University School of Sciences M.SC. Mathematics 1st sem ...basicsciences.rayatbahrauniversity.edu.in/Curriculam/M.Sc Mathema… · 5 MS7305 Special Functions 5 0 0 5 5 6. MS7306 UGC

University School of Sciences M.SC. Mathematics 1st sem syllabus 2015

Rayat Bahra University

University School of Sciences

Study scheme and Syllabus

Batch 2015 Onwards

Programme : Sciences

Level : Postgraduate

Course : M.Sc. Mathematics



(Semester-1)

(Semester-2)

M.Sc. Mathematics Semester:I

S.No. Subject Code Subject Name Teaching Schedule

L T P Total Credits

1 MS6101 Real Analysis 5 0 0 5 5

2 MS6102 Groups and Rings 5 0 0 5 5

3 MS6103 Linear Programming 5 0 0 5 5

4 MS6104 Complex Analysis-I

5 0 0 5 5

5 MS6105 Integral Equations 5 0 0 5 5

6 MS6106 UGC Pattern(OTQ) Self Study 0 0 0 0 2

Total 25 0 0 25 27

M.Sc. Mathematics Semester:II

S.No Subject

Code

Subject Name Teaching Schedule Credits

L T P Total

1 MS6201 Functional Analysis 5 0 0 5 5

2 MS6202 Rings and Modules 5 0 0 5 5

3 MS6203 Non-Linear Programming 5 0 0 5 5

4 MS6204 Complex Analysis-II 5 0 0 5 5

5 MS6205 Boundary Value Problems 5 0 0 5 5

6 MS6206 UGC Pattern(OTQ) Self Study 0 0 0 0 2

Total 25 0 0 25 27



(Semester-3)

(Semester-4)

M.Sc. Mathematics Semester:III

S.No Subject Code Subject Name Teaching Schedule Credits

L T P Total

1 MS7301 Fluid Mechanics-I 5 0 0 5 5

2 MS7302 Differential Geometry-I 5 0 0 5 5

3 MS7303 Elasticity-I 5 0 0 5 5

4 MS7304 Topology 5 0 0 5 5

5 MS7305 Special Functions 5 0 0 5 5

6. MS7306 UGC Pattern(OTQ)Self study 0 0 0 0 2

7. MS7307 Project Mathematics-I 0 0 0 0 2

Total 25 0 0 25 29

M.Sc. Mathematics Semester:IV

S.No Subject Code Subject Name Teaching Schedule Credits

L T P Total

1 MS7401 Fluid Mechanics-II 5 0 0 5 5

2 MS7402 Differential Geometry-II 5 0 0 5 5

3 MS7403 Elasticity-II 5 0 0 5 5

4 MS7404 Integral Transforms and their

Applications

5 0 0 5 5

5 MS7405 Probability & Mathematical

Statistics

5 0 0 5 5

6. MS7406 UGC Pattern(OTQ)Self study 0 0 0 0 2

7 FS7407 Soft Skills 2 0 0 2 2

8 MS7408 Project Mathematics-II 0 0 0 0 2

Total 27 0 0 27 31



Syllabus

Semester-1



M.Sc. Mathematics (Semester –I) SYLLABUS

Sub Code Subject Name L T P C

MS 6101 Real Analysis 5 0 0 5

Course Objectives The aim of this course is to make the students learn fundamental concepts of metric spaces, the

Riemann-Stieltjes integral as a generalization of Riemann Integral, the calculus of several

variables and basic theorems.

UNIT-01 (12HRS)

Basic Topology: Finite, countable and uncountable sets, metric spaces, compact sets, perfect

sets, connected sets.

UNIT-02 (10 HRS)

Sequences and series: Convergent sequences, subsequences, Cauchy sequences(in metric

spaces), completion of a metric space, absolute convergence, addition and multiplication of

series, rearrangements of series of real and complex numbers.

UNIT-03 (10HRS)

Continuity: Limits of functions (in metric spaces), continuous functions, continuity and

compactness, continuity and connectedness, monotonic functions. The Riemann-Stieltjes

integral: Definition and existence of the Riemann-Stieltjes integral, properties of the integral,

integration of vector-valued functions, rectifiable curves.

UNIT-04 (10 HRS)

Sequences and series of functions: Problem of interchange of limit processes for sequences of

functions, Uniform convergence, Uniform convergence and continuity, Uniform convergence

and integration, Uniform convergence and differentiation, equicontinuous families of functions,

Stone Weierstrass Theorem.

Learning Outcomes:

Upon completing this course students will

1. Determine the basic topological properties of subsets of the real numbers,

2. Determine the continuity, differentiability, and integrability of functions defined on

subsets of the real line

3. Determine the Riemann integrability and the Riemann-Stieltjes integrability of a bounded

function and prove a selection of theorems concerning integration,

4. Illustrate the effect of uniform convergence on the limit function with respect to

continuity, differentiability, and integrability.



Text Book :

01. Shanti Narayan, ‘A Course of Mathematical Analysis’, S.Chand and Co. Ltd., New

Delhi, 12th Revised Edition 1986.

Reference Books:

01. Goldberg, R.R.: Methods of Real Analysis, Oxford and IHB Publishing Company,

New Delhi.

02. Knopp, K.: ‘Theory and Applications of Infinite series’, Blackie and Sons Ltd. London

and Glasgow,2nd Edition 1951 (Reprinted 1957).

03 Rudin, Walter: ‘Principles of Mathematical Analysis’. 3rd Edition (International Student

dition) McGraw-Hill Inc. 1976





MS 6102 Groups and Rings 5 0 0 5

Course Objectives This course covers some advanced topics of Group and Ring Theory, which are two most

important branches of Algebra. It also covers classification of groups, rings, integral domains

and field

UNIT-01 (10HRS)

Review of basic property of Groups, Dihedral groups, Symmetric groups and their congugacy

classes, Simple groups and their examples. Simplicity of )5( nAn , Sylow Theorems and their

applications,

UNIT-02 (12HRS)

Direct Products, finite Abelian Groups, Fundamental Theorem on Finite Abelian Groups,

Normal and Subnormal Series, Derived Series, Composition Series, Solvable Groups,

Zassenhaus Lemma, Scherier’s refinement theorem and Jordan-Holder Theorem.

UNIT-03 (10HRS)

Review of Rings, Zero Divisors, Nilpotent Elements and Idempotents, Matrices, Quaternions,

Ring of endomorphisms, polynomial rings in many variables

UNIT-04 (10HRS)

Factorization of polynomials in one variable over a field. Unique factorization domains. Gauss

Lemma, Eisenstein’s Irreducibility Criterion, Unique Factorization in ][xR , Euclidean and

Principal ideal domains.

