors (moment of inertia, index of refraction, etc.), diagnolization 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-503 General Topology Credit Hours:03+0 Specific Objectives of the Course: 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: Introduction to topological spaces and metric spaces, open and closed sets.Neighbourhoods.Limit points. Closure of a set.Bases and sub-bases.First and second axiom of countability.Continuous functions and Homeomorphisms. Product spaces. Separation axioms. Completely regular spaces. Normed spaces.Compactness.Local compactness. Connected spaces. Convergence and completeness, Baire‟s theorem. RECOMMENDED BOOKS 1. J. R. Munkres, Topology (A first course), Prentice Hall Inc, 1975. 2. T. Hussain, Topology and Maps, Plenum Press NY, 1977. 3. A. Majeed, Elements of Topology & Functional Analysis, Ilmi Kitab Khana,1990. 4. S. Willard, General Topology, Addison Wesley NY,1970 . 5. F. Simmon, Introduction to Topology & Modern Analysis, McGraw Hill, NY, 1993

MATH-504 COMPLEX ANALYSIS-I Prerequisite: Real Analysis I Credit Hours: 03+0 Specific Objectives of the Course: This is an introductory course in complexanalysis, giving the basics of the theory along with applications, with anemphasis on applications of complex analysis and especially conformalmappings. Students should have a background in real analysis (as in thecourse Real Analysis I), including the ability to write a simple proof in ananalysis context. Course Outline: The algebra and the geometry of complex numbers, Cauchy-Riemann equations, harmonic functions, elementary functions, branches of the logarithm, complex exponents. Contours and contour integrals, the Cauchy-Goursat Theorem, Cauchy integral formulas, the Morera Theorem, maximum modulus principle, the Liouville theorem, fundamental theorem of algebra. Convergence of sequences and series, the Taylor series, the Laurent series, uniqueness of representation, zeros of analytic functions. Residues and poles and the residue theorem, evaluation of improper integrals involving trigonometric functions, integrals around a branch point., the argument principle, the Roche theorem. Recommended Text: 1. L.L.Pennisi; 1976. Elements of Complex Variables, Holt Rinehart & Winston NY. 2. R.V.Churchil; 1989, Brown JW: Complex Variables and Applications, McGraw Hill Co, New York.

4. L.V.Ahlfors; 1966. Complex Analysis, McGraw Hill Co. 5. S. Levinson and R. Redheffer, 1970. Complex Variables, Holden Day Inc. 6. S.B.H.Anthony; Complex Function Theory, Holland N Y. 7. M. Iqbal, 1998. Fundamental of Complex Analysis, Ilmi Kitab Khana, Lahore

MATH-505 ORDINARY-DIFFERENTIAL EQUATIONS Prerequisite(s): Calculus III, Computing Tools for Mathematicians Credit Hours: 03+0 Specific Objectives of the Course: 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. The fundamental existence theorem (without proof) Linear Operators, Fundamental solutions of the homogeneous equations, Reduction of order of a differential equation, Linear

homogeneous equations with constant coefficients, the method of undetermined coefficients, the method of variation of parameters, the Cauchy Euler equation. 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 Hermit equations, properties of the Bessel function, Hypergeometric, Lauguere equations and their solutions. Orthogonal polynomials. Green function for ordinary differential equations. Recommended Text: 1. L. L. Pennisi., Elements of Ordinary Differential Equations. 2. Boyce & Diprima: 1986 Elemenary Differential Equations and Boundary Value Problems. John Wiley. 2. Robert M. Martin Jr. 1983 Elementary Differential Equations with Boundary Value

Problems, McGraw Hill Book Company Inc. 3. L.R.Shepley: Introduction to Ordinary Differential Equations, John Wiley and Sons. 4. Colmb & Shanks, Elements of Ordinary Differential Equations McGraw Hill. 5. Zill DG, Cullen MR, Differential Equations with Boundary-Value Problems, (3rd

Edition), 1997, PWS Publishing Co. MATH-506 REAL ANALYSIS I Prerequisite: Calculus III Credit Hours: 03+0 Specific Objectives of the Course: This is the first rigorous course inanalysis and has a theoretical emphasis. It rigorously develops thefundamental ideas of calculus and is aimed to develop the students‟ ability todeal 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, John Wiley, New York

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

Gaughan ED, Introduction to Analysis (5th edition), 1997, Brooks/Cole 3. Rudin W, Principles of Mathematical Analysis (3rd edition), 1976, McGraw Hill,

New York 4. Rudin. 1976, W. Principles of Mathematical Analysis, 3rd Ed, McGraw Hill. 5. S.C.Malik, 1984. Mathematical Analysis, Wiley Eastern Ltd.

MATH-551 ALGEBRA II Prerequisite: Algebra I Credit Hours: 03+0 Specific Objectives of the Course: This is a course in advanced abstractalgebra, 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: Allenby RBJT,Rings, Fields and Groups: An Introduction to Abstract Algebra, 1983,Edward Arnold Farleigh JB, A First Course in Abstract Algebra (7th edition), Addison-Wesley, Reading, Ma., USA Macdonald ID, The Theory of Groups, 1975, Oxford Clarendon Press, Ma., USA MATH-552 CLASSICAL MECHANICS Prerequisite: Vector and Tensor Analysis Credit Hours: 03+0 Specific Objectives of the Course:

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, No ether‟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) 5. Synge JL, Griffith BA, Principles of Mechanics, McGraw Hill, New York

MATH-553 COMPLEX ANALYSIS-II Prerequisite: Complex Analysis I Credit Hours: 03+0 Specific Objectives of the course: 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 complex -1 , including the ability to write a simple proof in an analysis context. Course outline: Mapping by Elementary functions, conformal mapping and its properties. Univalent functions and their subclasses, Functions with positive real part, Convex and starlike functions and their basic properties. Gamma and Beta functions and their basic properties , linear functions, and the functions Zn, the function 1/z, the point at Infinity.

The linear functional transformation, Special linear functional transformations, The function z 1/2 , the transformation W = ez , The transformation W = sinz. RECOMMENDED BOOKS: 1. R. V. Churchil, Complex Variables and Applications, McGraw Hill Co, 1960. 2. L. V. Ahlfors, Complex Analysis, McGraw Hill Co, 1966. 3. M. Iqbal, Fundamental of Complex Analysis, Ilmi Kitab Khana, Lahore, 1998.

