subject: applied mathematics (amat) general course...

1

1 Compulsory for the Physical Science students. 2 Compulsory also for the students offering Physics as a subject. 3 Compulsory for the students in the Special Degree Programme in Mathematics (Applied Mathematics

stream)

Subject: Applied Mathematics (AMAT)

General

Course Units Status Pre-requisite Co-requisite

Year 1

Sem 1

AMAT 11032 Vector Algebra1 C A/L Combined

Mathematics

Year 1

Sem 2

AMAT 12042 Elementary Ordinary Differential

Equations1 C

A/L Combined

Mathematics

AMAT 12053 Vector Analysis2 C AMAT11032

AMAT 12062 Mechanics I C AMAT 12053

Year 2

Sem 1 AMAT 21035 Mechanics II C AMAT12062

Year 2

Sem 2 AMAT 22045 Numerical Methods C

PMAT 12062

PMAT 12073

Year 3

Sem 1

AMAT 31053 Numerical Methods using

Appropriate Software O AMAT22045

AMAT 31063 Mechanics III O AMAT21035

AMAT 31073 Mathematical Modelling O PMAT 12073

AMAT 31083 Mathematics for

Finance I O PMAT 12062

PRPL 31012 Professional Placement O

Year 3

Sem 2

AMAT 32093 Computational Mathematics O AMAT31053

AMAT 32103 Introduction to Fluid Dynamics O PMAT 31073 PMAT 32113

AMAT 32113 Mathematics for

Finance II O AMAT31083

Special


Year 3

Sem 1

AMAT 41053 Qualitative and Quantitative

Behaviour of the Solutions of Ordinary

Differential Equations

C AMAT22045

AMAT 41063 Advanced Mathematical

Modelling O PMAT 12073

Year 3

Sem 2

AMAT 42073 Advanced Computational

Mathematics C AMAT41053

AMAT 42083 Fluid Dynamics C PMAT 41063 PMAT 42093

AMAT 42093 Financial Mathematics O PMAT 12062

Year 4

Sem 1

AMAT 41244 Boundary Values

Problems3 C/O PMAT 41073

AMAT 41254 Quantum Mechanics C AMAT31063

AMAT 43288 Research/Study Project C

Year 4

Sem 2

AMAT 42264 Quantum Field

Theory C

AMAT41254/

PHYS 44014

AMAT 42274 Tensors and General

Relativity C

PHYS 12194

PMAT 21035

2

Applied Mathematics

Level – 1

Course Code : AMAT 11032

Title : Vector Algebra

Pre-requisites : A/L Combined Mathematics

Learning Outcomes:

At the end of the course the student should be able to apply the concepts and theorems of Vector Algebra and to

apply them in geometry and trigonometry.

Course Contents:

Vector Algebra: Introduction to vectors, Condition for coplanarity of three vectors, Orthogonal triads of

unit vectors, Scalar and vector products, Triple products, Solution of vector equations, Applications in

plane and solid geometry, and in plane and spherical trigonometry.

Method of Teaching and Learning: A combination of lectures and tutorial discussions.

Assessment: Based on tutorials, tests and end of course examination.

Recommended Reading : 1. Davis, H.F. & Snider, A.D., (1992). An Introduction to Vector Analysis, C. Brown, New York.

2. Chatterjee, D., (2005). Vector Analysis, Prentice Hall, India.


Title : Elementary Ordinary Differential Equations


Learning Outcomes: At the end of this course, the student should be able to distinguish linear and non-linear ordinary differential

equations and to solve non-linear first order and linear ordinary differential equations using appropriate methods.

Also the student will be able to solve the Legendre and Bessel equations.

Course Contents:

First Order Linear Differential Equations: Formation of differential equations,Variable separable,

Homogeneous, Exact equations, Integration factor method, Bernoulli equations.

First Order Non-Linear Differential Equations: Riccarti, Clairaut types.

Linear differential equations of higher degree: Equations with constant coefficients, Operator

methods.



Recommended Reading : 1. Raisinghania, M.D., (1991). Advanced Differential Equations, S.Chands, India.

2. Mondal, C. R., (2003). Ordinary Differential Equations, Prentice Hall, India.


Title : Vector Analysis

Pre-requisites : AMAT 11032

Learning Outcomes:

3

At the end of the course the student should be able to demonstrate knowledge of the concepts and theorems of

Vector Analysis and to apply them in classical field theories in Physics.

Course Contents: Vector Analysis: Scalar and vector fields, Differentiation of vector functions, Frenet-Serret formulae,

Surfaces and normals, Gradient, Divergence and Curl operators and identities involving them,

Divergence and Stokes’ theorems, Conservative and solenoidal fields, Scalar and vector potentials,

Applications of vector analysis in gravitational, electrostatic and magneto static fields.



Recommended Reading :

1. Davis, H.F. & Snider, A.D., (1992). An Introduction to Vector Analysis, C. Brown, New York.

2. Chatterjee, D., (2003). Vector Analysis, Prentice Hall, India.


Title : Mechanics I

Co-requisites : AMAT 12053

Learning Outcomes: At the end of this course, the student should be able to demonstrate knowledge of basic concepts of Kinematics

and Newtonian dynamics and also be able to apply Newton’s laws in solving simple problems related to

the motion of a particle.

Course Contents:

Newtonian Kinematics: Inertial frames, Transformations between inertial frames, Relative motion of

particles, Relative motion of frames of reference.

Motion of a Particle: Mass, Momentum, Torque and angular momentum, Equation of motion in

vectorial form, One dimensional motion, Integrals of motion, Work, kinetic energy & potential energy,

Impulse, Motion under conservative forces, Motion under a central force, Kepler’s laws, Rotating frames

of reference, Motion relative to rotating earth.




1. Chorlton, F., (1969). Text book of Dynamics, D. Van Nostrand.

2. Desloge, E.A., (1982). Classical Mechanics, John Wiley, New York.

Level - 2


Title : Mechanics II


Learning Outcomes:

4

At the end of this course, the student should be able to apply Newton’s laws in solving simple problems

related to the motion of a particle, a system of particles and rigid bodies, and the student will gain a

sound knowledge of Lagrangian approach to mechanics.

Course Contents: System of Particles: Centre of mass, External and internal forces, Integrals of motion, Momentum,

Angular momentum, Work, kinetic energy & potential energy, Conservative systems, Constants of

motion.

Rigid Body Motion: Rigid bodies, Moments and products of inertia, Principal axes, Equimomental

systems, Motion of a lamina, Instantaneous centre, Body and space centrodes, Uniplanar motion of a

rigid body, Impulsive motion, Euler’s equations of Motion.

Lagrangian Mechanics: Generalized coordinates, Lagrange’s equations of motion for elementary

systems, Constraint forces, Lagrange’s equation of motion for holonomic systems, Determination of

holonomic constraint forces, generalized force functions, Lagrange equations, Constants of motion in the

Lagrangian formalism.