Learning Outcomes:


1. Assess properties implied by the definitions of a group and rings,

2. Use various canonical types of groups (including cyclic groups and groups of permutations)

and canonical types of rings (including polynomial rings and modular rings),

3. Analyze and demonstrate examples of subgroups, normal subgroups and quotient groups

4. Produce rigorous proofs of propositions arising in the context of abstract algebra

Text Book

01. V.K.Khanna & S.K.Bhambri “A Course in Abstract Algebra” Vikas Publishing House

Pvt Limited

Reference Books:

01. I. N. Herstein: “Topics in Algebra” Wiley Eastern Limited, New Delhi.

02. Surjeet Singh and Q. Zameeruddin, “Modern Algebra” Vikas Publishing House, New

Delhi, 1993.

03. B. Hartley and T. O. Hawkes: “Rings, Modules and Linear Algebra” Chapman and Hall.




Sub Code Subject Name L P T C

6103 Linear Programming 5 0 0 5

Course Objectives The objective of this course is to acquaint the students with the concept of convex sets, their

properties and various separation theorems so as to tackle with problems of optimization of

functions of several variables over polyhedron and their duals. The results, methods and

techniques contained in this paper are very well suited to the realistic problems in almost every

area.

UNIT-01 (10HRS)

Linear Programming and examples, Convex Sets, Hyperplane, Open and Closed half-spaces,

Feasible, Basic Feasible and Optimal Solutions, Extreme Point & graphical methods

UNIT-02 (10HRS)

Simplex method, Charnes-M method, Two phase method, Determination of Optimal solutions,

unrestricted variables, Duality theory, Dual linear Programming Problems, fundamental

properties of dual Problems, Complementary slackness, Unbounded solution in Primal. Dual

Simplex Algorithm, Sensitivity analysis.

UNIT-03 (10HRS)

Parametric Programming, Revised Simplex method, Transportation Problems,Balanced and

unbalanced Transportation problems, U-V method, Paradox in Transportation problem,

Assignment problems,

UNIT-04 (10HRS)

Integer Programming problems:Pure and Mixed Integer Programming problems,0-1

programming problem, Gomary’s algorithm, Branch & Bound Technique.Travelling salesman

problem

Learning Outcome:

Upon successful completion of the course a student will :

1. Formulate and model a linear programming problem from a word problem and solve

them graphically in 2 and 3 dimensions, while employing some convex analysis,

2. Place a Primal linear programming problem into standard form and use the Simplex

Method or Revised Simplex Method to solve it,

3. Find the dual, and identify and interpret the solution of the Dual Problem

4. Be able to modify a Primal Problem, and use the Fundamental Insight of Linear

Programming to identify the new solution, or use the Dual Simplex Method to restore

feasibility,

Text Book

01. Kanti Swaroop & P.K.Gupta “ Operations Research” Sultan Chand & Sons, New Delhi

References:

01. N.S. Kambo: “Mathematical Programming Techniques” Affiliated East-West

Press Pvt.Ltd. New Delhi.



02. Suresh Chandra,Jayadeva and Aparna Mehra: “Numerical Optimization with

Applications”, Narosa Publishing House,New Delhi.

03. S.M.Sinha: “Mathematical Programming,Theory and Methods” Elsevier.

04. G.Hadley: “Linear Programming”, Narosa Publishing House,New Delhi.





6104 Complex Analysis-I 5 0 0 5

Course Objectives The objective of the course is to provide foundation for Complex Analysis. The students are

expected to understand complex numbers algebraically and geometrically, define and analyze

limits and continuity for complex functions as well as consequences of continuity. This course is

in accordance with UGC NET syllabus guidelines.

UNIT-01 (1HRS)

Complex plane, geometric representation of complex numbers, joint equation of circle and

straight line, stereographic projection and the spherical representation of the extended complex

plane. Topology on the complex plane, connected and simply connected sets. Complex valued

functions and their continuity. Curves, connectivity through polygonal lines.

UNIT-02 (10HRS)

Limit, continuity and differentiability of complex functions.Analytic functions, Cauchy-Riemann

equations in cartesian and polar co-ordinates. Construction of harmonic functions and Milne

Thomson method. Entire functions,Applications of analytic functions to flow problems and

velocity potentials.

UNIT-03 (10HRS)

Sets and curves in complex plane .Simply and multiply connected domains. Parametric

representations of curves. ML inequality .Complex Line integrals, Cauchy theorem,

Independence of path. Existence of Indefinite Integrals . Fundamental theorem of Integal

Calculus. Principle of path deformation. Cauchy Integral formula. Cauchy Integral formula for

higher order derivatives and multiply connected domains .

UNIT-04 (10HRS)

Converse of Cauchy Integral theorem. Liouvile’s theorem Fundamental theorem of Algebra.

Taylor and Laurrent series. Zeros and poles of analytic functions. Singularities and residues.

Cauchy Resiudes theorem .Contour Integration ..

Learning Outcome:

Upon successful completion of the course, a student will be able to:

1. Apply the concept and consequences of analyticity and the Cauchy-Riemann equations

and of results on harmonic and entire functions including the fundamental theorem of

algebra,

2. Analyze sequences and series of analytic functions and types of convergence,

3. Evaluate complex contour integrals directly and by the fundamental theorem, apply the

Cauchy integral theorem in its various versions, and the Cauchy integral formula, and



4. Represent functions as Taylor, power and Laurent series, classify singularities and poles,

find residues and evaluate complex integrals using the residue theorem.

Text Book:

01.Shanti Narayan, “ Complex Analysis” Sultan Chand & Sons, New Delhi

References :

01. D.V. Ahlfors,, “Complex Analysis,” (InternationalStudent edition), McGraw-Hill

International Book Company.

02. C.P.Gandhi, “Engineering Mathematics for Graduates” Laxmi Publications, New Delhi

03. E. T. Copson: “An Introduction to the Theory of Functions of a Complex Variable”, The

English Language Book Society and Oxford University Press,

04. S. Ponnusamy: “Foundations of Complex Analysis” by Narosa Publising House, New

Delhi

05. J.B. Conway: “Function of One Complex Variable”, Graduate Texts, Springer-Verlag,

Indian edition by Narosa Publising House, New Delhi





6105 Integral Equations 5 0 0 5

Course Objectives The main objective of this paper is to understand the behavior of integral equations and other

related problems in related mathematical sciences. Also, to apply these techniques to solve real

life problems.