4. E. D. Rainvill, Special Functions, McGraw Hill, 1992.

5. N. W. Lebedev, Special functions and their applications, Dover Publishing Inc.,

1972.

MATH-554 Set Theory & Mathematical Logic Credit Hour: 03+0 Course Outline: Set Theory: Functions, Relations, Partially and totally ordered sets, Axiom of choice, Hausdorff‟s Maximal Principle, Zorn‟s Lemma, Well-Ordering theorem, Zormelo‟s theorem and their equivalences, Finite, Infinite and Denumerable sets, Cardinal Arithmetic, Ordering of the cardinal numbers and Schroder-Bernstein theorem, Arithmetic and ordering of the ordinal numbers. Mathematical Logic: Statement forms and connectives, Tautology and contradiction, Logical equivalence, Algebra of propositions, Logical Implication, Arguments, Adequate systems of connectives. Applications: Definition and properties of a Boolean algebra, partial orders in a Boolean algebra, switching circuit design, Lattices (modular, distributive, complemented). RECOMMENDED BOOKS: 1. R. R. Stoll, Set theory and Logic, Dover Publication Inc. N.Y., 1971.

2. C. C. Pinter, Set theory, Addison- Wesley Publishing Company, Inc. N.Y., 1963.

3. P. R. Halmos, N. J. Naïve, Set Theory. D.van Nostrand Co. Princeton, 1968.

4. E. Mendelson, Boolean algebra and Switching Circuits Mc Graw Hill Inc., 1970.

5. J. D. Monk, Mathematical Logic, Springer-Verlag N.Y., 1976.

MATH-555 Differential Geometry Credit Hours: 03+0 Specific Objectives of the Course: To prepare the students, not majoring in mathematics, with the essential tools of geometry to apply the concepts and the techniques in their respective disciplines. Course Outline: Space curves. Osculating plane. The moving trihedron. Serret-Frenet formulae. Osculating circle. The concept of surface curves. Spherical and Cylindrical helices. Spherical Indicatrix, Involutes and Evolutes. First fundamental form of a surface. The

second fundamental form. Normal curvature. Principal directions and principal curvatures. Gaussian and mean curvature. Euler's Theorem. Gauss-Weingarten and Gauss-Godazzi equations. RECOMMENDED BOOKS: 1. E. Weatherburn, Differential Geometry of Three Dimensions, Cambridge Uni., 1961. 2. Millman & Parker, Elements of Differential Geometry, Prentice Hall, 1977.

3. D. J. Struik, Lectures on Classical Differential Geometry, Addison Wesley, 1962. 4. M.P.Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc.,Englewood,NewJersey,1985. 5. B.O.Neil, Elementary Differential Geometry, Academic Press, 1966. 6. A.Goetz, Introduction to Differential Geometry, Addison-Wesley, 1970. MATH-556 REAL ANALYSIS II Prerequisite: Real Analysis I Credit Hours: 03+0 Specific Objectives of the Course: A continuation of Real Analysis I, thiscourse rigorously develops integration theory. Like Real Analysis I, RealAnalysis II emphasizes proofs. Course Outline: Series of numbers and their convergence.Series offunctions and their convergence.Dabroux upper and lower sums andintegrals; Dabrouxintegrability. 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 ofbounded variation. The length of a curve in Rn Recommended Books:

1. Bartle RG, Sherbert DR, Introduction to Real Analysis (3rd edition), 1999,John Wiley, New York

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 edition), 1976, McGraw

Hill,New York MATH-601 FUNCTIONAL ANALYSIS-I Prerequisite: Complex Analysis Credit Hours: 03+0 Specific Objectives of the Course:

This course extends methods of linear algebra and analysis to spaces of functions, in which the interaction betweenalgebra and analysis allows powerful methods to be developed. The coursewill be mathematically sophisticated and will use ideas both from linearalgebra 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 functional, 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 equations Inner-Product Spaces: Inner-product space, Hilbert space, orthogonal and orthonormal sets, orthogonal comple-ments, Gram-Schmidt orthogonalization process, representation of functional, Reiz-representation theorem, weak and weak* Convergence. 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 3. Kreyszig E, Introductory Functional Analysis with Applications, John Wiley,

New York 4. Rudin W, Functional Analysis, 1973, McGraw Hill,

MATH-602 PARTIAL-DIFFERENTIAL EQUATIONS Prerequisite: Real Analysis I, Ordinary-Differential Equations Credit Hours: 03+0 Specific Objectives of the Course: 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 coordinates; 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:

5. N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill Book

Company, 1987.

3. R. Dennemyer, Introduction to Partial Differential Equations and Boundary Value

Problems, McGraw-Hill Book Company, 1968.

4. M. Humi, W. B. Miller, Boundary Value Problems and Partial Differential

Equations, PWS-Kent Publishing Company, Boston, 1992.

5. C. R. Chester, Techniques in Partial Differential Equations, McGraw-Hill Book

Company, 1971.

6. R. Haberman, Elementary Applied Partial Differential Equations, Prentice Hall,

Inc.New Jersey, 1983.

7. E. Zauderer, Partial Differential Equations of Applied Mathematics, John Wiley &

Sons, Englewood Cliff, New York, 1983.

8. Myint UT, Partial Differential Equations for Scientists and Engineers (3rd edition),

1987, North Holland, Amsterdam

MATH-603 NUMERICAL ANALYSIS-I Prerequisite: Computing Tools for Mathematicians Credit Hours: 03+0 Specific Objectives of the Course: This course is designed to teach thestudents about numerical methods and their theoretical bases. The studentsare expected to know computer programming to be able to write program foreach numerical method. Knowledge of calculus and linear algebra would helpin learning these methods Course Outline: Computer arithmetic, approximations and errors; methodsfor 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. Numericalsolution of systems of algebraic linear equations: Gauss-elimination method Gauss-Jordan method; matrix inversion; LU-factorization; Doolittle‟s, Crount‟sCholesky‟s methods; Gauss-Seidel and Jacobi methods Recommended Books:

1. Atkinson KE, An Introduction to Numerical Analysis (2nd edition), 1989, John Wiley, New York (suggested text)

2. Burden RL, Faires JD, Numerical Analysis (5th edition), 1993, PWS PublishingCompany

3. Chapra SC, Canale RP, Numerical Methods for Engineers, 1988, McGraw Hill,New York

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

5. Garvan F, The Maple Book, 2002, Chapman & Hall/CRC 6. Kaufmann S, MathematicaAs a Tool: An Introduction with Practical Examples,