2. Desloge, E.A., (1982). Classical Mechanics, John Wiley, New York.

3. Ramsey, A.S., (1975). Dynamics,Parts I & II, Cambridge University Press.

4. Goldstein, H., (1977). Classical Mechanic , Addison Wesley.


Title : Numerical Methods

Pre-requisites : PMAT 12062/PMAT 12073

Learning Outcomes: At the end of the course, the student should be able;

1. to implement numerical methods for a variety of multidisciplinary applications

2. to establish the limitations, advantages and disadvantages of numerical methods

3. to develop and use algorithms and theorems to find numerical solutions and

bounds on their error to various types of problems including root finding,

polynomial approximation, numerical differentiation, numerical integration.

Course Contents: Introduction: Floating point number system, Error in numerical computation, Notion of algorithm.

Solution of equations with one variable: Numerical solution of nonlinear equations, Solutions of polynomial

equations.

Interpolation: Polynomial interpolation, Spline interpolation.

Approximation of functions: Least square method, Best fit.

Solution of System of Linear Equations (Direct and Iterative Methods): Gaussian eliminations with partial

pivoting, Ill conditioning, Operation counts, Jacobi and Gauss-Seidel iterative methods and their convergence.

Numerical Differentiation and Integration: Trapezoidal, Simpson quadratic formulae, Romberg integration

method, Gaussian quadrature.

Numerical Solutions of Ordinary Differential Equations: Explicit and Implicit numerical schemes, Euler’s

method, Computation of error bound, Stability of methods, Predictor-Corrector methods.

5

Method of Teaching and Learning: A combination of lectures and tutorial discussions

Assessment: Based on tutorials, tests and end of course examination


1. Sastry, S. S., (2003). Introductory Methods of Numerical Analysis, Prentice Hall India.

2. Kreyszig, E., (1983). Advanced Engineering Mathematics, John Wiley, New York.

3. Burden, R.L. & Faireu, J.D., (1993). Numerical Analysis, PWS-KENT, Boston.

4. Gerald, C.F.; Wheatley, P.O., (1994). Applied Numerical Analysis, Addison Wesley.

Level – 3


Title : Numerical Methods using Appropriate Software


Learning Outcomes: On completion of this unit, the student should be able to write computer programs to solve structured engineering

and mathematical problems using appropriate numerical techniques in the MATLAB environment.

Course Contents: Introduction to MATLAB: Basic procedures in using MATLAB, Handling numbers and matrices, Control

structures in MATLAB, Program design in MATLAB language, Plotting.

Implementation of Numerical Algorithms : Algorithms studied in AMAT 22045 will be implemented using

MATLAB.

Method of Teaching and Learning: A combination of lectures and computer laboratory sessions.



1. Vanloan C.F., (2000). Introduction to scientific computing: a matrix-vector approach using

MATLAB, Prentice Hall, New York.

2. Hanselman, D. & Littlefield, B.R., (2000). Mastering MATLAB 6, Prentice Hall, New York.

3. Chapman S.J., (2000). MATLAB Programming for Engineers, Brookkole.


Title : Mechanics III


Learning Outcomes: At the end of this course, the student should be able to explain, the motion of a dynamical system using Lagrange

and Hamilton formalism, and the classical analogue of quantum mechanics.

Course Contents:

Eularian angles, Motion of a symmetrical top, Normal modes, Lagrange equation of motion for

impulsive motion, D’Alambert’s principle. Hamilton’s equations of motion, Canonical transformations,

Poisson brackets, Lagrange brackets. All forms of generating functions.





2. Ramsey, A.S., (1975). Dynamics, Parts I & II, Cambridge University Press.

3. Goldstein, H., (1977). Classical Mechanics, Addison Wesley.

6


Title : Mathematical Modelling

Pre-requisites : PMAT 12073

Learning Outcomes:

At the end of this course the student should be able;

1. to explain how the general principals arise in the context of Mathematical Modelling

2. to explain and apply the basic concepts of Mathematics and their uses in analyzing and solving

real-world problems

3. to analyze some existing mathematical models and construct simple models for real world

situations.

Course Contents:

Philosophy of modelling, Modelling Methodology, Problem formulation, Mathematical Descriptio n,

Analysis, Interpretation, Case studies from the real world situation.

Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

Assessment: Based on projects, tutorials, tests and end of course examination.


1. Kapur, J.N., (1994). Mathematical Modeling, Wiley Eastern.

2. Real E.A., (2000). Bender , An introduction to Mathematical Modelling , Dover.


Title : Mathematics for Finance I


Learning Outcomes : On successful completion of the course the student should be able;

1. to explain the goal of Financial Management and the conflicts of interest between managers and owners

2. to show a systematic knowledge, understanding and critical awareness of basics in Financial

Mathematics such as derivatives markets and risk management, including credit risk and credit

derivatives.

Course Contents: Options, futures and derivative securities: Fundamental concepts in securities markets: equities, currencies,

commodities and debt markets. Introduction to derivatives products: forward contracts, futures, warrants and

options. Exchange-traded and OTC products. Hedging using forward and futures. Swaps, cost-of-carry, financing

forward positions. Common hedging strategies using options and futures. Futures and derivative securities,

Arbitrage pricing theory.

The Black Scholes analysis: Black-Scholes formula and its sensitivity on the model’s parameters. Delta,

Gamma, Vega, Rho. Hedging strategies. Implied and historical volatilities. Discrepancies between market prices

and Black Scholes.

Options: American options and other instruments with early-exercise Features. Asian Options. Introduction to

Monte-Carlo evaluation of exotics on the binomial tree, Stochastic volatility, volatility risk.

Method of Teaching and Learning: A combination of lectures and tutorial discussions, computer

practical.

Assessment: Based on tutorials, tests, presentations and end of course examination .


1. J. Hull , (1993). Options and other Derivative Securities , Prentice Hall.

2. John. C. Hull, (2007). Options, future and other derivatives, Prentice Hall, India.

7


Title : Computational Mathematics


Learning Outcomes: Upon satisfactorily completing this course the student should be able;

1. to solve system of linear equations using various numerical methods and implement them using a

programming language and explain the type of numerical error which may o ccur.

2. to investigate the criteria such as convergence, rate of convergence, accuracy and where

appropriate consistency and stability.

3. to apply several numerical algorithms to solve boundary initial value problems in the form of

partial differential equations.

Course Contents: Numerical Linear Algebra: Vector Norms, Matrix norms, General properties of vector and matrix norms.

Modern Methods for Solving Linear Systems of Equations: Relative error bound, Condition number, Matrix

Decomposition Techniques: LU and QR Factorizations, Iterative and Relaxation Methods: Jacobi, Gauss-Siedel,

Richardson, SOR Iterative , Gradient Methods : Conjugate Gradient Method.

Finite Difference Methods: Finite Difference schemes for solving some elliptic, parabolic and hyperbolic

equations and implementation with MATLAB.


practicals.



1. Burden, R.L., Fairs, J.D. & Reynolds, A.C., (1981). Numerical Analysis, Prindle, Weber & Schmidt.

2. Trefethen L.N. & Bau D., (1997). Numerical Linear Algebra, Philadelphia, USA.

3. Golub H., Vanloan C.F., (1996). Matrix computations, Johns Hopkins University Press.

4. Vanloan C.F., (2000). Introduction to scientific computing: a matrix-vector approach using




Title : Introduction to Fluid Dynamics


Co-requisites : PMAT 32113

Learning Outcomes: At the end of this course, the student should be able to recognize the difference between the discrete mass points

and continuous matter in mechanics and to explain the two dimensional and the axi-symmetrical motion of a

perfect fluid.