Syllabus

UNIT-01 (10 HRS) Integral Equations and Boundary Value Problems: Definitions of Integral Equations and their

classification. Eigen values and Eigen functions. Fredholm integral equations of second kind

with separable kernels.Reduction to a system of algebraic equations.

UNIT-02 (10 HRS) An Approximate Method.Method of Successive Approximations. Iterative Scheme for Fredholm

Integral equations of the second kind.Conditions of uniform convergence and uniqueness of

series solution.Resolvent kernel and its results.Application of iterative Scheme to Volterra

integral equations of the Second kind.

UNIT-03 (12 HRS)

Classical Fredholm Theory: Fredholm Theorems.Integral Transform Methods. Fourier

Transform. Laplace Transform. Convolution integral.Application to Volterra integral equations

with convolution-type kernels.Abel's equations. Inversion formula for singular integral equation

with kernel of the type (h(s)-h(t)-a, 0<a<1. Cauchy's Principal Value of singular

integrals.Solution of the Cauchy-type singular integral equation.The Hilbert kernel.Solution of

the Hilbert-Type singular integral equation.

UNIT-04 (10 HRS)

Integral Equations with symmetric kernel: Symmetric kernels.Complex Hilbert

Space.Orthonormal system of functions. Fundamental properties of eigen values and eigen

functions for symmetric kernels. Expansion in eigen function and bilinear form. Hilbert Schmidt

Theorem and some immediate consequences.Solutions of integral equations with symmetric

kernels.

Learning Outcome

Upon completing this course students will ablt to

1. Apply Integral techniques to specific research problems in mathematics or other fields

2. Solve problems of mathematical models related to integral equations

3. Use to Fourier Transform, Lapalace Transform to solve different Integral Equations.

Text Book:

M.D.Raisinghania, “ Integral Equations & Boundary Value Problems, Sultan Chand & Sons,

New Delhi

References:



01. S.G. Mikhlin, “Linear Integral Equations (translated from Russian)”, Hindustan Book

Agency, 1960.

02. 3. I.N. Sneddon, “Mixed boundary value problems in potential theory”, North Holland,

1966.

03. A.J. Jerri. “Introduction to Integral Equations with Applications”. 2nd

ed. New York:

John Wiley, 1999..

04. L. Elsgolts: “Differential equations and the calculus of variations”, Mir Publication

05.R.P. Kanwal, “Linear Integral Equation. Theory and Techniques”, Academic

Press, New York.



Semester: I M. Sc (Mathematics)

Subject Code: MS6106 L T P C

2 0 0 2 Subject : UGC Pattern [Objective Type Question] Self Study Credits: 2



Syllabus

Semester-2



M.Sc. Mathematics (Semester –II) SYLLABUS


MS 6201 Functional Analysis 5 0 0 5

Course Objectives The objective of this course is to introduce Banach and Hilbert spaces. The various operators on

Hilbert and Banach spaces are also included. The contents of the course are according to the

UGC NET syllabi/UGC guidelines.

UNIT01: (10HRS)

Normed linear spaces.Banach spaces and examples.Quotient space of normed linear spaces and

its completeness, equivalent norms.Riesz Lemma, basic properties of finite dimensional normed

linear spaces and compactness. Weak convergence and bounded linear transformations, normed

linear spaces of bounded linear transformations.Baire Category theorem and its applications.

UNIT02: (10HRS)

Dual spaces with examples. Uniform boundedness theorem and some of its consequences. Open

mapping and closed graph theorems. Hahn-Banach theorem for real linear spaces, complex linear

spaces and normed linear spaces. Banach Steinhauns theorem (uniform boundedness

principle)Reflexive spaces.Weak Sequential Compactness.Compact Operators.Solvability of

linear equations in Banach spaces. The closed Range Theorem.

UNIT03: (10HRS)

Inner product spaces.Hilbert spaces.Orthonormal Sets.Bessel's inequaJity. Complete orthonormal

sets and Parseval's identity. Structure of Hilbert spaces.Projection theorem.Riesz representation

theorem.Adjoint of an operator on a Hilbert space.Reflexivity of Hilbert spaces.Self-adjoint

operators, Inner product spaces, orthonormal sets, Approximation and optimization, Positive

,normal and unitary operators.

UNIT04: (10HRS)

Bounded Operators on Hilbert spaces: Bounded operators and adjoints; normal, unitary and

isometric operators,

Learning Outcome: Upon successful completion of the course, a student will be able to:

1. Explain the fundamental concepts of functional analysis and their role in modern

mathematics and applied contexts

2. Demonstrate accurate and efficient use of functional analysis techniques

3. Demonstrate capacity for mathematical reasoning through analyzing, proving and

explaining concepts from functional analysis

4. Apply problem-solving using functional analysis techniques applied to diverse situations

in physics, engineering and other mathematical contexts

Text Book:

01. B.V. Limaye, Functional Analysis, Wiley eastern Ltd.



References:

02. S.K. Berberian: Introduction to Hilbert Spaces, (N.Y. O.W.P.), 1996.

03. A.H.Siddiqui: Functional Analysis (Tata-McGraw Hill), 1987.

04. Walter Rudin: Real and Complex Analysis(McGraw-Hill) 3rd Edition, 1986.

05. P.K.Jain, O.P. Ahujaa Khalia Ahaed:Functional Analysis ,New Age International (P) Ltd.

06. C. Goffman & G. Pedrick “First Course in Functional Analysis, PHI, India

07.F.K.Riesz and Bela Sz Nagy: Functional Analysis(N.Y.wingar)





MS 6202 Rings and Modules 5 0 0 5

Course Objectives This course is a basic course in Algebra for students who wish to pursue research work in

Algebra. Contents have been designed in accordance with the UGC syllabi in mind.

UNIT01: (12HRS)

Modules, Submodules, Quotient Modules, Free Modules, Difference between Modules and

Vector Spaces, Homomorphisms, Simple Modules, Structure Theorem of finitely generated

modules over a P.I.D., Artinian and Noetherian Modules.

UNIT02: (10HRS)

Fields, examples, characteristic of a field. Algebraic extensions, The degree of a field extension,

Adjunction of roots, splitting fields, finite fields, Algebraically closed fields,

UNIT03: (10HRS)

Separable and purely inseparable extensions. Perfect fields, primitive elements. Langrange’s

theorem on primitive elements, normal extensions

UNIT04: (10HRS)

Galois extensions, the fundamental theorem of Galois theory. Cyclotomic extensions. Cyclic

extensions, Quintic equations and solvability by radicals.