1994, Springer, New York MATH-604 ALGEBRA III Prerequisite(s): Algebra II Credit Hours: 03+0 Specific Objectives of the Course: This is a course in abstract linear algebra. The majority of follow up courses in both pure and applied mathematics assume the material covered in this course. Course Outline: Vector spaces; sums and direct sums of subspaces of a finite dimensional vector space, Dimension theorem, linear transformation, null space, image space of linear transformation, rank and nullity of a linear transformation, relation between rank, nullity and dimension of the domain of a linear transformation, matrix of linear transformation, change of basis, inner product spaces, projection of a vector along another vector, norm of a vector, Cauchy Schwartz inequality, Orthogonal and orthonormal basis, similar matrices and diagonalization of a matrix, Home (V,W), dimension and basis of Home (V.W), dual space and dual basis, annihilators. Recommended Books:

1. Axle SJ, Linear Algebra Done Right, Undergraduate Texts in Mathematics, 1996, Springer, New York

2. Birkhoff G, Maclane S, A Survey of Modern Algebra (4th edition), AKP Classics 3. Perry WL, Elementary Linear Algebra, 1988, McGraw-Hill, New York

MATH-605 Dynamics Credit Hours: 03+0 Specific Objectives of the Course: In this area we study the motion of particles under different certain forces and physical concepts

Course Outline: Particle Dynamics: Projectile motion under gravity, constrained particle motion, angular momentum of a particle. Orbital Motion: Motion of a particle under a central force, use of reciprocal polar co-ordinates, use of pedal co-ordinates and equations, Kepler's laws of planetary motion. Motion of a system of Particles: Linear momentum of a system of particles, angular momentum and rate of change of angular momentum of a system, use of centroid, moving origins, impulsive forces, elastic impact. Introduction to Rigid Body Dynamics: Moments and products of inertia, the theorems of parallel and perpendicular axes, angular momentum of a rigid body about a fixed point and about fixed axes, principal axes. Kinetic energy of a rigid body rotating about a fixed point, general motion of a rigid body, momental ellipsoid, equimomental system, coplanar distribution. RECOMMENDED BOOKS: 1. F. Chorlton, Text book of Dynamics, Ellis Horwood Ltd., 1983.

2. L. A. Pars, Introduction to Dynamics, Cambridge Uni. Press, 1953.

3. A. S. Remsey, Dynamics Part-I, Cambridge Uni. Press, 1962. 4. J. L. Synge and B. A. Griffith, Principle of Mechanics, McGraw Hill Book Co., 1970. MATH-651 Functional Analysis II Prerequisite(s): Functional Analysis-I Credit Hours: 03+0 Specific Objectives of the Course: 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, Hilbert spaces, orthonormal bases, convexity in Hilbert spaces, operators in Hilbert spaces, Invariant sub-spaces. Decomposition of Hilbert spaces. Finite dimensional spectral theory, and spectral mapping theorem. ECOMMENDED BOOKS:

1. E. Taylor, D. C. Lay, Introduction to Functional Analysis, John Wiley & Sons,

1980.

2. E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley &

Sons, 1978.

3. W. Rudin, Functional Analysis, McGraw Hill, 1978.

4. G. F. Simmons, Introduction to Topology & Modern Analysis, McGraw Hill, 1963.

5. K. Yosida, Functional Analysis, Springer Verleg NY, 1967.

6. A. Majeed, Elements of Topology and Functional Analysis, Ilmi Kitab Khana,

Lahore, 1997.

MATH-652 OPTIMIZATION THEORY Prerequisite(s): Algebra I, Real Analysis I Credit Hours: 03+0 Specific Objectives of the Course: The main objective is to teach the basicnotions and results of mathematical programming and optimization. The focuswill be to understand the concept of optimality conditions and the constructionof solutions. Students should have a good background in analysis, linearalgebra and differential equations. Course Outline: Linear programming: simplex method, duality theory, dualand primal-dual simplex methods. Unconstrained optimization: optimality conditions, one-dimensional problems, multi-dimensional problems and the method of steepest descent. Constrained optimization with equality constraints: optimality conditions, Lagrange multipliers, Hessians and bordered Hessians. Inequality constraints and the Kuhn-Tucker Theorem. The calculus of variations, the Euler-Lagrange equations, functional depending on severalvariables, vibrational problems in parametric form, transportation models andnetworks. 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 MATH-653 Numerical Analysis-II Prerequisite: Numerical Analysis-I Credit Hours: 03+0 Specific Objectives of the Course: 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 content: Numerical differentiation: Forward difference formulas, Central difference Formulas, error in numerical differentiation, extrapolation to the limit. Numerical solution of Difference Equations: Formation of difference equations, numerical solutionof linear homogeneous and non-homogeneous difference equations with constant coefficients. Numerical solution of Ordinary Differential Equations: Initial Value Problems; Taylor's methods and truncation error. Euler‟s method and error estimates. Modified Euler method. Runge-Kutta methods. Predictor and corrector methods; Milne-Simpson method, Adams-Bash forth-Moulton method and Hamming's method. Numerical Solution of Partial Differential Equations: Finite difference method for solving Parabolic, Hyperbolic and Elliptic Equations. RECOMMENDED BOOKS: 1. W. A. Smith, Elementary Numerical Analysis, Harper & Row Pub., 1979.

2. C. E. Froberg, Introduction to Numerical Analysis, Addsion-Wesley Co., 1974.

3. M. K. Jain et al., Numerical Methods for Scientific and Engineering Computation,

Wiley Ltd.