Course Contents: Further Vector Analysis: Orthogonal curvilinear coordinates, Gradient, Divergence and curl.

Basic Principles of Fluid Dynamics: Fluid pressure, Velocity, Acceleration, Stream lines, Equation of

continuity, Euler’s equation of Motion, Vorticity, Irrotational motion under conservative forces,

Bernoulli’s equation, Helmholtz vorticity equation, Vortex lines, Velocity circ ulation round a closed

curve, Cyclic and acyclic motions, Kinetic energy in irrotational motion, Kelvin’s theorem, Uniqueness

theorems.

8

Two Dimensional Motion: Stream function and plotting stream lines, Complex potential, Sources and

sinks, Vortices, Doublets and image systems, Milne-Thompson theorem, Flow past a cylinder,

Applications of conformal transformations, Blassius theorem.

Axi-symmetric Motion: Stokes’ stream function (3D).



Recommended Reading:

1. Thompson, L.E.M., (1949). Theoretical Hydrodynamics, MacMillon.

2. Chorlton, F., (1985). Textbook of Fluid Dynamics, CBS, India.

3. Bachelor, G.K., (1977). An Introduction to Fluid Dynamics, Cambridge University Press.


Title : Mathematics for Finance II


Learning Outcomes:

On successful completion of the course the student should be able to explain the recent developments and

methodologies in financial mathematics and the links between the theory of financial mathematics and

their practical application and to critically evaluate such methodologies.

Course Contents: Finite-difference equations to Partial Differential Equations: Analysis of finite-difference equations in the

limit of many trading periods, Derivation of the Black-Scholes PDE for European-style derivatives and the PDE

for American-style options.

Arbitrage pricing theory in continuous-time finance: Martingale measures, Geometric Brownian Motion,

exponential martingales.

The Black-Scholes analysis in continuous time: Riskless portfolios, replication, Risk-neutral valuation, Option

formulas revisited, optimal exercise boundary for American-Style options, options on indices and index futures,

Early-exercise premium and approximations, Portfolio insurance.

Options and Boundary-value problems: Stopping times, Closed-form solutions, Parity relations, Two-asset

options, Correlations and Margrabe’s formula, Barriers depending on a second traded asset, Lookback options.


practicals.

Assessment: Based on tutorials, tests, presentations and end of course examination.


1. J. Hull, (1993). Options and other Derivative Securities, Prentice Hall.

2. John.C.Hull., (2007). Options, future and other derivatives, Prentice Hall, India.

9

Level – 4


Title : Qualitative and Quantitative Behavior of the Solutions of Ordinary Differential

Equations


Learning Outcomes:

At the end of this course, the student should be able to obtain the numerical solutions of differential

equations and their implementations using MATLAB.

Course Contents:

Introduction to MATLAB: Basic procedures in using MATLAB, Handling numbers and matrices, Control

structures in MATLAB, Program design in MATLAB language, Plotting.

Basic Properties of the Solutions of Ordinary Differential Equations: Stability of the solution and State-

Space Analysis, Qualitative behavior of the solution of Ordinary Differential Equations.

Numerical Solutions of Ordinary Differential Equations: Concept of consistency, Stability and convergence

properties of numerical schemes, Single and multistep methods, Solving Stiff systems, Shooting method, Gear's

implementation of automatic ordinary differential equation solver, Finite difference discretizations for second

order boundary value problems.

Lab work using MATLAB: Algorithms studied in AMAT 22045 and in this unit will be implemented using

MATLAB.

Method of Teaching and Learning: A combination of lectures, computer laboratory sessions, tutorial

discussions and presentations.


Recommended Reading : 1. Vanloan, C.F., (2000). Introduction to scientific computing: a matrix-vector approach using



3. Ferdinand, V., (2000). Nonlinear differential equations and dynamical systems, Springer Verlag.


Title : Advanced Mathematical Modelling


Learning Outcomes: At the end of this course the student should be able;

1. to explain how the general principals arise in the context of Mathematical Modelling

2. to explain and apply the basic concepts of Mathematics and their uses in analyzing and solving

real-world problems

3. to analyze some existing mathematical models and construct simple models for real world

situations.

Course Contents:

Modelling Methodology, Problem formulation, Analysis & interpretation, Case studies from the real

world situation illustrating the different aspects of the modelling process .

Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations .

Assessment: Based on tutorials, presentations and end of course examination.


1. Kapur, J.N., (1994). Mathematical Modelling, Wiley Eastern.

10


Title : Advanced Computational Mathematics


Learning Outcomes: Upon satisfactorily completing this course the student should be able;

1. to solve system of linear equations using various numerical methods and imp lement them using a

programming language and explain the type of numerical error which may occur

2. to have an appreciation of criteria such as convergence, rate of convergence, accuracy and where

appropriate consistency and stability

3. to apply several numerical algorithms to solve boundary initial value problems in the form of


Course Contents: Numerical Linear Algebra: Vector Norms, Matrix Norms, General Properties of Vector Norms.

Modern Methods for Solving Linear Systems of Equations: Relative error bound, Condition number, Matrix

Decomposition Techniques: LU and QR Factorization, Iterative and Relaxation Methods: Jacobi, Gauss-Siedel,

SOR, Richardson, Gradient Method: Conjugate gradient method, steepest decent method .

Finite Difference Method: Consistency, Stability and convergence of finite difference schemes, Introduction to

finite difference schemes for solving some parabolic and hyperbolic equations.

Introduction to Finite Element Methods: Variational formulation of problem, Reitz-Galerkin formulation,

Basic functions.

Practicals using MATLAB:


practical.



1. Golub, H.& Vanloan, C.F., (1996 ). Matrix Computations, Johns Hopkins.


3. Rainer Krem, (1995). Numerical Analysis, Springer Verlay Newyork Inc.

4. J.W.Thomas, (1998). Numerical PDE, Finite Difference Methods, Springer.

5. Chun, T.J., (2002). Computational Fluid Dynamics, Cambridge University Press.


Title : Fluid Dynamics



Learning Outcomes:

At the end of this course, the student should be able to recognize the difference between discrete mass-

points and continuous matter in mechanics and to explain the fundamental properties of two-dimensional

and Axi-symmetric motion, three dimensional motion of a perfect fluid, and motion of a viscous fluid .

Course Contents: Further Vector Analysis: Orthogonal curvilinear coordinates, Gradient, divergence and curl.

Basic Principles of Dynamics: Fluid pressure, Velocity, Acceleration, Stream lines, Equation of

continuity, Euler’s equation of motion, Vorticity, Irrotat ional motion under conservative forces,

Bernoulli’s equation, Helmholtz vorticity equation, Vortex lines, Velocity circulation round a closed

curve, Cyclic and acyclic motions, Kinetic energy in irrotational motion, Kelvin’s theorem, Uniqueness

theorems.