Learning Outcome:


1. Demonstrate understanding of the concepts of a field and a module and their role in

mathematics.

2. Demonstrate familiarity with a range of examples of these structures.

3. Prove the basic results of field theory and module theory.

4. Explain the structure theorem for finitely generated modules over a principal ring and its

applications to abelian groups and matrices.

Text Book :

01. C. Musili: Rings and Modules, 2nd Revised Edition, Narosa Publishing House, New

Delhi, 1994.

Reference Books:

01.I. N. Herstein: Topics in Algebra, 2nd Edition, Vikas Publishing House, New

Delhi, 1976.

02.Surjeet Singh and Q. Zameeruddin, Modern Algebra, 7th Edition, Vikas

Publishing House, New Delhi, 1993..

03.P.B. Bhattacharya, S.K. Jain and S.R. Nagpal: Basic Abstract Algebra, 2nd

Edition, Cambridge University Press,2002





MS 6203 Non Linear Programming 5 0 0 5

Course Objectives To acquaint the students with the concepts of convex and non-convex functions, their properties,

various optimatility results, techniques to solve nonlinear optimization problems and their duals

over convex and non-convex domains and also with the game theory. This is extended course

work of course MS6103.

UNIT01: (10HRS)

Nonlinear Programming: Convex functions,Concave functions,Definitions and basic

properties,subgradients of convex functions,Diffrentiable convex functions,Minima and Maxima

of convex function and concave functions. Generalizations of convex functions and their basic

properties.

UNIT02: (12HRS)

Unconstrained problems,Necessary and sufficient optimality criteria of first and second

order.First order necessary and sufficient Fritz John conditions and Kuhn-Tucker conditions for

Constrained programming problems with inequality constraints,with inequality and equality

constraints.Kuhn Tucker conditions and linear programming problems.

UNIT03: (10HRS)

Duality in Nonlinear Programming, Weak Duality Theorem, Wolfe’s Duality Theorem,

Hanson-Huard strict converse duality theorem, Dorn’s duality theorem, strict converse duality

theorem, Dorn’s Converse duality theorem, Unbounded dual theorem, theorem on no primal

minimum .Duality in Quadratic Programming.

UNIT04: (10HRS)

Quadratic programming:Wolfe’s method, Beale’s method for Quadratic programming.Linear

fractional programming,method due to Charnes and Cooper. Nonlinear fractional,programming,

Dinkelbach’s approach.Game theory - Two-person, Zero-sum Games with mixed strategies,

graphical solution, solution by Linear Programming.

Learning Outcome:


1. Estimate the actual complexity of Nonlinear Optimization problems.

2. Apply lower complexity bounds, which establish the limits of performance of optimization

method.

3. Explain the main principles for constructing the optimal methods for solving different types

of minimization problems.

4. Use the main problem classes (general nonlinear problems, smooth convex problems,

nonsmooth convex problems, structural optimization ' polynomial-time interior-point

methods).



Text Book :

01. Kanti Swarup, P.K. Gupta & Man Mohan: “Operations Research” Sultan Chand & Sons,

New Delhi 9th Edition, 2001

References:

01. Mokhtar S.Bazaraa & C.M. Shetty: “Nonlinear Programming,Theory & Algorithms,2nd

Edition,Wiley, New-York,2004

02. S.M.Sinha: “Mathematical Programming,Theory and Methods,”Elsevier,1st

Edition,2006

03. O. L. Mangasarian: “Nonlinear Programming”, TATA McGraw Hill Company

Ltd.(Bombay, New Delhi), 1st Edition, 1969.





MS 6204 Complex Analysis-II 5 0 0 5

Course Objectives This course is designed to provide follow up to Course-MS6104. This course will provide some

extended topics needed for students to pursue research in Mathematics. This course is designed

in accordance to syllabus guidelines of UGC NET.

UNIT01: (15HRS)

Bilinear and conformal mappings (transformations). Definitions and examples. Conformality of

sinz,cosz,sinhz,coshz etc.

UNIT02: (10HRS)

Maximum and Minimum Modules principle, Schwarz Lemma. Rouche’s theorem, Zeros and

poles of meromorphic functions Number of zeros and poles .Argument Principle. Infinite

products, Weierstrass theorem, Mittagleffer’s theorem, Canonical products.

UNIT03: (17HRS)

Analytic Continuation through power series(basic ideas), Natural boundary, The Gamma

function and Riemann Zeta function. Elliptic functions

UNIT-04 (10HRS)

Convergence of Power Series.Radius of convergence. Cauchy Hadmard Theorem.Power series

representations of analytic functions

Learning Outcome Upon successful completion of the course, a student will be able to:

1. Understand the significance of differentiability for complex functions and be familiar

with the Cauchy-Riemann equations;

2. Evaluate integrals along a path in the complex plane and understand the statement of

Cauchy's Theorem;

3. Compute the Taylor and Laurent expansions of simple functions, determining the nature

of the singularities and calculating residues;

4. Use the Cauchy Residue Theorem to evaluate integrals and sum series.

Text Book:

01.J.B. Conway: “Function of One Complex Variable”, Graduate Texts, Springer-Verlag,

Indian edition by Narosa Publising House, New Delhi.

References :

01.D.V. Ahlfors,, “Complex Analysis,” (InternationalStudent edition), McGraw-Hill

International Book Company.

02.C.P.Gandhi, “Engineering Mathematics for Graduates” Laxmi Publications, New Delhi

03.E. T. Copson: “An Introduction to the Theory of Functions of a Complex Variable”, The

English Language Book Society and Oxford University Press,

04.S. Ponnusamy: “Foundations of Complex Analysis” by Narosa Publising House, New

Delhi,





MS 6205 :Boundary Values Problems 5 0 0 5

Course Objectives This Course is developed to provide follow up to Course-MS6105. This course will provide

basic topics needed for students to pursue research in pure Mathematics. This course is designed

in accordance to syllabus guidelines of UGC NET.

Syllabus

UNIT-01 (10 HRS)

Introduction to Boundary Values Problems, Definition of a boundary value problem for an

ordinary differential equation of the second order and its reduction to a Fredholm integral

equation of the second kind.Dirac Delta F' Jnction.