4. R. L. Burden, J. D. Faires, Introduction to Numerical Analysis. Seventh Ed.

MATH-654 MODELING AND SIMULATION Prerequisite(s): Partial-Differential Equations Credit Hours: 03+0 Specific Objectives of the Course: 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 modelling, consisting of three parts: modelling 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 modelling; objective, background, approximation and idealization, model validation, compounding. Modelling wave phenomena (wave equation); shallow water waves, uniform transmission line, traffic flow, RC circuits. Modelling the heat equation and some application to heat conduction problems in rods, lamina, cylinders etc. Modelling the potential equation (Laplace equation), applications in fluid mechanics, gravitational problems. Equation of continuity. Simulation: Techniques of simulation (students are required to simulate atleast one system) Recommended Books: 1. Giordano FR, Weir MD, Differential Equations: A Modeling Approach, 1994, Addison-Wesley, Reading, Ma, USA (suggested text) 1. Jerri AJ, Introduction to Integral Equations with Applications, 1985, Marcel Dekker, New York 2. Myint UT, Debnath L, Partial Differential Equations for Scientists and Engineers (3rd edition), 1987, North Holland, Amsterdam MATH-675 QUANTUM MECHANICS Prerequisite: FSc and A-level Modern Physics & electronics Credit Hours: 03+0 Objectives 1. Understanding the behaviour of quantum mechanical particle and development of Schrodinger equation in one and three dimensions 2. introduction to Quantum mechanical operators and determination of angular momentum of a quantum mechanical particle Quantum Mechanics of One Dimensional Problems: Review of concepts of classical mechanics, State of a system, Properties of one dimensional potential functions, Functions and expectation values, Dirac notation, Hermitian operators, Solutions of Schrodinger equation for free particles, The potential barrier problems, The linear harmonic oscillator, Particle in a box. Formalism of Quantum Mechanics:

The state of a system, Dynamical variables and operators, Commuting and non-commuting operators, Heisenberg uncertainty relations, Time evolution of a system, Schrodinger and Heisenberg pictures, Symmetry principles and conservation laws. Angular Momentum:

Orbital angular momentum, Spin, The eigenvalues and eigen functions of L2

and Lz,

Matrix representation of angular momentum operators, Addition of angular momenta. Schrodinger Equation in Three Dimensions: Separation of Schrodinger equation in Cartesian coordinates, Central potentials, The free particle, Three dimensional square well potential, The hydrogen atom, Three dimensional square well potential, The hydrogen atom, Three dimensional isotopic oscillator. Recommended Books:

1. David J. Griffiths, Introduction to Quantum Mechanics, PRENTICE Hall, Int., Inc. 2. R.L. Liboff, 'Introductory Quantum mechanics', Addison Wesley Publishing Company, Reading Mass. (1980) 3. B.H. Bransden & C.J. Joachain, 'Introduction to Quantum Mechanics' Longman Scientific & Technical London (1990) 4. J.S. Townsend, 'A Modern Approach to Quantum Mechanics', McGraw Hill Book Company, Singapore (1992) 5. W. Greiner, 'Quantum Mechanics: An Introduction', Addison Wesley Publishing Company, Reading Mass. (1980) 6. Bialynicki-Birula, M. Cieplak & J. Kaminski, 'Theory of Quantua', Oxford University Press, New York (1992) 7. W. Greiner, 'Relativistic Quantum Mechanics', Springer Verlag, Berlin (1990) 8. F. Schwable, 'Quantum Mechanics', Narosa Publishing House, New Delhi (1992) 9. Sureh Chandra, Quantum Mechanics, CBS Publishers New Delhi Banglore

The BS Electives (Alphabetical Listing)

MATH-604 Algebra-III MATH-654 Modelling & Simulation MATH-605 Dynamics MATH-606 Abstract Algebra I MATH-655 Abstract Algebra II MATH-607 Advanced Calculus I MATH-656 Advanced Calculus II MATH-608 Advanced Numerical- MATH-657 Advanced Numerical- Analysis I Analysis II MATH-609 Astronomy I MATH-658 Astronomy II MATH-610 Electromagnetism I MATH-659 Electromagnetism II MATH-611 Fluid Dynamics I MATH-660 Fluid Dynamics II MATH-612 Group Theory I MATH-661 Group Theory II

(Fundamentals) (Study of Symmetries) MATH-613 History and Philosophy of MATH-662 History and Philosophy Mathematics I of Mathematics II MATH-614 Measure Theory I MATH-663 Measure Theory II MATH-615 Modern Algebra I MATH-664 Modern Algebra II

(Galios Theory & Applications) (Commutative Rings &Fields)

MATH-616 Nonlinear Systems I MATH-665 Nonlinear Systems II MATH-617 Numerical Solutions of MATH-666 Numerical Solutions of

PDEs I PDEs II MATH-618 Operations Research I MATH-667 Operations Research II MATH-619 Projective Geometry I MATH-668 Projective Geometry II MATH-620 Relativity I MATH-669 Relativity II MATH-621 Software Engineering I MATH-670 Software Engineering II

(Design & Development) (Analysis) MATH-622 Theory of Processes I MATH-671 Theory of Processes II

(Stochastic Processes) (Renewal Processes & Theory of Ques)

MATH-623 Theory of Splines I MATH-672 Theory of Splines II MATH-624 Topology I MATH-673 Topology II

(Topological-Dimension Theory) (Differential Topology) MATH-625 Integral Equations MATH-674 Fluid Mechanics MATH-675 Quantum Mechanics MATH-626 Analytical Mechanics-I MATH-676 Analytical Mechanics-II

COMPULSORY MATHEMATICS COURSES FOR BS (4 YEAR) (FOR STUDENTS NOT MAJORING IN MATHEMATICS)

1. MATH-321 BASIC ALGEBRA

Prerequisite: Mathematics at secondary level Credit Hours: 03+0

Specific Objectives of the Course: To prepare the students, not majoring in mathematics, with the essential tools of algebra to apply the concepts and the techniques in their respective disciplines. Course Outline: Preliminaries:Real-number system, complex numbers, introduction to sets, set operations, functions, types of functions. Matrices: Introduction to matrices, types, matrix inverse, determinants, system of linear equations, Cramer‟s rule. Quadratic Equations:Solution of quadratic equations, qualitative analysis of roots of a quadratic equations, equations reducible to quadratic equations, cube roots of unity, relation between roots and coefficients of quadratic equations. Sequences and Series:Arithmetic progression, geometric progression, harmonic progression. Binomial Theorem: Introduction to mathematical induction, binomial theorem with rational and irrational indices. Trigonometry: Fundamentals of trigonometry, trigonometric identities. Recommended Books: 1. Dolciani MP, Wooton W, Beckenback EF, Sharron S, Algebra 2 and Trigonometry, 1978, Houghton & Mifflin, Boston (suggested text) 2 Kaufmann JE, College Algebra and Trigonometry, 1987, PWS-Kent Company, Boston 3.Swokowski EW, Fundamentals of Algebra and Trigonometry (6th edition), 1986, PWS-Kent Company, Boston 2. MATH-371 BASIC CALCULUS

Prerequisite: MATH-321 BASIC ALGEBRA Credit Hours: 03+0

Specific Objectives of the Course: To prepare the students, not majoring in mathematics, with the essential tools of calculus to apply the concepts and the techniques in their respective disciplines.