11

Two Dimensional Motion: Stream function and plotting stream lines, Complex potential, Sources and

sinks, Vortices, Doublets and image systems, Milne-Thompson theorem, Flow past a cylinder,

Applications of conformal transformations including Schwarz-Christoeffel transformation, Blassius

theorem.

Axi-symmetric Motion: Stokes’ stream function (3D).

Three Dimensional Motion: Irrotaional motion, Laplace’s Equation, Spherical Harmonics, Flow of a

stream past a fixed sphere, Motion of a sphere in a fluid, Impulsive motion.

Compressible Fluids: Sound waves in gasses.

Viscous fluids: Naviers -Stokes theorem, Motion of viscous fluids.


Assessment: Based on tutorials, tests, presentations and end of course Examination.

Recommended Reading: 1. Raisinghania, M.D., (1995). Hydrodynamics, Schaum’s.

2. Thompson, L.E.M., (1949). Theoretical Hydrodynamics, MacMillon.

3. Chorlton, F., (1985). Textbook of Fluid Dynamics, CBS, India.

4. Bachelor G.K., (1977). An Introduction to Fluid Dynamics, Cambridge University Press.


Title : Financial Mathematics


Learning Outcomes :

On successful completion of the course the student should be able to;

1. explain the Goal of Financial Management and the conflicts of interest between managers and owners

2. show a systematic knowledge, understanding and critical awareness of basics in Financial Mathematics

such as derivatives markets and risk management, including credit risk and credit derivatives

3. develop skills in the use of relevant information technology.

Course Contents:

Options, futures and derivative securities, Futures and derivative securities, Arbitrage pricing theory, The Black

Scholes analysis, Options, Finite-difference equations to PDEs, Arbitrage pricing theory in continuous-time

finance, The Black-Scholes analysis in continuous time, Options and Boundary-value problems.



Recommended readings:

1. Daniel J. Duffy Finite, (2006). Difference Methods in Financial Engineering: A Partial Differential

Equation Approach , John Wiley & sons,

2. John.C.Hull, (2007). Options, future and other derivatives, Prentice Hall, India.


Title : Boundary Value Problems


Learning Outcomes:

At the end of this course, the student should be able to solve two dimensional and three dimensional partial

differential equations, which are physically important, under given boundary conditions.

12

Course Contents:

Partial Differential Equations in two variables: Linear second-order equations in two independent

variables, Normal forms, Hyperbolic, parabolic and elliptic equations, Boundary value problems in

rectangular and cylindrical coordinates, Applications to heat flow, Vibrations and waves, Laplace and

Poisson equations in two dimensions, Bessel functions.

Boundary Value Problems in Three Dimensions: Green's theorem in three dimensions, Uniqueness of

solutions with Dirichlet and Neumann boundary conditions, Formal solutions of boundary value

problems in electrostatics, Method of images, Laplace equation in spherical polar coordinates, Boundary

value problems with spherical symmetry; Heat and wave equations, Three dimensional boundary value

problems with azimuthal symmetry, Legendre functions and applications to gravitation and

electrostatics, Potentials of circular rings and discs.




1. Habernsann, R., (1987). Elementary Applied Partial Differential Equations, Prentice-Hall.

2. Raisinghania, M.D., (1991). Ordinary and Partial Differential Equations, S. Chands, India.


Title : Quantum Mechanics


Learning Outcomes: At the end of this course, the student should be able to demonstrate knowledge of concepts and principles of

quantum mechanics and to apply them to solve simple problems.

Course Contents: Quantum mechanics in Hilbert space, Axiomatic structure of quantum mechanics, The Shrödinger

picture, Heisenberg and interaction picture, Complete set of observables, formalism of wave mechanics

and its applications, Completely continuous operators, uncertainty principle, potential well, simple harmonic

oscillator, scattering theory of two particles, potential scattering, approximate methods



Recommended Reading : 1. Schiff, L.I., (1987). Quantum Mechanics, McGraw-Hill, New York.

2. Prugovecki, E., (1981). Quantum Mechanics in Hilbert Space, Academic Press.

3. Liboff R, L., (2002). Introductory Quantum Mechanics, Pearson Education.

4. Baggot, J., (1997). The meaning of Quantum Theory, Oxford University Press.

5. Sakurai, J.J., (2001). Advanced Quantum Mechanics, Replica Press (P) LTD, India.


Title : Quantum Field Theory

Pre-requisites : AMAT 41254/PHYS 44014

Learning Outcomes:

At the end of this course, the student should be able to demonstrate knowledge of basic properties of

relativistic local field theory and the quantization of spin zero and spin half fields.

Course Contents: Relativistic wave equation, Review of mechanics of a finite system, Quantisation, General Theorems,

Quantisation of spin zero fields and spin ½ fields, Momentum and angular momentum operators, Phase

factor, Conventions between the spinners, Two - component theory.

13



Recommended Reading : 1. Schiff, L.I., (1989). Quantum Mechanics, McGraw-Hill, New York.

2. Prugovecki, E., (1981). Quantum Mechanics in Hilbert Space, Academic Press.

3. Lee, T. D., (1981). Particle Physics and Introduction to Field Theory, Taylor and Francis.


Title : Tensor Analysis and General Relativity

Pre-requisites : PHYS 12194/PMAT 21035

Learning Outcomes:

At the end of the course the student should be able to demonstrate knowledge of concepts and theorems in tensor

algebra, tensor analysis and the formalism of general relativity, and to solve Einstein’s field equations in simple

cases, and to apply the solutions in Astrophysics.

Course Contents: Covariant and contravariant vectors, Metric tensor, Invariants, Inner products, Differential -forms, Tensor

analysis, Covariant differentiation of tensors, Einstein’s field equations, Schwarzchild interior and

exterior solutions, Rotating systems, Dragging of inertial frames, Gravitation red shift, Bending of light

and gravitational lenses.



Recommended Reading: 1. Adler R., Bazin M. and Schiffer M., (1970). Introduction to General Relativity, McGraw Hill, New

York.

2. Misner, C.W.,Thorne, K.S. & Wheeler, J.A., (1973). Gravitation, W. H. Freeman.

3. Hawking S.W. & Ellis G.F.R., (1973). The Large Scale Structure of Space-time, Cambridge

University Press.

4.. Willmore, T.J., (1964). An Introduction to Differential Geometry , Oxford University Press.

5. James B.Hartle, (2006). Gravity, Pearson Education.


Title : Research/Study Project

Learning Outcomes:

At the end of this course, the student should be able to demonstrate competence in research/independent-

study in an area in Applied Mathematics.

Method of Teaching and Learning:

A research/study project under the supervision of a senior staff member of the department .

Assessment: Submission of a research/study project report and an oral presentation.

Recommended Reading : Required reading material will be recommended by the supervisor depending

on the relevant project.