UNIT-02 (10HRS) Greens functions and Boundary Values Problems: Green's function approach to reduce

boundary value problems of a self-adjoint differential equation with homogeneous boundary

conditions to integral equation forms. Auxiliary problem satisfied by Green's function. Integral

equation formulations of boundary value problems with more general and inhomogeneous

boundary conditions. Modified Green's function.(10 Hours)

UNIT-03 (10HRS) Integral representationof Boundary Values Problems

Integral representation formulas for the solution of the Laplace's and Poisson's

equations.Newtonian single-layer and double layer potentials. Interior and exterior Dirichlet and

Neumann boundary value problems for Laplace's equation. Green's function for Laplace's

equation in a free space as well as in a space bounded by a ground vessel.Integral equation

formulation of boundary value problems for Laplace's equation.

UNIT-04 (12 HRS) Applications of Boundary Values Problems: Poisson's integral formula., Green's function for the

space bounded by grounded two parallel plates or an infinite circular cylinder. Perturbation

techniques and its applications to mixed boundary value problems. Two-part and three-part

boundary value problems. Solutions of electrostatic problems involving a charged circular disk

and annular circular disk, a spherical cap, an annular spherical cap in a free space or a bounded

space

Learning Outcome:


1. Demonstrate a working knowledge of classical mechanics and its application to standard

problems such as central forces;

2. Understand and apply Lagrange’s equations to simple physical systems;

3. Solve dynamical problems involving classical particles by using the Lagrangian and

Hamiltonian formulation.



Text Book: M.D.Raisinghania, “ Integral Equations & Boundary Value Problems, Sultan

Chand & Sons, New Delhi

References:

01. 2. S.G. Mikhlin, Linear Integral Equations (translated from Russian), Hindustan Book

Agency, 1960.

02. 3. I.N. Sneddon, Mixed boundary value problems in potential theory, North Holland,

1966.

03.A.J. Jerri. Introduction to Integral Equations with Applications. 2nd

ed. New York:

John Wiley, 1999..

04.R.P. Kanwal, Linear Integral Equation. Theory and Techniques, Academic Press,

New York, 1971.



Syllabus

Semester-3



M.Sc. Mathematics (Semester –III) SYLLABUS


MS7301 Fluid Mechanics-I 5 0 0 5

Course Objectives The Course Objective of this course is to introduce the fluids flow and to solve the basic

problems of fluid mechanics by mathematical modeling. The contents of the course are

according to the UGC NET syllabi/UGC guidelines.

UNIT-01 (12HRS)

Real fluids and ideal fluids, velocity of fluid at a point, streamlines, pathlines, streaklines,

Velocity potential, vorticity vector, local and particle rate of change, equation of continuity,

irrotational and rotational motion, acceleration of fluid, conditions at rigid boundary.

UNIT-02 (10 HRS)

Euler’s equation of motion, Bernoulli’s equation, their applications, Potential theorems, axially

symmetric flows, impulsive motion, Kelvin’s Theorem of circulation, equation of vorticity.

UNIT-03 (10HRS)

Some three dimensional flows: sources, sinks and doublets, images in rigid planes, images in

solid sphere, Stoke’s stream function.

UNIT-04 (10 HRS)

Two dimensional flows: complex velocity potential, Milne Thomson Circle Theorem and

applications, Theorem of Blasius, vortex rows, Karman vortex street.

Learning Outcomes:


1. Give an introduction to the basic laws and principles used to describe equilibrium and

motions of fluids.

2. Apply control volume analysis, potential flow theory Bernoulli equation, to solve

problems in fluid mechanics.

3. Extend their knowledge of laminar and turbulent boundary layer fundamentals by further

research and industry work.

4. Apply the concepts developed for fluid flow analysis to issues in aerospace design.

Text Book:

01. Chorlton, F. (Text Book of Fluid Dynamics), CBS Publishers, Indian Edition, 2004.

References: 1. L.D.Landau & E. N. Lipschitz (Fluid Mechanics), 2

nd Edition, Vol.6 (Course of

Theoretical Physics).

2. G. K. Batchelor (An Introduction to Fluid Mechanics), Cambridge University

Press(1967).

3. Kundu and Cohen (Fluid Mechanics), 2003, Indian Reprint Published by

Harcourt(India) Pvt. Ltd.





MS7302 Differential Geometry-I 5 0 0 5

Course Objective This course has been designed keeping in view the importance of tensors in mechanics and other

branches of mathematical sciences. The course contents are such that the basics of tensors and

their properties are introduced and their use in differential geometry.

UNIT-01 (10 HRS)

Tensors: Notations and Summation Convention, Transformation law for vectors, Cartesian

tensors, Algebra of Cartesian tensors, Differentiation of Cartesian tensors,

UNIT-02 (12HRS)

The metric tensor, Transformation of curvilinear co ordinates, General tensors, Contravariant,

Covariant derivative of a vector, Physical components, Christoffel symbol, Relation with the

metric tensor, Covariant derivative of a tensor, Riemann – Christoffel curvature tensor.

UNIT-03 (10 HRS)

Curves with Torsion: Tangent, Principal normal, Curvature, Binormal, Torsion, Serret-Frenet

formulae, Locus of Center of curvature, Circle of curvature, torsion of a curve, Involutes,

Evolutes and Bertrand curves.

UNIT-04 (10 HRS)

Envelopes and Developable surfaces: Surfaces, Tangent plane, normal, Envelop, Edge of

regression, Developable surfaces, Curvilinear co ordinates on a surface: Fundamental

Magnitudes

Learning Outcomes: Upon completing this course students will

1. Explain the concepts and language of differential geometry and its role in modern

mathematics

2. Appreciate the distinction between intrinsic and extrinsic aspects of surface geometry

3. Understand the curvature and torsion of a space curve, how to compute them, and how

they suffice to determine the shape of the curve.

4. Explain the definition of a smooth surface, and the means by which many examples may

be constructed.

5. Understand the first and second fundamental forms of a surface, how to compute them,

and how they suffice to determine the local shape of the surface.

Text Book:

01. C. E. Weatherburn: Differential Geometry of three dimensions, RPH Calcutta, 1st Edition,

1988.



References:

01. Shanti Narayan: Text Book of Cartesian Tensors, S. Chand and Company, New Delhi,

1950.

02. E. C. Young: Vectors and Tensor Analysis, Pure and Applied Mathemaics, A program of

Monographs, Textbooks and Lecture Notes, Marcel Deccer (1994)

03. A. Goetz: Introduction to Differential Geometry: Addision Wesley Publishing Company,

(1970)





MS7303 Elasticity-I 5 0 0 5

Course Objective This course is designed to make aware the use of mathematical methods in the problems of

elasticity and wave propagation due to existing research in such field. The contents of the

curriculum has been developed keeping in view the UGC syllabus guidelines.

