MSC Mathematics Syllabus outline

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Books Recommended
1. M.Saleem and M.Rafique, Special Relativity, Ellis Horwood, 1992.
2. Rosser, Special Relativity, 1972.
3. W. Ringler, Introduction to Special Relativity, Oxford, 1982.
4. D.T. Greenwood, Classical Dynamics, Prentice-Hall, Inc., 1965.
5. H.Goldstein, Classical Mechanics, Addison-Wesley;, 1964.
6. L.A. Pars, Treatise of Analytical Dynamics, Heninemann Press, London, 1965.
Paper (IV-VI) option (ix): Electromagnetic Theory
Five question to be attempted, selecting at least one question from each section
Section I (3/9)
Electrostatics
Coulomb’s law. Electric Field and potential. Lines of force and equipotential surfaces. Gauss’s law and deductions. Conductors and condensers. Dipoles. Dielectrics. Polarisation and apparent charges. Electric displacement. Energy of the field. Minimum energy.
Magnetostatic Field
The Magnetostatic Law of Force. Magnetic Doublets. Magnetic shells. Force on Magnetic doublets. Magnetic induction. Para and dia magnetism.
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Section II (3/9)
Steady and Slowly Varying Currents
Electric current. Linear conductors. Conductivity. Resistance, Kirchhoff’s laws. Heat production. Current density vector. Magnetic field of straight and circular current. Magntic flux. Vector potential. Forces on a circuit in magnetic field. Magnetic field energy. Law of electromagnetic induction coefficients of self and mutual induction. Alternating current and simple I.C.R. circuits in series and parallel. Power factor.
Section III (3/9)
Potential Problems
The equations of electromagnetism
Maxwell’s equations in free space and material media. Solution of Maxwell’s equations. Plane electromagnetic waves in homogeneous and isotropic media. Reflection and Refraction of plane waves, Wave guides. Laplace’s equation in plane, polar and cylindrical coordinates. Simple introduction to the Legendre polynomials. Method of Images; Images in a plane, Images with spheres and cylinders.
Books Recommended
1. Ferraro, Electromagnetic Theory, Athlone Press, 1954.
2. J.R. Reitz & Milford, Foundations of Electromagnetic Theory, Addison-Wesley, 1967.
3. Pugh & Pugh, Electricity and Magnetism.
4. J.D. Jackson, Classical Electrodynamics, John Wiley, 1963.
Paper (IV-VI) option (x): Operations Research
Five questions to be attempted, selecting at least one question from each section.
Section I (3/9)
Linear Programming Formulations and Graphical Solution. Simplex Method. M-Technique and Two-phase Technique. Special Cases. Sensitivity Analysis. The Dual Problem. Primal-Dual Relationships. Dual Simplex Method. Sensitivity and Postoptimal Analysis.
Section II (3/9)
Transportation Model. North-West Corner, Least-Cost and Vogel’s Approximations Methods. The Method of Multipliers. The Assignment model. The Transhipment Model. Network Minimization. Shortest-Route Algorithms for Acyclic Networks. Maximal-flow
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problem. Matrix Definition of LP Problem. Revised Simplex Method. Bounded Variables. Decomposition Algorithm. Parametric Linear Programming.
Section III (3/9)
Applications of Integer Programming. Cutting-plane Algorithms. Branch-and-Bound Method. Zero-one Implicit Enumeration. Elements of Dynamic Programming. Problem of Dimensionality. Solution of Linear Programmes by Dynamic Programming.
Books Recommended
1. Hamdy A. Taha, Operations Research-An Introduction, Macmillan Publishing Company Inc., New York, 1987.
2. B.E. Gillett, Introduction to Operations Research, Tata McGraw Hill Publishing Company Ltd., New Delhi.
3. F.S. Hillier & G.J. Liebraman, Operations Research, CBS Publishers and Distributors, New Delhi, 1974.
4. C.M. Harvey, Operations Research, North Holland, New Delhi, 1979.
Papers (IV-VI) option (xi) Differential Equations and Dynamical Systems
Attempt any five questions.
Systems of linear differential equations
Linear systems, solution matrix, fundamental solution matrix, phase space analysis, autonomous systems, definition of stability, stability for linear and almost linear systems, basic concepts of Liapunov’s method.
Dynamical Systems
What is a dynamical system? Phase space, dynamical equations, flows, iterated maps, examples, Reduction to first order autonomous systems. Return maps. Examples. Fixed points and periodic orbits. Notions of stability, invariant sets and attractors. Examples of attractors, strange attractors and chaos, introduction to symbolic coding. Linear systems and their behaviour. Linearizing nonlinear systems. Linear stability. Floquet theory. Co-ordinate changes and conjugacies between dynamical systems. Hartman-Grossman theorem (without proof). Stable and unstable manifolds for fixed points and periodic orbits. Introduction to bifurcations, the distinction between local and global bifurcations. Introduction to symbolic dynamics, the shift map and the Smale horseshoe. Local bifurcation theory: the saddle node, transcritical, period doubling and Hopf bifurcations.
Books Recommended
1. Richard E. Williamson, Introduction to Differential Equations and Dynamical Systems, Mcgraw Hill, international edition, 1997.
2. V.I. Arnol’d, Ordinary Differential Equations, Springer, 1988.
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3. V.I. Arnol’d, Geometrial Methods in the Theory of Ordinary Differential Equations, Springer, 1988.
4. D.K. Arrowsmith and C.M. Place, Introduction to Dynamical Systems, Cambridge University Press, 1990.
5. P.G. Drazin, Nonlinear Systems, Cambridge University Press, 1992.
6. P.A. Glendinning, Stability, Instability and Chaos, Cambridge University Press, 1994.
7. R. Grimshaw, Nonlinear Ordinary Differential Equations, CRC Press, 1991.
8. M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1974.
9. D.W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, 1987.
10. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, 1990.