14

Subject: Pure Mathematics (PMAT)

General


Year 1

Sem 1

PMAT 11042 Discrete Mathematics I1, 2 C

PMAT 11083 Topics in Basic

Mathematics3 A

PMAT 14102 Logic and Reasoning A

Year 1

Sem 2

PMAT 12052 Calculus I1 C A/L Combined

Mathematics

PMAT 12062 Discrete Mathematics II2

C

PMAT 11042

PMAT 12073 Calculus II

C

PMAT 12052

PMAT 12093 Introduction to Calculus3 A

Year 2

Sem 1

PMAT 21035 Linear Algebra

C

PMAT 12062

Year 2

Sem 2

PMAT 22045 Infinite Series and Series of

Functions C PMAT 12073

Year 3

Sem 1

PMAT 31073 Introduction to Functions

of Several Variables O PMAT 22045

PMAT 31083 Algebraic Structures O PMAT 21035

PMAT 31093 Ordinary Differential

Equations

O

PMAT 12073

PMAT 31103 Riemann Theory of

Integration O PMAT 22045

PRPL 31012 Professional Placement O

Year 3

Sem 2

PMAT 32113 Complex Variables O PMAT 31073

PMAT 32123 Geometry O PMAT 21035

PMAT 32133 Partial Differential

Equations and Integral Transforms O

PMAT 31093

PMAT 22045

1 Compulsory for Physical Science students 2 Compulsory for Management and Information Technology students 3 Available only for students who have not offered Combined Mathematics for G.C.E. (A/L)

Examination

15

Pure Mathematics

Level-1

Course code : PMAT 11042

Title : Discrete Mathematics I

Learning Outcomes: At the end of the course the student should be able to demonstrate knowledge of the concepts of Set Theory,

Relations and Matrix Algebra and to apply them in Modular Arithmetic and solving systems of linear equations.

Course Contents:

Set theory: Sets, Operation on sets, ordered pairs and Cartesian Products.

Relations: Relations, Order relations, Equivalence relations, Module Arithmetic, Functions.

Matrices: Matrix algebra, Special types of square matrices, Determinant of a matrix.

Systems of Linear Equations: Homogeneous and Non-homogeneous types, Method of solving such systems.


Assessment: Based on the tutorials, tests, and end of course examination.


1. Johnsonbaugh, R., (1990). Discrete Mathematics, MacMillan.

2. Lipschutz, S., (1976). Discrete Mathematics, McGraw-Hill, New York.


Title : Topics in Basic Mathematics

Learning Outcomes :

At the end of this course, the student should be able to demonstrate knowledge in basic discrete

mathematical concepts stated in the content and apply these concepts in appropriate manner in given

problems.

Course Contents:

Set theory, Product sets, Relations, Functions and graphs, Determinants of order two and three, Matrices, Linear

Equations. Finite series, Binomial theorem, Exponential, Trigonometric and logarithmic functions.




1. Lipschutz, S., (1976). Discrete Mathematics, McGraw-Hill, New York.

2. Nachman, L.J., (1978). Fundamental Mathematics, John Wiley, New York.


Title : Logic and Reasoning

Learning Outcomes: At the end of this course, the student should be able to identify valid reasoning and to construct coherent and

persuasive arguments themselves.

16

Course contents:

Argument reconstruction, Validity and soundness, Deductive reasoning and formal logic, Informal logic and

everyday reasoning, Basic symbolic logic, Truth tables, Informal fallacies, Conceptual theories, Empirical

theories.





2. Lipschutz, S., (1964). Set Theory and Related Topics, McGraw-Hill.

3. Aggarwal, R.S. and Matharu, R.S., (1989). A text book on Discrete Mathematics, S. Chands.

4. Vatssa, B.S., (1988). Discrete Mathematics, Wishwa Prakashan.


Title : Calculus I


Learning Outcomes: On successful completion of the course, the student should be able;

1. to demonstrate knowledge of the axiomatic description of the field of real numbers and prove theorems

from the given set of axioms

2. to explain the concepts of limits continuity and differentiation of real-valued functions

3. to use the concepts of limits, continuity and the derivatives to describe the qualitative behavior of a

graph of a single variable function

4. to use derivatives to solve applied problems.

Course Contents:

Real Numbers: Supremum and infimum of a set, completeness axioms, symbols and

Functions and Limits: Limits of Functions, Continuous Functions.

Derivative and Applications: Derivative of a function, Function composition, Chain rule, Implicit functions and

Implicit differentiation, Higher order derivatives, Rolle’s Theorem, Mean Value Theorem, Monotonicity and

first derivative test, Concavity and second derivative test, Absolute extrema, Asymptotes and limits involving

infinity, Graph sketching using polar and Cartesian coordinates.

Indeterminate Forms: L’hospital Rule.



Recommended Readings:

1. Ayres, Jr. F. & Mendelson, E., (1990). Calculus, McGraw-Hill, New York.

2. Earl W. Swokowski, (1977). Calculus A First Course,Wadsworth International Group.

3. Arora, S. & Malik, S.C., (1994). Mathematical Analysis, Wiley Eastern.

4. Munem, M.A. & Foulis, D.J., (1984). Calculus, Worth Publishers, New York.


Title : Discrete Mathematics II


Learning Outcomes: At the end of the course the student should be able to demonstrate knowledge of the concepts of Mathematical

Logic, Methods of Proofs and basic Graph Theory and to apply them in developing mathematical arguments in a

logical manner.

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Course contents: Mathematical Logic: Propositional Calculus, Predicate Calculus.

Boolean algebra: Boolean algebra and its properties, Algebra of propositions, Boolean functions, Algebra of

electric circuits and its applications.

Methods of Proof: Direct proof, Proof by contrapositive, Proof by contradiction. Mathematical induction, Case

analysis, Counter examples.

Cardinality: Finite sets, Denumerable sets, Uncountable sets, Cardinal numbers.

Graph Theory: Graphs and Multigraphs, Subgraphs, Matrices and graphs, Isomorphic and homeomorphic

Graphs, Planar graphs, Kuratowski's theorem.

Method of Teaching and Learning: A combination of lectures and tutorial discussions




2. Lipschutz, S., (1976) Discrete Mathematics, McGraw-Hill, New York.

3. West, D. B., (2002). Introduction to Graph Theory, Prentice Hall, India.


Title : Calculus II


Learning Outcomes: On successful completion of the course, the student should be able;

1. to carefully define the meaning of convergence of a real sequence of real numbers and use definition to

discuss the behavior of a given sequence

2. to use derivatives to solve applied problems

3. to evaluate a definite integral as a limit and using integration techniques

4. to demonstrate knowledge of techniques of integration.

Course Contents:

Sequences: Limits and limit theorems for sequences, Monotone sequences.

Conic sections: Parabolas, Ellipses, Hyperbolas, Translations and rotations of axes.

Transcendental functions: Exponential and logarithmic functions, Hyperbolic functions

Anti-differentiation: Anti-derivatives, Method of change of variable.

Definite Integral and Applications: Definite integral and its properties, Fundamental theorem of calculus,

Volumes of solids of revolution, Method of cylindrical shells and slicing, Arc length and surface area.

Techniques of Integration: Integrals of powers of sines and cosines, and of other trigonometric functions,

Integration by trigonometric substitution, by parts, by partial fractions for linear and quadric cases and by

special substitutions, Reduction formulae.