UNIT-01 (10 HRS)

Tensors: Summation convention, Coordinate transformation, Cartesian tensors of different

orders, Sum, product and quotient laws, Contraction, Symmetric and skew symmetric tensors,

Relation between alternate and Kronecker tensors, Eigen values and Eigen vectors of a tensor of

order two,

UNIT-02 (10 HRS)

Three scalar invariants of a tensor of order two, Eigen vectors and values of symmetric tensors,

Orthogonality of Eigen vectors and reality of Eigen values, Gradient, Divergence and Curl in

tensor notations, Gauss divergence theorem.

UNIT-03 (10 HRS)

Analysis of Strain: Affine transformation, infinitesimal affine transformation, Geometrical

interpretation of component of strain, Strain quadric of Cauchy.Analysis of Strain: Principal

strains and Invariants, general infinitesimal deformation, Example of Strain, Equations of

Compatibility, Finite deformations.

UNIT-04 (10 HRS)

Analysis of Stress: Stress tensor, Equation of equilibrium, Stress quadric of Cauchy, Principal

stress and invariants, Maximum normal and shear stress, Plane Stress, generalized plane stress,

Airy stress function, General solution of biharmonic equation, stresses and displacements in

terms of complex potentials, simple problems.

Learning Outcomes:

Upon completing this course students will able to

1. Calculate stresses, strains and tractions, and formulate boundary value problems;

2. Explain constitutive relations for elastic solids and compatibility constraints;

3. Solve various two-dimensional problems (plane strain) using the Airy stress function.

Text Book:

01. Shanti Narayan: Text Book of Cartesian Tensor, Sultan Chand and Company, N Delhi,

1950.



References:

01. I. S. Sokolnikoff: Mathematical Theory of Elasticity, Mc-Graw Hill Inc., 1956.

02. A. E. H. Love: A Treatise on Mathematical Theory of Elasticity, Dover Publications,

1944.

03. K. E. Bullen and B. A. Bolt: An Introduction to the Theory of Seismology, Cambridge

University Press, Cambridge (1985)





MS7304 Topology 5 0 0 5

Course Objectives The course is an introductory course on point-set topology. This will help the student to

understand further deeper topics in topology like Differential/Algebraic Topologies etc. This

course is designed as to help other mathematical courses.

UNIT-01 (10 HRS)

Topological Spaces, bases for a topology, the order topology, the product topology on, the

subspace topology, closed sets and limit points, continuous functions,

UNIT-02 (10HRS)

The product topology, the metric topology, the quotient topology.Connected spaces, connected

subspaces of the real line, components and local connectedness.

UNIT-03 (10 HRS)

Compact spaces, compact space of the real line, limit point compactness, local compactness,

nets.The countability axioms, the separation axioms, normal spaces,

UNIT-04 (10HRS)

The Urysohn Lemma, the Urysohn Metrization Theorem, the Tietze Extension Theorem, the

Tychonoff Theorem.

Learning Outcomes:


1. Define and illustrate the concepts of the Topological spaces, Topological Subspaces

2. Explain connectedness and compactness, and prove a selection of related theorems, and

its relation to product topology

3. Describe different examples distinguishing general, geometric, and algebraic topology.

Text Book:

01. James R. Munkers : Topology(Second Edition 2002), Prentice Hall of India.

References: 01. James Dugundji : Topology, UBS Publishers, 1

st Edition, 1990 .

02. John L. Kelley : General Topology (Van Nostrand), 1st Edition, 1955.

03. G.G. Simmons - Introduction to Topology and Modern Analysis Tokyo, McGraw Hill,

Kongakusha), 1st Edition, 1963..





MS7305 Special Functions 5 0 0 5

Course Objective This course is designed to make the students understand hyper geometric Legendre, Bessel,

Hermite and Laquerre functions and the applications of special functions. During this course

students are expected to understand the the general properties of Gamma and Beta functions;

solving linear differential equations by Laplace method; general properties of hypergeometric

equation and its solutions; classical orthogonal polynomials.

UNIT-01 (10HRS)

Hypergeometric Functions: The hypergeometric series, An integral formula for the

hypergeometric series, The hypergeometric equation, Linear relations between the solutions of

the hypergeometric equation, Relations of contiguity, The confluent hypergeometric function,

Generalised hypergeometric series.

UNIT-02 (10 HRS)

Legendre Functions: Legendre polynomials, Recurrence relations for the Legendre polynomials,

The formulae of Murphy and Roderigues, Series of Legendre polynomials, Legendre’s

differential equation, Neumann’s formula for the Legendre functions, Recurrence relations for

the functions Qn (µ),

UNIT-03 (10 HRS)

The use of Legendre functions in potential theory, Legendre’s associated functions, Integral

expression for the associated Legendre function, Surface spherical harmonics, Use of associated

Legendre functions in wave mechanics.

UNIT-04 (12HRS)

Bessel Functions: The origin of Bessel functions, Recurrence relations for the Bessel

coefficients, Series expansions for the Bessel coefficients, Integral expressions for the Bessel

coefficients, The addition formula for the Bessel coefficients, Bessel’s differential equation,

Spherical Bessel functions, Integrals involving Bessel functions, The modified Bessel functions,

The Ber and Bei functions, Expansions in series of Bessel functions, The use of Bessel functions

in potential theory, Asymptotic expansion of Bessel functions.

The Functions of Hermite And Laguerre: The Hermite polynomials, Hermite’s differential

equation, Hermite functions, the occurrence of Hermite functions in wave mechanics, The

Laguerre polynomials, Laguerre’s differential equation, The associated Laguerre polynomials

and functions, The wave functions for the hydrogen atom.

Learning Outcomes:

Upon completing this course students will be able to

1. Understand the general properties of Gamma and Beta functions;

2. Explain methods of studying asymptotic behaviour of functions;

3. Solve linear differential equations by power series; classical orthogonal polynomials



Text Book:

01. George E Andrews, Richard Askey & R. Roy, Special Functions,The University Press,

Cambridge,

References:

01. L. Andrews, Special Functions for Engineers and Applied Scientists, Macmillan, 1985.

02. N. N. Lebedev, Special Functions & Their Applications, Revised Edition, Dover, 1976.

03. W. W. Bell, Special Functions for Scientists and Engineers, Dover, 1968.



Semester: III M. Sc (Mathematics)





Semester: III M. Sc (Mathematics)


0 0 0 2 Subject : Project Mathematics-I



Syllabus

Semester-4



M.Sc. Mathematics (Semester –IV) SYLLABUS


MS7401 Fluid Mechanics-II 5 0 0 5

Course Objective The Course Objective of this course is to introduce the real problems of fluid flow through tubes

of different cross sections. The basic concepts of wave motion in fluids have been introduced to

tackle more realistic dynamical problems. The contents of the course are according to the UGC

NET syllabi/UGC guidelines. This course is designed to provide follow up to Course-MS7301.