Course Outline: Preliminaries: Real-number line, functions and their graphs, solution of equations involving absolute values, inequalities. Limits and Continuity: Limit of a function, left-hand and right-hand limits, continuity, continuous functions. Derivatives and their Applications: Differentiable functions, differentiation of polynomial, rational and transcendental functions, derivatives. Integration and Definite Integrals: Techniques of evaluating indefinite integrals, integration by substitution, integration by parts, change of variables in indefinite integrals. 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. Swokowski EW, Calculus and Analytic Geometry, 1983, PWS-Kent Company, Boston

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

edition), 2005, Addison-Wesley, Reading, Ma, USA 3. MATH-421 ANALYTIC GEOMETRY

Prerequisite: MATH-371 BASIC CALCULUS Credit Hours: 03+0

Specific Objectives of the Course: To prepare the students, not majoring in mathematics, with the essential tools of geometry to apply the concepts and the techniques in their respective disciplines Course Outline: Geometry in Two Dimensions: Cartesian-coördinate mesh, slope of a line, equation of a line, parallel and perpendicular lines, various forms of equation of a line, intersection of two lines, angle between two lines, distance between two points, distance between a point and a line. Circle: Equation of a circle, circles determined by various conditions, intersection of lines and circles, locus of a point in various conditions. Conic Sections: Parabola, ellipse, hyperbola, the general-second-degree equation

Recommended Books:

1. Abraham S, Analytic Geometry, Scott, Freshman and Company, 1969 2. Kaufmann JE, College Algebra and Trigonometry, 1987, PWS-Kent Company, Boston

3. Swokowski EW, Fundamentals of Algebra and Trigonometry (6th

edition), 1986, PWS-Kent Company, Boston

4. COURSE FOR NON-MATHEMATICS MAJORS IN SOCIAL SCIENCES MATH-324ELEMENTRY ALGEBRA & STATISTICS

Discipline: BS (Social Sciences). Pre-requisites: SSC (Metric) level Mathematics Credit Hours: 03+0 Minimum Contact Hours: 40

Assessment:written examination; Effective:2008 and onward Aims: To give the basic knowledge of Mathematics and prepare the students not majoring in mathematics. Objectives: After completion of this course the student should be able to: • Understand the use of the essential tools of basic mathematics; • Apply the concepts and the techniques in their respective disciplines; • Model the effects non-isothermal problems through different domains Contents:

1. Algebra: Preliminaries: Real and complex numbers, Introduction to sets, set operations, functions, types of functions. Matrices: Introduction to matrices, types of matrices, inverse of matrices, determinants, system of linear equations, Cramer‟s rule. Quadratic equations: Solution of quadratic equations, nature of roots of quadratic equations, equations reducible to quadratic equations. Sequence and Series: Arithmetic, geometric and harmonic progressions. Permutation and combinations: Introduction to permutation and combinations, Binomial Theorem: Introduction to binomial theorem. Trigonometry: Fundamentals of trigonometry, trigonometric identities. Graphs: Graph of straight line, circle and trigonometric functions.

2. Statistics: Introduction: Meaning and definition of statistics, relationship of statistics with social science, characteristics of statistics, limitations of statistics and main division of statistics. Frequency distribution: Organization of data, array, ungrouped and grouped data, types of frequency series, individual, discrete and continuous series, tally sheet method, graphic presentation of the frequency distribution, bar frequency diagram histogram, frequency polygon, cumulative frequency curve. Measures of central tendency: Mean medium and modes, quartiles, deciles and percentiles. Measures of dispersion: Range, inter quartile deviation mean deviation, standard deviation, variance, moments, skewness and kurtosis.

Books Recommended:

1. Swokowski. E. W., „Fundamentals of Algebra and Trigonometry‟, Latest Edition. 2. Kaufmann. J. E., „College Algebra and Trigonometry‟, PWS-Kent Company, Boston,

Latest Edition. 3. Walpole, R. E., „Introduction of Statistics‟, Prentice Hall, Latest Edition. 4. Wilcox, R. R., „Statistics for The Social Sciences’,

1. MATHEMATICS FOR CHEMISTRY Follow the course MATH-371. BASIC CALCULUS, Because the course outline are same. MATH-325 ELEMENTRY CALCULUS & STATISTICS

Credit Hours: 03+0 Course Objective To help the students to get knowledge about various mathematical and statistical concepts and techniques used in geology. Course Contents Mathematics Indices and logarithm, and their application, the principles of algebra, solution of quadratic equation, solution of two simultaneous equations, both linear, one linear one quadratic, both quadratic, basic trigonometry definition, Trigonometric ratios of general angle, Trigonometric identities, Multiple angle and half angle formula, Sum and difference formula, Graph of trigonometric functions, Inverse trigonometric functions, Coordinate geometry, Coordinates, Change of coordinates, Graph-Log and exponential, The straight line, distance between two points, Circle, Parabola, Differential calculus, Limits, Definition and properties of limits, Continuity, Derivatives, Rules for differentiation (algebraic, logarithmic, exponential, and inverse functions), Integration, Introduction, Integration of algebraic, trigonometric, exponential functions, and their combinations, Integration by substitution, Integration by parts, Differential equations, Definition and classification of differential equations of Ist order and Ist degree, Solution of ordinary differential equations and of second order equations with constant coefficients Statistics Descriptive statistics, The meaning of statistics The role of statistics in Geology Limitations and characteristics of statiticsGrouped and ungrouped data, Frequency distribution, Relative and cumulative frequency distribution, Histogram, Frequency polygon and frequency curve, Cumulative frequency polygon and cumulative frequency curve, Measures of central tendency(A.M., G.M.H., Median, Mode, Percentile measures variability, range, quartile deviation, mean deviation, standard deviation, coefficient of

deviation), Statistical sampling study, Sample and population, Need of samples, Designing and conducting the sampling study, Simple stratified and systematic sampling (theoretical approach only) Recommended Books:

1. Basic Concepts of Mathematics, by Elias Zakon, ISBN 1-931705-00-3, published by The Trillia Group, 2001.

2. New Mathematics and Applied Mathematics Books July - August 2000 HG6024.A3.W554 1995 - Wilmott, Paul. Mathematics of financial derivatives: a student introduction. Cambridge University Press, Oxford; New York S-BKS. Elementary Statistics, Ninth Edition by Mario.F 1995.