Recommended Reading: 1. Ayres, Jr. F. & Mendelson, E., (1990). Calculus, McGraw-Hill, New York.

2. Earl, W. Swokowski, (1977). Calculus A First Course,Wadsworth International Group.

3. Robert, G. Bartle, Donald, R. Shebert, (2000). Introduction to Real Analysis, John Wiley & Sons, Inc.

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5. Munem, M.A. & Foulis, D.J., (1984). Calculus, Worth Publishers, New York.


Title : Introduction to Calculus

Learning Outcomes: At the end of this course, the student should be able to demonstrate knowledge in basic mathematical

concepts of calculus required for the undergraduate studies and further studies as well.

Course Contents:

Limits, Continuity, Differential Calculus, Maxima and Minima, indeterminate Forms,Methods of

Integration, Improper Integrals, Taylor's formula, Newton's methods, Ordinary Differential Equations.



Recommended reading:

1. Ayres, Jr .F. & Mendelson, E., (1990). Calculus, McGraw-Hill, New York.


Level – 2

Course Code : PMAT 21035

Title : Linear Algebra


Learning Outcomes:

At the end of the course the student should be able to demonstrate knowledge of the concepts and theorems of

Vector Spaces and Linear Transformations and to apply them to solve problems in diverse areas.

Course Contents:

Vector Spaces: Vector Spaces, Subspaces, Linear independence, Basis and dimension, Linear transformations,

Kernel and image of a linear transformation, dimension theorem, Inner product, Gram-Schmidt orthogonalization

process, Orthogonal complement, Orthogonal projections.

Eigenvalues and Eigenvectors: Polynomials of matrices and linear operators, Eigenvalues and

eigenvectors, Diagonalization, Characteristic polynomial , Cayley-Hamilton theorem, Minimum

polynomial, Jordan Canonical Form, Rational canonical form.

Applications of Linear Transformations: Lines and planes, Quadratic forms, Conic sections, Quadratic

surfaces, Least squares, Differential equations and other applications.





2. Khanna, V.K. & Bhambri, S.K., (1998). A Course in Abstract Algebra, Vikas, India.

3. Aggarwal, R.S. & Matharu, R.S., (1991). Linear Algebra, S. Chands, India.

4. Narayan, S. & Pal, S., (1992). A text book of Modern Abstract Algebra , S.Chands, India.


Title : Infinite Series and Series of Functions

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Learning Outcomes: At the end of this course, the student should be able to demonstrate knowledge of the nature of the

convergence of infinite series and conditions under what differentiation and integration can be

performed.

Course Contents: Infinite Series: Properties of infinite series, Series of non-negative terms, Alternating series, Absolute and

conditional convergence, Power Series, Taylor and Maclaurine Series, Binomial Series, Fourier Series.

Series of Functions: Pointwise convergence of a sequence of functions, Uniform convergence of

sequences and series of functions, Consequences of uniform convergence, Differentiation and integration

of infinite series.




1. Ross, K.A., (1980). The theory of calculus, Springer-Verlag, New York.


3. Gupta, S.L. & Rani, N., (1999). Principles of Real Analysis, Vikas, India.

4. B.S.Vatsa., (2002). Principles of Mathematical Analysis,Oscar Publications, India.

Level – 3


Title : Introduction to Functions of Several Variables

Pre- requisites : PMAT 22045

Learning Outcomes: At the end of this course, the student should be able to demonstrate knowledge of extending the ideas and

concepts they have studied for functions of a single variable to functions of s everal variables. Also they

shall be able to demonstrate knowledge of limits, partial derivatives and extrema for functions of several

variables, and how to compute double integrals.

Course Contents: Limits, Continuity, Partial derivatives, Differentiability, Sufficient conditions for differentiability,

Composite functions and the chain rules, Higher order partial derivatives, Changing the order of

differentiation, The mean value theorem, Taylor’s formula and series, Extrema for functions of several

variables, Lagrange multipliers, Plane transformations, The double integral, Changing the order of

integration, The change of variables theorem.




1. Spiegel, M.R., (1974). Advanced Calculus, McGraw-Hill, New York.



Title : Algebraic structures


20

Learning Outcomes:

At the end of the course the student should be able to demonstrate knowledge of structures in Algebra and to

apply the knowledge to solve problems in different areas of algebra.

Course Contents:

Groupoids: Elements of Groupoids.

Group Theory: Groups and subgroups, Normal subgroups, Factor groups, Group Homomorphism.

Rings: Rings and subrings, Characteristic of a ring, Ideals, Quotient Rings, Integral Domains, Euclidean

Domains.

Fields: Finite and infinite fields, Field of fractions.

Method of Teaching and Learning: A combination of lectures, tutorial discussions.

Assessment: Based on tutorials, tests, and end of course examination.


1. Fraleigh, J.B., (1994). A first Course in Abstract Algebra, Addison Wesley.

2. Khanna, V.K. & Bhambri, S.K., (1999). A Course in Abstract Algebra,Vikas,India.

3. Singh, S. & Zameeruddin, Q., (1997). Modern Algebra, Vikas, India.

4. Cohn, P.M., (1974). Algebra, John Wiley, New York.

5. Johnsonbaugh, R., (1990) Discrete Mathematics, MacMillan.


Title : Ordinary Differential Equations.


Learning Outcomes:

At the end of this course, the student should be able;

1. to develop mathematical models of some real-life systems or phenomenon from physical,

sociological or economical problem, solve the mathematical model and interpret the

mathematical results back into the context of the original problem

2. to determine the type of a given differential equation, determine the existence of a solution

and if a solution can be obtained, select the appropriate analytical technique for finding the

solution.

Course Contents:

Laplace transform method, Euler’s equation, Other non-linear equations, Systems of linear equations

with constant coefficients, Linear equations with variable coefficients, Method of variation of

parameters, Series solutions of ordinary differential equations and the method of Frobenius, Legendre

and Bessel equations.


Assessment: Based on tutorials, tests, and end of course examination.

Recommended Reading: 1. Raisinghania, M.D., (1991). Advanced Differential Equations, S.Chands,India.

2. Mondal, C. R., (2003). Ordinary Differential Equations, Prentice Hall, India.


Title : Riemann Theory of Integration


21

Learning Outcomes :

At the end of the course the student should be able to demonstrate knowledge of the basic concepts of Riemann

Integration and to apply them in solving integration problems.

Course Contents:

Riemann Integration: The Riemann integral, Properties of the Riemann integral, Fundamental theorem

of calculus.

Improper integrals: Properties of Improper Integrals, Leibnitz’s rule.




1. Munem, M.A. & Foulis D.J., (1984). Calculus, Worth Publishers.

2. Somasundaran, D. & Choudhary, B., (1996). A first Course Mathematical Analysis, Narosha

Publishing House, India.

3. Malik, S.C. & Arora, S., (1994). Mathematical Analysis, Wiley Eastern.


Title : Complex Variables


Learning Outcomes: At the end of this course, the student should be able to demonstrate knowledge of

the usage of applications of complex numbers and complex valued functions.