Syllabus

UNIT-01 (10HRS)

Viscous Flows: Stress components, Stress and strain tensror, coefficient of viscosity and Laminar

flow, plane Poiseuille flows and Couette flow. Flow through tubes of uniform cross section in

the form of circle,

UNIT-02 (12HRS)

Ellipse, equilateral triangle, annulus, under constant pressure gradient.Diffusion of vorticity.

Energy dissipation due to viscosity, steady flow past a fixed sphere, dimensional analysis,

Reynold numbers, Prandtl’s boundary layer, Boundary layer equation in two dimensions,

Karman integral equation.

UNIT-03 (10HRS)

Elements of wave motion, waves in fluids, surface gravity waves, standing waves, dispersion

relation,

UNIT-04 (10HRS)

Path of particles, waves at the interface of two liquids, equipartition of energy, group velocity,

energy of propagation of waves.

Learning Outcomes:


1. Give an introduction to the Viscous Flows, Waves and can used to describe equilibrium.

2. Use the subject knowledge in other academic fields like chemical engineering, physics

,meteorology, oceanography, hydrolog.

3. Develop approximations to the exact solution by eliminating negligible contributions to

the solution using scale analysis

4. Formulate models for turbulent flow problems using Reynolds decomposition

5. Evaluate friction forces on objects using the boundary layer approximation in laminar and

turbulent flows

Text Book:

01. Chorlton, F. (Text Book of Fluid Dynamics), CBS Publishers, Indian Edition 2004.



References 01. L.D.Landau & E. N. Lipschitz (Fluid Mechanics), 2

nd Edition, Vol.6 (Course of

Theoretical Physics), 1987, Pergamon Press Ltd.

02. G. K. Batchelor (An Introduction to Fluid Mechanics), Cambridge University Press

(1967).)

03 Kundu and Cohen (Fluid Mechanics), Indian Reprinted published by Harcourt(India)

Private Ltd.(2003).





MS7402 Differential Geometry-II 5 0 0 5

Course Objective This course is the continuation of Course MS7302 on basic problems of differential

geometry. The students are get acquainted through various important formulae and results

of differential geometry and its relations to Quadric surfaces.

UNIT-01 (10HRS)

Curves on a surface: Principal directions and curvature, First and second curvature,

Euler’s theorem, Dupin theorem, Dupin’s indicatrix,

UNIT-02 (10HRS)

Normal curvature, Mean curvature, Umblic points, Conjugate directions, conjugate system,

asymptotic lines, Curvature and Torsion, Isometric lines, Null lines.

UNIT-03 (10 HRS)

Equations of Gauss and of Codazzi: Gauss’s formulae for r11, r12, r22, Gauss Cgharacteristic

equation, Mainardi-Codazzi relation, Bonnet’s theorem,

UNIT-04 (12 HRS)

Quadric Surfaces: Geodesics, Geodesic property, equation of geodesics, surface of revolution,

Torsion of geodesic, Central quadrics, Fundamental magnitudes, The fundamental theorem of

surface theory, Liouville’s equation, Joachimsthal’s theorem.

Learning Outcomes:


1. Solve the problems related to curvature, asymptotes.

2. Explain the definition of a smooth surface, and the means by which many examples may

be constructed.

3. Appreciate the distinction between intrinsic and extrinsic aspects of surface geometry.

4. Analyze and solve complex problems using appropriate techniques from differential

geometry

Text Book:

01. C. E. Weatherburn: Differential Geometry of three dimensions, RPH Calcutta, 2008.

References:

01. A. Goetz: Introduction to Differential Geometry: Addision Wesley Publishing Company,

(1970)

02. A. W. Joshi: Tensors and Riemanian Geometry





MS7403 Elasticity-II 5 0 0 5

Course Objective This course is designed to provide follow up to Course-MS7303. This course will provide basic

topics needed for students to pursue research in pure Mathematics. The contents of the

curriculum has been developed keeping in view the UGC guidelines.

UNIT-01 (10 HRS)

Equations of Elasticity: Generalized Hook’s Law, Homogeneous isotropic media, Equilibrium

and dynamical equations for isotropic media, Strain energy function, Uniqueness of solution,

Beltrami-Michell Compatibility equations, Saint Venant’s Principal,

UNIT-02 (12 HRS)

D’Alembert’s method of one dimensional wave equation, Waves in three dimensions, Harmonic

waves, Spherical waves, Superposition of waves and stationary waves, Solution of equation of

wave motion of stationary type by method of separation of variables, Cartesian, plane polar and

spherical polar coordinates.

UNIT-03 (10 HRS)

Elastic Waves: Wave propagation in isotropic elastic solid medium, Waves of dilation and

distortion, Rayleigh waves, Love waves,

UNIT-04 (10 HRS)

Reflection of P, SV and SH-waves from free surface of a half-space, Reflection and refraction of

elastic waves (P, SV and SH-waves) at Solid-Solid and Solid-Liquid interface.

Learning Outcomes:


1. Explain the definition of a Elaticity and its applications with examples.

2. use computational tools to model and analyze Wave components,

3. use for further research work in various disciplines including physics, chemistry.

Text Book:

01. I. S. Sokolnikoff: Mathematical Theory of Elasticity, Mc-Graw Hill, Inc., 1956.

References:

01. A. E. H. Love: A Treatise on Mathematical Theory of Elasticity, Dover Publications,

1944.

02. K. E. Bullen and B. A. Bolt: An Introduction to the Theory of Seismology, Cambridge

University Press, Cambridge (1985)

03. P. M. Shearer: Introduction to Seismology, Cambridge University Press (1999).





MS7404 Integral Transforms And Their Applications 5 0 0 5

Course Objective This is an extended course work of Course MS6105. This course help to make understand the

students Laplace, finite Laplace, Hankel, fourier, Finite fourier, Cosine and sine transformations

and Mellin’s transformation and application of integral transformations.

UNIT-01 (10 HRS)

Laplace Transforms: Definition and examples, Existence theorem and basic properties,

Convolution theorem and properties of convolution, Differentiation and Integration of Laplace

transform, The inverse Laplace transform and examples, Tauberian theorems for Laplace

transforms and Watson’s Lemma, Laplace transforms of fractional integrals and fractional

derivatives.