3. Mathematics by S. M. Yousaf. 4. Statistics by Bhattey

2. MATHEMATICS FOR PHYSICS

MATH-323 Advanced Calculus

Contents 1. Preliminary calculus.

• Differentiation Differentiation from first principles; products; the chain rule; quotients; implicit Differentiation; logarithmic differentiation; Leibnitz‟ theorem; special points of a Function; theorems of differentiation. • Integration Integration from first principles; the inverse of differentiation; integration by inspection; sinusoidal function; logarithmic integration; integration using partial fractions; substitution method; integration by parts; reduction formulae; infinite and improper integrals; plane polar coordinates; integral inequalities; applications of integration.

2. Complex numbers and hyperbolic functions

• The need for complex numbers • Manipulation of complex numbers Additions and subtraction; modulus and

argument; multiplication; complex conjugate; division • Polar representation of complex numbers Multiplication and division in polar

form • de Moivre‟s theorem Trigonometrical identities; finding the nth roots of unity;

solving polynomial equations • Complex logarithms and complex powers • Applications to differentiation and integration • Hyperbolic functions

Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions

3. Series and limits • Series • Summation of series

Arithmetic series; geometric series; arithmetic-geometric series; the difference method; series involving natural numbers; transformation of series

• Convergence of infinite series Absolute and conditional convergence; convergence of a series containing only real positive terms; alternating series test

• Operations with series • Power series Convergence of power series; operations with power series • Taylor series

Taylor‟s theorem; approximation errors in Taylor series; standard Maclaurin series

• Evaluation of limits

4. Partial differentiation • Definition of the partial derivative • The total differential and total derivative • Exact and inexact differentials • Useful theorems of partial differentiation • The chain rule • Change of variables • Taylor‟s theorem for many-variable functions • Stationary values of many-variable functions • Stationary values under constraints

5. Multiple integrals

• Double integrals • Triple integrals • Applications of multiple integrals Areas and volumes; masses, centers of

mass and centroids; Pappus‟ theorems; moments of inertia; mean values of functions

• Change of variables in multiple integrals, Change of variables in double integrals;

6. Vector algebra • Scalars and vectors • Addition and subtraction of vectors

• Multiplication by a scalar • Basis vectors and components • Magnitude of a vectors • Multiplication of vectors

Scalar product; vector product; scalar triple product; vector triple product • Equations of lines and planes • Using vectors to find distances

Point to line; point to plane; line to line; line to plane • Reciprocal vectors

7. Matrices and vector spaces

• Vectors spaces, Basic vectors; the inner product; some useful inequalities • Matrices • The complex and Hermitian conjugates of a matrix • The determinant of a matrix

Properties of determinants • Special square matrices • The inverse of a matrix • The rank of a matrix • Simultaneous linear equations

N simultaneous linear equations in N unknowns Diagonal; symmetric and Skew-symmetric; orthogonal; Hermitian; unitary normal

• Eigen vectors and Eigen values Of a normal matrix; of Hermitian and Skew-Hermitian matrices; of a unitary matrix; of a general square matrix

• Determination of Eigenvalues and Eigenvectors Degenerate Eigenvalues

8. Vector calculus

• Differentiation of vectors Composite vector expressions; differential of a vector

• Integration of vectors • Space curves • Vector functions of several arguments • Surfaces • Scalar and vector fields • Vector operators

Gradient of a scalar field; divergence of a vector field; curl of a vector field • Vector operator formulae

Vector operators acting on sums and products; combinations of grad, div and curl

• Cylindrical and spherical polar coordinates Cylindrical polar coordinates; spherical polar co

ELECTIVE COURSES MATH-607 ADVANCED CALCULUS-I (NUMERICAL SOLUTION OF PDE) Credit Hours: 03+0 Course Outline: Numerical Methods for Parabolic PDEs; finite difference methods, explicit methods, Crank-Nicolson implicit method, Local Truncation Error, Stability, Consistency and convergence, Fourier stability methods, alternating directions implicit method, higher level schemes, nonlinear equations, predictor corrector methods, computer problems, computer problems. Numerical methods for hyperbolic PDEs; computer implementations. Numerical Methods for Elliptic PDEs; finite-difference methods, Poisson Equation, Laplace Equations, Curved boundary, finite-differences in Polar co-ordinates.

RECOMMENDED BOOKS 1. C. Jhonson, Numerical Solutions of Partial Differential Equations by the finite

methods, Cambridge University Press 2. W. F. Ames, Numerical methods for P.D.Es, Academic Press. 3. G. D. Smith, Numerical Solutions of P.D.Es finite difference methods, Clarendon

Press, Oxford. 4. G. W. Thomas, Numerical Solutions of P.D.E‟s. MATH- 656 ADVANCED CALCULUS-II (INTEGRAL EQUATION) Credit Hours: 03+0 Course Outline: Classification of integral equations. Voltera integral equations. Relation between linear differential equations and Voltera's integral equations. Solution of the integral equation of second kind in series. The method of successive approximation and substitution. Method of Laplace Transform. Iterated Kernels. Reciprocal kernel. Voltera's solution of the Fredholm's equation. Fredholm's two fundamental relations. Fredholm's solution of

the integral equation when D()=0, Solution of the homogeneous equation when

D()=0, D/() 0, Solution of the homogeneous integral equation when D()=0. Characteristic constants and fundamental functions. Associated homogeneous integral

equation. Kernels of the form ai(x) bi(y) Existence of at least one characteristic constant for a symmetric kernel. RECOMMENDED BOOKS: 1. W. V. Lovitt, Linear Integral Equations, Dover Pub. Inc., 1950. 2. A. J. Jerri, Introduction to Integral Equations with Applications, Marcel Dekker Inc.,

1999. 3. R. P. Kanwal, Linear Integral Equation, Academic Press.N.Y., 1996. 4. H. Hochstadt, Integral Equations, John Wiley N.Y., 1973.