11. R. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, 1989.
12. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, 1983.
13. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
14. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, 1995.
Paper (IV-VI) option (xii): Waves and Solitons
Attempt any five questions.
Weakly dispersive nonlinear waves. Water waves in a rectangular channel and the Korteweg-de Vries (KdV) equation. Solitary waves and their interactions. Conservation laws: mass, momentum, energy and others. The modified KdV equations and the Miura transformation to the KdV. Derivation of the KdV, mKdV and nonlinear Schrodinger equations as models of physica systems. The Lax pair representation for the KdV, scattering theory and the inverse transform, leading to solution of the initial value problem.
Books Recommended
1. P.G. Drazin, Nonlinear Systems, Cambridge University Press, 1992.
2. R. Grimshaw, Nonlinear Ordinary Differential Equations, CRC Press, 1991.
3. D.W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, 1987.
4. P.G. Drazin, Solitons, Cambridge University Press, 1983.
5. P.G. Drazin & R.S. Johnson, Solitons: An Introduction, Cambridge University Press, 1989, 1990.
6. P.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, 1982.
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Paper (IV-VI) option (xiii) Solid Mechanics
Five questions to be attempted, selecting at least two questions from each section.
Section I: Elasticity (5/9)
Analysis of stress and strain, Generalized Hook’s law. Differential equations of equilibrium in terms of stress and in terms of displacements. Boundary conditions. Compatibility equations. Plane stress. Plane strain. Stress function. Two-dimensional problems in rectangular and polar co-ordinates. Torsion problems.
Section II: Elastodynamics (4/9)
Equations of wave propagation in elastic solids, Primary and secondary waves. Reflection and transmission at plane boundaries. Surface waves: Love waves and Raleigh waves. Dispersion relations. Geophysical applications.
Books Recommended
1. S.P. Timoshonko, & J.N. Goodier, Theory of Elasticity, McGraw-Hill, 1970.
2. I.S. Sokolnikoff, Mathematical Theory of Elasticity, McGraw Hill, 1956.
3. W.Prager, Introduction to Mechanics of Continua, Gim and Co., 1961.
4. J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Company, Amsterdam, 1973.
5. W.M. Ewing, W.S. Jardetsky, and F. Press, Elastic Waves in Layered Media, McGraw Hill.
6. K.F. Graff, Wave Motion in Elastic Solids, Clarendon Press, 1975.
7. J. Jaffreys, The Earth, Fifth edition, Cambridge University Press, 1970.
8. K.E. Bullen, An Introduction to the Theory of Seismology, Cambridge University Press, 1965.
Paper (IV-VI) option (xiv): Theory of Optimization
Attempt any five questions.
Introduction to optimization, Single variable optimization. Multivariable optimization. Linear Programming Problem. Geometry of linear programming problems. Solution of system of linear simultaneous equations. Simplex method, Revised simplex method. Duality in linear programming. Decomposition Principle. Transportation problem. Unimodel function. Elimination methods, Unrestricted search. Exhaustive search. Dichotomous search. Fibonacci method. Quadratic interpolation method, Cubic interpolation method. Direct root method. Direct search method. Random search methods. Univariate method. Descent methods. Steepest descent method. Conjugate gradient method. Quasi Newton methods. Cutting plane method. Methods of feasible directions. Penalty function method, Dynamic programming.
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Books Recommended
1. B.S. Gottfried & W. Joel, Introduction to Optimization Theory, Prentice-Hall, 1973.
2. S.S. Rao, Optimization Theory and Applications.
3. M.J. Fryer, Optimization Theory: Applications in Operations Research and Economics.
4. K.V. Mital, Optimization Methods in Operations Research and Systems Analysis, Second Edition, 1983.
5. R.K. Sudaram, A First Course in Optimization Theory, Cambridge University Press, 1996.
Paper (IV-VI) option (xv): Theory of Approximation and Splines
Five questions to be attempted, selecting at least two questions from each section.
Section I: (4/9)
Euclidean Geometry
Basic concepts of Euclidean Geometry, Scalar and Vector functions, Barycentric Coordinates, Convex Hull, Matrices of Affine Maps: Translation, Rotation, Scaling. Reflection and Shear.
Approximation using Polynomials
Linear Interpolation, Least squares polynomial curve fitting, Lagrange’s Method. Hermite’s Methods, Divided Differences Methods.
Section II (5/9)
Parametric Curves (Scalar and Vector Case)
Algebraic Form. Hermite Form. Control Point Form, Bernstein Bezier Form, Matrix Forms of Parametric Curves, Algorithms to compute B.B. Form, Convexhull Property, Affine invariance property, variation diminishing property, Rational Quadratic Form, Rational Cubic Form.
Spline Functions
Splines, Cubic Splines, End Conditions of Cubic Splines: Clamped conditions, Natural conditions, 2nd Derivative conditions, Periodic conditions, Not a knot conditions, General Splines: Natural Splines, Periodic Splines, Truncated Power Function, Representation of spline in terms of truncated power functions, Odd degree interpolating splines.
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Books Recommended
1. Curves and Surfaces for Computer Aided Geometric Design A Practical Guide, Academic Press. Inc. by Gerald Farin.
2. Computational Geometry for Design and Manufacture, Ellis Horwood by I.D. Faux.
3. An Introduction to Spline for use in Computer Graphics and Geometric Modeling, Morgan Kaufmann Publisher, Inc. By Richard H. Bartels.
4. A Practical Guide to Splines, Springer Verlag 1978 by Carl de Boor.
5. Spline Functions: Basic Theory, John Wiley 1981 by Schumaker.
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