Course Contents:

Algebra and geometry of complex numbers, Curves and regions in the complex plane, Complex valued

functions, Limits, Continuity, Derivatives, Analytic functions, Cauchy Riemann equations, Rational

functions, Exponential functions, Trigonometric and hyperbolic functions, Logarithm, General power,

Line integral in the complex plane, Cauchy’s integral theorem, Cauchy’s integral formula, The

derivatives of an analytic function, Taylor and Laurent series, Analyticity at infinity, Zeros and

singularities, Residues, The Residue theorem, Evaluation of real integral.




1. Murray, R.S., (1993). Complex Variables, McGraw-Hill, New York.

2. Churchill, R.V., (1960). Complex Variables and Applications, McGraw-Hill, New York.

3. Ponnasamg, S., (1995). Foundations of complex variables, Narosa, ndia.


Title : Geometry



two and three dimensional Analytical Geometry involving pairs of straight lines, conics and to apply the

standard results to solve problems related to lines, planes and conicoids.

Course Contents:

Analytical Geometry in Two Dimensions: Pairs of straight lines, General equation of second degree,

Change of origin and rotation of axes, Joachimsthal’s ratio equation, Equations of tangents, Pairs of

tangents and chords of contact, Harmonic conjugates, Pole and Polar, Invariance, Reduction to standard

forms of conic, Parametric treatment, Degenerate conic, Properties of conic, Matrix methods.

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Analytical Geometry in Three Dimensions: Equations of the line, plane, sphere, cone, ellipsoid and

hyperboloid, Tangent plane, Normal, Pole and polar, Ruled surfaces, General equation of the second

degree, Properties of quadrics.




1. Gibson, G.A. & Pinkerton, P., (1951). Analytical Geometry, MacMillan.

2. Maxwell, E.A., (1962). Elementary Coordinate Geometry, Oxford University press.

3. Chatterjee, D., (2003). Analytic Solid Geometry, Prentice Hall, India.

4. Jain, P.K. & Ahmad, K., (1994). Analytical Geometry of Two Dimensions, Wiley Eastern.

5. Shanti Narayan, (2005). Analytic Solid Geometry, S.Chand and Company.


Title : Partial Differential Equations and Integral Transforms

Pre-requisites : PMAT 31093/PMAT22045


solving problems involving partial differential equations including integral transform methods.

Course Contents: Partial Differential Equations: Introduction to first order partial differential equations, geometrical

problems, The general solution, Lagrange system, Theory of second order partial differential equations.

Integral Transforms: Laplace, Fourier and other integral transform methods for partial differential

equations.




1. Raisinghania, M.D., (1991). Advanced Differential Equations, S.Chand, India.

2. Pinsky, M.A., (2003). Partial Differential Equations and Boundary Value Problems with Applications

Waveland Press, New York.

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Subject: Pure Mathematics (PMAT)

Special


Year 3

Sem 1

PMAT 41063 Functions of Several

Variables C PMAT 22045

PMAT 41073 Mathematical Methods C PMAT 22045

PMAT 41083 Advanced Theory of

Riemann Integration C PMAT 22045

Year 3

Sem 2

PMAT 42093 Complex Analysis C PMAT 31073

PMAT 42103 Differential Geometry O PMAT 22045

Year 4

Sem 1

PMAT 41284 Topology1 C PMAT 41083

PMAT 41294 Functional Analysis C PMAT 31073

PMAT 41304 Group Theory C PMAT 21035

Year 4

Sem 2

PMAT 42314 Measure Theory C PMAT 41083

PMAT 42324 Ring Theory and Field

Theory O PMAT 41304

PMAT 42334 Graph Theory and

Number Theory O PMAT 41304

PMAT 42344 Special Topics in

Mathematics and Statistics C PMAT 41083

PMAT 43358 Research/Study Project C

1 Students in the Mathematical Physics program are strongly advised to attend these lectures.

24

Level - 4


Title : Functions of Several Variables


Learning Outcomes:

At the end of this course, the student should be able to demonstrate knowledge of extending the ideas and

concepts they have studied in respect of functions of a single variable to functions of several variables,

and compute multiple integrals.

Course Contents: Limits, Continuity, Differentiability, Mean Value Theorem, Taylor’s Formula and Series, Extrema,

Lagrange Multipliers, Plane Transformations, The Double Integral, The Triple Integral, Leibniz’s Rule.





2. Spiegel, M.R., (1974). Advanced Calculas, Mcgraw-Hill, NewYork.

3. Gupta, S.L & Rani, N., (1999). Principles of Real Analysis, Vikas, India.

4. Bowman, F. & Gerard, F.A., (1967). Higher Calculus, Cambridge University Press.


Title : Mathematical Methods



solving problems involving partial differential equations including integral transform methods.

Course Content:

Partial Differential Equations: Introduction to first order partial differential equations, Theory of

second order partial differential equations.

Integral Transforms: Laplace Transforms, Fourier Transforms, Hankel Transforms, Fourier method for





1. Rainville, & Bedient, (1981). Elementary Differential Equations, Macmillan.

2. Pinsky, M.A., (2003). Partial Differential Equations and Boundary Value Problems with Application ,

Waveland press, New York.

3. Raisinghania, M.D., (1991). Advanced Differential Equations, S.Chands, India.


Title : Advanced Theory of Riemann Integration


Learning Outcomes: At the end of the course the student should be able to demonstrate knowledge of the

concepts and theorems of Riemann Integration and to apply them in solving advanced integration problems.

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Course Content:

Riemann Integration: The Riemann integral, Properties of the Riemann integral, Fundamental theorem of

calculus.

Improper Integrals: Properties of Improper Integrals, Leibnitz’s rule.




1. Ross, K.A. (1980). Elementary Analysis: The Theory of Calculus, Springer-Verlag, New York.

2. Gupta, S.L. & Rani, N. (1994). Principles of Real Analysis, Vikas, India.

3. Bartle, R.G. (1976). The Elements of Real Analysis, John-Wiley, New York.


Title : Differential Geometry


Learning Outcomes: At the end of this course the student should be able to demonstrate the fundamental

knowledge of curves and surfaces in space and the importance of the two factors curvature and torsion,

and their intrinsic properties.

Course Contents: Theory of Curves: Concept of a curve, Arc length, Curvature and torsion, Frenet-serret formulae,

General helix, intrinsic equations, Fundamental existence and uniqueness theorems for space curves,

Canonical representation of a curve. Involutes and Evolutes, Theory of contact.

Theory of Surfaces: Concept of a surface, Topological properties of a surface, Surface of revolution,

Ruled surfaces, Length of arc on a surface, Vector element of an area, First and second fundamental

forms, Curves on a surface, Direction coefficients, Direction ratios, Family of curves on a surface,

Double family of curves. Umbilical point, Intrinsic properties of a surface, Geodesies.

Curvature: Principle curvature and directions, Gaussian and mean curvatures, Lines of curvature

Rodrigues formula.