MATH-608 ADVANCE NUMERICAL ANALYSIS I Credit Hours: 03+0 Course Outline: Numerical solution of Difference Equations: Formation of difference equations, numerical solution of linear homogeneous and non-homogeneous difference equations with constant coefficients. Numerical solution of Ordinary Differential Equations: Initial Value Problems; Taylor's methods and truncation error. Euler‟s method and error estimates. Modified Euler method. Runge-Kuttamethods. Predictor and corrector methods; Milne-Simpson method, Adams-Bashforth-Moulton method and Hamming's method. RECOMMENDED BOOKS: 1. W. A. Smith, Elementary Numerical Analysis, Harper & Row Pub. Int., 1979. 2.C. E. Froberg, Introduction to Numerical Analysis, Addsion-Wesley Co., 1974. 3.M. K. Jain, Numerical Methods for Scientific and Engineering Comp., Wiley E.Ltd, 1985. 4.R. L. Burden, J. D. Faires, Introduction to Numerical Analysis. 8th Ed, 2004. MATH-657 ADVANCE NUMERICAL ANALYSIS-II Credit Hours: 03+0 Course Outline: Interpolation: introduction; efficient evaluation of the interpolating polynomial; interpolating at equally spaced points; polynomial interpolation with derivative data; interpolation by cubic splines; interpolation by trigonometric functions. Orthogonal polynomials and least-squares approximation: least square polynomial/approximation; polynomial approximation by use of orthogonal polynomials; approximation with trigonometric functions; with exponential functions; and their error estimates, solution of non-linear equations: newton‟s methods for systems of equations. RECOMMENDED BOOKS: 1. W. A. Smith, Elementary Numerical Analysis, Harper & Row Pub. Int., 1979. 2.C. E. Froberg, Introduction to Numerical Analysis, Addsion-Wesley Co., 1974. 3.M. K. Jain, Numerical Methods for Scientific and Engineering Comp., Wiley E.Ltd, 1985. 4.R. L. Burden, J. D. Faires, Introduction to Numerical Analysis. 8th Ed, 2004.

MATH-626 ANALYTICAL MECHANICS-I Credit Hours: 03+0 Course Outline: Generalized Co-ordinates. Constraints. Holonomic and Non-holonomic system. Conservation theorem. Principle of virtual work. D'Alembert's principle. Konning theorem. Euler dynamical equation of motion of rigid body about a fixed point. Properties of rigid body under no force. The rotating earth. Lagrange's equation for holonomic and non-holonomic system and their application. Energy Integral. Application of Noether's theorem. Equation of motion with or without Lagrange multipliers. Lagrange‟s equation for impulsive force. Motion of a symmetrical top. Eulerian angles. RECOMMENDED BOOKS: 1. H. Goldstein, Classical Mechanics, Addison Wesley Co., 1987. 2. F. Chorlton, Text Book of Dynamics, John Wiley & Sons, 1983. 3. L. A. Pars, A Treatise on Analytical Dynamics, Cambridge Uni. Press. 4. E. T. Whittaker, A Treatise on Analytical Dynamics of Particles and Rigid Bodies,

CUP, 1970. MATH-676 ANALYTICAL MECHANICS-II Credit Hours: 03+0

Course Outline: Hamiltonian theory, Cyclic co-ordinates and Routh's procedure. Variational principle. Canonical transformation. Lagrange's and Poisson's brackets. Hamilton Jacobi theory. Theory of small oscillations. Normal co-ordinates and normal modes of vibration. RECOMMENDED BOOKS: 1. H. Goldstein, Classical Mechanics, Addison Wesley Co., 1987. 2. F. Chorlton, Text Book of Dynamics, John Wiley & Sons, 1983. 3. Gupta & Satya, Classical Mechanics, KRN, Meerath. 4. L. A. Pars, A Treatise on Analytical Dynamics, Cambridge Uni. Press. 1. E. T. Whittaker, A Treatise on Analytical Dynamics of Particles and Rigid Bodies.

CUP, 1970. MATH-627 DIFFERENTIAL GEOMETRY-I Credit Hours 03

Course Outline:

Historical background; Motivation and applications. Index notation and summation

convention; Space curves; The tangent vector field; Arc length; Curvature; Principal

normal; Binormal; Torsion; The osculating, the normal and the rectifying planes; The

Frenet-Serret Theorem; Spherical images; Sphere curves; Spherical contacts;

Fundamental theorem of space curves; Local surface theory; Coordinate

transformations; The tangent and the normal planes; Parametric curves; Normal and

geodesic curvatures; Gauss‟s formulae; Christoffel symbols of first and second kinds;

Principal, Gaussian, Mean and Normal curvatures; Isometries and the fundamental

theorem of surfaces.

RECOMMENDED BOOKS:

1. R. S. Millman, G. D. Parker, Elements of Differential Geometry, Prentice-Hall Inc., New Jersey, 1977.

2. D. J. Struik, Lectures on Classical Differential Geometry, Addison-Wesley Publishing Company, Inc., Massachusetts, 1977.

3. M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood, New Jersey, 1985.

4. B. O. Neil, Elementary Differential Geometry, Academic Press, 1966. 5. A. Goetz, Introduction to Differential Geometry, Addison-Wesley, 1970. 6. F. Charlton, Vector and Tensor Methods, Ellis Horwood, 1976.

MATH-677 DIFFERENTIAL GEOMETRY-II Credit Hours 03 Course Outline: Definition and examples of manifolds; Differential maps; Sub-manifolds; 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 connections; 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. M. P. Do Carmo, , Riemannian Geometry, Birkhauser, Boston, 1992. 3. D. Lovelock, H. Rund, Tensors., Differential Forms and Variational Principles, John-

Willey, 1975. 4. D. Langwitz, , Differential and Riemannian Geometry, Academic Press, 1970. 5. R. Abraham, J. E. Marsden, T. Ratiu, Manifolds, Tensor Analysis and Applications,

Addison-Wesley, 1983.

MATH-611 FLUID DYNAMICS-I Credit Hours 03 Course Outline: Properties of Liquids and Gas:Ideal and real fluids; properties and gases; viscosity and compressibility of fluids; fluid pressure. Fluid Dynamics: One dimensional in-viscid flow (flow filament theory); equation of continuity; Euler‟s equations of motion; Bernoulli‟s equation; impulse and momentum. one dimensional viscous flow; generalized Bernoulli‟s equation; laminar and turbulent flow in circular pipes; pipe flow problems; flow in open channels.