UNIT-02 (10 HRS)

Applications of Laplace Transform to solve/evaluate: Ordinary and partial differential equations,

Initial and boundary value problems, Integral equations, Definite integrals,

Difference equations and Differential-difference equations.Finite Laplace Transforms:

Definition and examples, Basic operational properties, Applications, Tauberian theorems for

finite Laplace transforms.

UNIT-03 (10 HRS) Hankel Transforms: Definition and examples, operational properties, Applications to solve

partial differential equations.Fourier Transforms: Fourier Integral formulas, Definition and

examples, Basic properties, Fourier cosine and sine transforms and examples, Basic properties of

Fourier cosine and sine transforms, Multiple Fourier transforms.

UNIT-04 (12 HRS)

Applications of Fourier Transform to solve/evaluate: Ordinary and Partial differential equations,

Integral equations, Definite integrals. Applications of Multiple Fourier transform.Finite Fourier

Cosine and Sine Transforms: Definition and examples, Basic properties, Applications, Multiple

finite Fourier transforms and their applications.Mellin Transforms: Definition and examples,

Basic operational properties and Applications.

Learning Outcomes:


1. Be familiar with the notation and terminology related to differential equations, Laplace

Transform, Fourier Transform and

2. Able to differentiate between ODE and PDE, know the methods to solve differential

equations and be able to solve ODE and PDE of special type.

3. Understand the utility of Laplace Transform and Fourier series in solving PDE.

4. Be able to integrate and differentiate the functions with complex variables. Understand

the use of complex integrations in evaluation of some real integrals.

5. Examine and solve the theory of integral equations



Text Book: 01. Brian Davies, Integral Transforms and their Applications, 3rd Edition, Springer-Verlag,

New York, Inc, 2001.

References:

01. Loknath Debnath, Integral Transforms and Their Applications, CRC Press, Inc., 1995.

02. P.P.G. Dyke, An Introduction to Laplace Transforms and Fourier Series, Springer-

Verlag, London, 2001.

03. Austin Keane, Integral transforms, Science Press, 1965.





MS7405 Probability and Mathematical Statis 5 0 0 5

Course Objective The course will also lay the foundation to probability theorem through various probability

distributions. This will help to students familiar with various techniques used in summarization

and analysis of data.

UNIT-01 (12 HRS)

Nature of Data and methods of compilation: Measurement scales, Attribute and variable,

Discrete and continuous variables. Collection, Compilation and Tabulation of

data.Representation of data: Histogram, Frequency Polygon, Frequency Curve, Ogives.Measures

of central tendency: Mean, Median, Mode, Geometric Mean, Harmonic Mean and their

properties.

UNIT-02 (10 HRS)

Measuring variability of data: Range, Quartile deviation, Deciles and Percentiles. Standard

deviation, Central and non-central moments, Sample and Population variance.. Skewness and

Kurtosis, Box and Whisker plot.Correlation & Regression Analysis: Scatter diagram. Karl

Pearson’s and Spearman’s rank correlation coefficient. Linear Regression and its properties.

Theory of attributes, independence and association.

UNIT – 03 (10 HRS)

Probability: Intuitive concept of Probability, Combinatorial problems, conditional probability

and independence, Bayes’ theorem and its applications. Random Variables and Distributions:

Discrete and Continuous random variables. Probability mass function and Probability density

function. Cumulative distribution function. Expectation of single and two dimensional random

variables. Properties of random variables. Moment generating function and probalility generating

functions.

UNIT-04 (10 HRS)

Distributions: Bernoulli distribution. Binomial distribution. Poisson distribution, Negative

Binomial and Hypergeometric distributions. Uniform, Normal distribution. Normal

approximation to Binomial and Poisson distributions. Beta, Gamma, Chi-square and Bivariate

normal distributions. Sampling distribution of mean and variance (normal p[opulation).

Chebyshev’s inequality, weak law of large numbers, Central limit

theorems.

Learning Outcomes:


1. Use various methods to compute the probabilities of events, Analyze and interpret

statistical data using appropriate probability distributions, e.g. binomial and normal,

apply central limit theorem to describe inferences,



2. Construct and interpret confidence intervals to estimate means, standard deviations and

proportions for populations,

3. Perform parameter testing techniques, including single and multi-sample tests for means,

standard deviations and proportions, and

4. Perform a regression analysis, and compute and interpret the coefficient of correlation.

Text Book:

01. S.C.Gupta & V.K.Kapoor, “Fundamental of Mathematical Staistics” Sultan Chand &

Sons, New Delhi.

References: 01. Goon, A.M., Gupta,M.K., Dasgupta,B: Fundamentals of Statistics, Vol-1& Vol-II

02. Sheldon Ross :A First Course In Probability, 6th edition, Pearson Education Asia

03. Meyer, P.L: Introductory Probability and Statistical Applications



Semester: IV M. Sc (Mathematics)





Semester: IV M. Sc (Mathematics)


2 0 0 2 Subject : Project Mathematics-II





FS7408 Campus to Workplace 2 0 0 2

Course Objective:

To help students to develop confidence to face interviews and to groom them for workplace.

Unit1-Speaking (10 hours)

Activities- Group Discussion, Mock Interview, Extempore, Declamation , Presentation.

Unit2-Personality Development (2 hours)

Activity-SWOT Analysis, Grooming and Work Ethics

Unit3-Listening Skills (1 hour)

Activity- Listening to comprehend.

Unit4-Writing Skills (6 hours)

Activity-Business letter, Cover Letter and Resume writing

Unit5-Reading (4 hours)

Activity- Reading comprehension exercises from competitive tests.

Unit6-Vocabulary Enhancement (4hours)

Activity- Exercises on Synonyms& Antonyms, One word substitution

Unit7-Grammar (6hours)

Activity- Exercises based on Narration, Change of voice and errors.

Learning Outcome

By the end of the course the students will display ability to face interviews confidently. They

become capable of cracking job oriented or higher studies’ entrance test. This course will help

them to adapt easily to their workplace.

Text Book



Sanjay Kumar& Pushp Lata , Communication Skills, Oxford University Press(2014)

Reference Books:

Barron’s Vocabulary Builder- Educational Series, Bright Publishers

Wren & Martin, High School Grammar, S.Chand & Company

Dr. T. Kalyana Chakravarthi & Dr. T. Latha Chakravarthi, Soft Skills for Managers, Bizantra

Documents

University School of Sciences M.SC. Mathematics 1st sem ...basicsciences.rayatbahrauniversity.edu.in/Curriculam/M.Sc Mathema… · 5 MS7305 Special Functions 5 0 0 5 5 6. MS7306 UGC