Introduction to Riemannian geometry




1. Lipschutz, L., (1969). Differential Geometry, McGraw-Hill, New York.

2. Willmore, T.J., (1964). An Introduction to Differential Geometry, Oxford University Press.

3. Do Carmo, M.P., (1976). Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey.


Title : Complex Analysis


Learning Outcomes: At the end of this course, the student should be able to demonstrate knowledge of the role of complex

numbers in Mathematics and similarities and the differences of the results in both real and complex

number systems. Also they should be able to demonstrate knowledge of usage of complex numbers,

complex functions and advanced theoretical aspects involving complex functions.

26

Course Contents:

Complex Numbers, Complex Valued Functions, Limits, Continuity, Differentiability, Cauchy-Riemann

Equations, Elementary Functions, Line Integrals, Cauchy’s Integral Theorem, Taylor and Laurent Series,

Singularities, Residue Theorem, Analytic functions, Maximum modulus theorem, Conformal mappings,

Schwarz Christoeffel Transformation, Rouche's theorem.



Recommended Reading : 1. Churchill, R.V., (1960). Complex Variables and Applications, McGraw-Hill, New York.

2. Murray, R.S., (1993). Complex Variables, McGraw-Hill, New York.

3. Pannasamy, S., (1995). Foundations of Complex Analysis, Narosa, India.


Title : Topology


Learning Outcomes: By the end of this course, the student should be able;

1. to demonstrate knowledge of definitions of topological and metric spaces and should be able to

demonstrate knowledge of the difference between standard topological and non-topological

properties

2. to explain the roles of open sets and their interconnections in topological spaces

3. to describe the topological notion of connectedness and its relation to path-connectedness

4. to describe the topological notion of compactness, and its significance in basic analysis.

Course Contents: Topological spaces, Basis for a topology, the subspace topology, Closed sets, Limit points, Continuous

functions, the product topology, the metric topology, Connected spaces, Compact spaces.




1. Munkres, J.R., (1999). Topology, a first course, Prentice-Hall, India.


Title : Functional Analysis



the functionals and its analysis under a topological background comprehending how algebra and analysis

combine to form a separate part of Pure Mathematics.

Course Contents: Metric spaces, Completion of metric spaces, Normed spaces, Banach spaces, Linear operators and

functionals, Inner product spaces, Hilbert spaces, Fundamental theorems for normed and Banach spaces.



27


1. Madox, I.J., (1992). Elements of Functional Analysis, Cambridge University Press.

2. Jain, P.K., Ahuja, O.P. & Ahmad, K., (1997). Functional Analysis, New Age International.

3. Kreyszig, E., (1978). Introductory Functional Analysis with Applications , John Wiley, New York.

4. Pannasamy, S., (1995). Foundations of Complex Analysis, Narosa, India.


Title : Group Theory


Learning Outcomes: At the end of the course the student should be able to demonstrate knowledge of the structure of Groups and to

apply the knowledge in solving problems in different areas in Algebra.

Course Contents: Groups, Subgroups, Normal subgroups, Quotient groups, Permutation groups, Derived group,

Homomorphisms, Automorphisms, Isomorphism theorems, Sylow's theorems, Internal direct product,

Structure theory of finite Abelian Groups, Groups of small order.



Recommended Reading : 1. Khanna, V.K. & Bhambri, S.K., (1998). A Course in Abstract Algebra, Vikas, India.

2. Frakeigh, J.B., (1976). A first course in Abstract Algebra, Addison-Wesley.

3. Baumslag, B. & Chandler, B., (1968). Group theory, McGraw-Hill, New York.

4. Narayan, S. & Pal, S., (1992). A Text Book of Modern Abstract Algebra, S.Chands, India.

5. Rotman, J.J., (1995). An Introduction to the Theory of Groups, Springer-Verlag.


Title : Measure Theory


Learning Outcomes: At the end of the course the student should be able to demonstrate knowledge of the

concepts and theorems of abstract Measure Theory and to apply them in Lebesgue integrals.

Course Contents:

Measure Theory: Algebra, -algebra, additivity properties of a set function, Measure, Borel sets,

Lebesgue measure, outer Measure, measurable subsets, measurable functions, Integral, Properties that

hold almost everywhere, integrable functions, Additivity Theorem, Monotone convergence theorem,

Dominated convergence theorem, Fatou's lemma, Relation of Riemann and Lebesgue integrals, Modes of

convergence.



Recommended Reading : 1. Cohn, D.L., (1997). Measure Theory, Birkhauser.

2. De Barra, G., (1991). Measure Theory and Integration, Wiley Eastern.


Title : Ring Theory and Field Theory


28

Learning Outcomes: At the end of the course the student should be able to demonstrate knowledge of the structures of Rings and

Fields and to apply the knowledge in solving problems in different areas in Algebra.

Course Contents: Ring Theory: Rings, Homomorphisms, Ideals, Quotient rings, Integral domains, Euclidean domains,

Factorization domains, Polynomial rings, Polynomials over the rational field.

Field Theory: Finite fields, Field of fractions, Field extensions.



Recommended Reading : 1. Khanna, V.K. & Bhambri, S.K., (1998). A Course in Abstract Algebra, Vikas, India.

2. Musili, C., (1994). Introduction to Rings & Modules, Narosa, India.

3. Fraleigh J. B., (1976). A first course in Abstract Algebra, Addison Wesley.

4. Cohn, P. M., (1992). Groups Rings and Fields, Springer.


Title : Graph Theory and Number Theory


Learning Outcomes: At the end of the course the student should be able to demonstrate knowledge of

elements of Graph Theory and Number Theory and to apply knowledge in solving problems in these areas.

Course Contents:

Graph Theory: Graphs and multigraphs, Subgraphs, Matrices and graphs, Isomorphic and

homeomorphic graphs, planar graphs, Coloured graphs, Colours and maps, Trees, Kuratowski's theorem.

Number Theory: Properties of integers, Prime numbers, linear theory of congruences, Number theoretic

functions, Some Diaphantine equations, Continued fractions.



Recommended Reading : 1. Johnsonbaugh, R., (1990). Discrete Mathematics, MacMillan.

2. Nivon, I., Zuckermann, H.S. & Montgomary, H.L., (1991). An Introduction to the Theory of

Numbers, John Wiley, New York.

3. Dickson, L.E., (2000). Introduction to the Theory of Numbers, Dover, New York.

4. Burton, D.M., (2001). Elementary Number Theory, McGraw-Hill, New York.


Title : Special Topics in Mathematics and Statistics


Learning Outcomes: At the end of this course, the student should be able to demonstrate knowledge of modern trends in

Mathematics and Statistics.

Course Contents:

This course is designed to include specialized modern topics in Mathematics and Statistics.

Method of Teaching and Learning: A combination of lectures, tutorial discussions and other methods.

29

Assessment: Based on tutorials, tests, presentations, end of course examination or other techniques.

Recommended Reading : Dependant on the topic.


Title : Research/Study Project

Learning Outcomes:

At the end of this course, the student should be able to demonstrate competence in research/independent-

study in an area in Mathematics/Statistics.

Method of Teaching and Learning:

A research/study project under the supervision of a senior staff member of the department.

Assessment: Submission of a research/study project report and an oral presentation.


Required reading material will be recommended by the supervisor depending on the releva nt project.

subject: applied mathematics (amat) general course...

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