RECOMMENDED BOOKS: 1. Chorlton, F., Textbook of fluid Dynamics, D. Van Nostrand Co. Ltd. 1967. 2. Thomson, M., Theoretical Hydrodynamics, Macmillan Press, 1979. 3. Jaunzemics, W., Continuum Mechanic, Machmillan Company, 1967. 4. Landau, L.D., and Lifshitz, E.M., Fluid Mehanics, Pergamon Press, 1966. 5. Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge University Press,1969.

MATH-660 FLUID DYNAMICS-II Credit Hours 03

Course Outline:

Constitutive equations; Navier-Stoke‟s equations; Exact solutions of Navier-Stoke‟s

equations; Steady unidirectional low; Poiseuille flow; Couette flow; Unsteady

unidirectional low; sudden motion of a plane boundary in a fluid at rest; Flow due to an

oscillatory boundary; Equations of motion relative to a rotating system; Ekman flow;

Dynamical similarity and the Reynold‟s number; Flow over a flat plate (Blasius‟ solution);

Reynold‟s equations of turbulent motion.

RECOMMENDED BOOKS:

1. L.D. Landau and E.M. Lifshitz., Fluid Mechanics, Pergamon Press, 1966.

2. Batchelor, G.K. , An Introduction to Fluid Dynamics, Cambidge University

Press,1969.

3. Walter Jaunzemis, Continuum Mechanics, MacMillan Company, 1967.

4. Milne-Thomson, Theoretical Hydrodynamics, MacMillan Company, 1967.

MATH-612 GROUP THEORY-I

Credit Hours 03

Course Outline:

Summery or a review of the fundamental Group theory, Center of a group, normalizers

and centralizers of a subset of a group, derived subgroup of a group, homomorphism

and isomorphism between groups, Homomorphism and isomorphism theorems, finite p-

groups, internal and external direct products, Sylow theorems and its applications.

Normal series and subnormal series of a group, refinement theorem, composition

series and Jordan Holder theorem, abelian Series, derived series, solvable groups and

solvable radical, nilpotent groups,upper and lower series and nilpotency length.

RECOMMENDED BOOKS:

1. J. B. Fraleigh, A First Course in Algebra, Addison Wesley Co., 1976.

2. I. N. Herstein, Topics in Algebra, Ginn & Co.

3. A. Majeed, Theory of Groups, University Grant Commission.

MATH-661 GROUP THEORY-II

Credit Hours 03

Course Outline:

Extension of Fields Elementary properties, Simple extensions, algebraic extensions,

Factorization ofpolynomials, splitting fields, algebraically closed fields, separable

extensions. Galois Theory,Automarphishs of fields, Normal extensions, the

fundamentals theorem of Galois Theory, Norms andtraces , The primitive element

theorem, Lagrange‟s Theorem, Nom 1al basis. Applications finite fieldscyclotomic

extensions of rational number field, cyclic extensions. Wedderburn‟s Theorem, Ruler

andCompasses construction, solution by radicals.

RECOMMENDED BOOKS:

1. Field extensions and Galois Theory by Julio R. Bastida.

2. Galois theory by David A. Cox.

3. Fields and Galois Theoryby John Mackintosh Howie.

4. Galois' theory of algebraic equations by Jean-Pierre Tignol.

MATH-618 OPERATIONS RESEARCH-I

Credit Hours 03

Course Outline:

Introduction: Formulation and graphical solution of two variables linear programming,

Simplex method, Method of Penality, the two-phase technique, Sensitivity analysis,

Dual Simplex method, Duality, Sensitivity and Parametric analysis, Transportation and

Assignment Models. Linear Programming: Advanced Topics.

RECOMMENDED BOOKS:

1. H. A. Taha, Introduction to Operations Research , 3rd Ed. McMillan Pub. Company

N.Y., 1982.

2. G. Hadley, Introduction to Linear Programming.

MATH-667 OPERATIONS RESEARCH-II

Credit Hours 03

Course Outline:

Matric definition of the standard LP problem, Foundation of Linear Programming,

Revised simplex Method, Bounded variables, Decomposition Algorithm, Parametric

Linear Programming. Application of Integer Programming, Cutting Methods, the

Fractional (pure Integer) algorithm, mixed algorithm, Game theory, graphical solutions

of two-person zero-sum games, mixed strategies, Graphical solution of (2xn) and

(mx2) games, Solution of (mxn) games by linear programming.

RECOMMENDED BOOKS:

1. H. A. Taha, Introduction to Operations Research , 3rd Ed. McMillan Pub. Company

N.Y., 1982.

2. H. Lieberman, Introduction to Operations Research, 8th Edition McGraw Hill Co.

MATH-620 RELATIVITY-I

Credit Hours 03

Course Outline

Historical background. Gallillean transformations. The postulates of special relativity.

Lorentz transformations. Simultaneity in special relativity. Length contraction and time

dilation. Velocity addition formulae. The four-vector formulism and Minkowski space.

Time four-vector and energy momentum four-vector. Equivalence of mass and energy.

Relativistic Kinematics. Doppler's effect.

RECOMMENDED BOOKS:

1. A. Bazin, Shiffer, Introduction to General Relativity, McGraw Hill Co., 1965.

2. Sears & Brehme, Introduction to the Theory of Relativity, Addison Wesley, 1968.

3. Derek & Lawden, An Introduction to Tensor & Relativity Science Paper Books.

4. W. Rindler, Essential Relativity.

5. A. P. French, Special Relativity, The ELBS Nelson.

1. A. Qadir, Relativity: An Introduction to the Special Relativity, World Scientific, 1989.

MATH-669 RELATIVITY-II

Credit Hours 03

Course Outline:

Particle production and particle decay. Threshold and binding energies. Accelerational

effects in special relativity. The principles of equivalence and gravitation effects in

special relativity. The clock paradox. Tachyaons. Electromagnetism in relativity.

RECOMMENDED BOOKS:

1. A. Bazin, Shiffer, Introduction to General Relativity, McGraw Hill Co., 1965.

2. Sears & Brehme, Introduction to the Theory of Relativity, Addison Wesley, 1968.

3. Derek & Lawden, An Introduction to Tensor & Relativity Science Paper Books.

4. W. Rindler, Essential Relativity.

5. A. P. French, Special Relativity, The ELBS Nelson.

6. A. Qadir, Relativity: An Introduction to the Special Relativity, World Scientific, 1989.