BHU M.Sc Mathematics syllabus

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[Arun Vats]

Ok, as you want the syllabus of M.Sc Mathematics of BHU so here I am providing you.

BHU M.Sc Mathematics syllabus

Semester I

MTM 101 Algebra-I Credits : 4

The class equation, Cauchy`s theorem, Sylow p-subgroups, Direct product of groups. Structure theorem for
finitely generated abelian groups. Normal and subnormal series. Composition series, Jordan-Holder theorem.
Solvable groups. Insolvability of Sn for n ³ 5.
Extension fields. Finite, algebraic, and transcendental extensions. Splitting fields. Simple and normal
extensions. Perfect fields. Primitive elements. Algebraically closed fields. Automorphisms of extensions. Galois
extensions.
Fundamental theorem of Galois theory. Galois group over the rationals.
References:
1. I. N. Herstein, Topics in Algebra, Wiley Eastern, 1975.
2. P. B. Bhattacharya, S. K. Jain and S. R. Nagpal, Basic Abstract Algebra (2nd Edition), Cambridge
University Press, Indian Edition 1977.
3. Ramji Lal, Algebra, Vol.1, Shail Publications, Allahabad 2001.
4. Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House 1999.
5. D. S. Malik, J. N. Mordeson, and M. K. Sen, Fundamentals of Abstract Algebra, McGraw-Hill
International Edition, 1997.
MTM 102 Real Analysis-I Credits : 4
Definition and existence of Riemann-Stieltjes integral, Conditions for R-S integrability. Properties of the R-S
integral, R-S integrability of functions of a function.
Series of arbitrary terms. Convergence, divergence and oscillation, Abel’s and Dirichilet’s tests. Multiplication
of series. Rearrangements of terms of a series, Riemann’s theorem.
Sequences and series of functions, pointwise and uniform convergence, Cauchy’s criterion for uniform
convergence. Weierstrass M-test, Abel’s and Dirichlet’s tests for uniform convergence, uniform convergence
and continuity, uniform convergence and Riemann-Stieltjies integration, uniform convergence and
differentiation. Weierstrass approximation theorem. Power series. Uniqueness theorem for power series, Abel’s
and Tauber’s theorems.

References:
1. Walter Rudin, Principle of Mathematical Analysis (3rd edition) McGraw-Hill Kogakusha, 1976,
International Student Edition.
2. K. Knopp, Theory and Application of Infinite Series.
3. T. M. Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, 1985.

MTM 103 Topology Credits : 4
Definition and examples of topological spaces. Closed sets. Closure. Dense sets. neighborhoods, interior,
exterior, and boundary. Accumulation points and derived sets. Bases and sub-bases. Subspaces and relative
topology.
Alternative methods of defining a topology in terms of Kuratowski closure operator and neighborhood systems.
Continuous functions and homeomorphism. First and second countable space. Lindelöf spaces. Separable
spaces.
The separation axioms T0, T1, T2, T3½, T4; their characterizations and basic properties. Urysohn’s lemma. Tietze
extension theorem.
Compactness. Basic properties of compactness. Compactness and finite intersection property. Sequential,
countable, and B-W compactness. Local compactness. One-point compactification.
Connected spaces and their basic properties. Connectedness of the real line. Components. Locally connected
spaces.
Tychonoff product topology in terms of standard sub-base and its characterizations.Product topology and
separation axioms, connected-ness, and compactness (incl. the Tychonoff’s theorem), product spaces.
Nets and filters, their convergence, and interrelation. Hausdorffness and compactness in terms of net/filter
convergence.

References:
1. J. L. Kelley, General Topology, Van Nostrand, 1995.
2. K. D. Joshi, Introduction to General Topology, Wiley Eastern, 1983.
3. James R. Munkres, Topology, 2nd Edition, Pearson International, 2000.
4. J. Dugundji, Topology, Prentice-Hall of India, 1966.
5. George F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1963.
6. N. Bourbaki, General Topology, Part I, Addison-Wesley, 1966.

7. S. Willard, General Topology, Addison-Wesley, 1970.
8. S.W. Davis Topology, Tata McGraw Hill, 2006
MTM 104 Differential Geometry of Manifolds-I Credits : 4
Tensor of the type(r,s). Definition and examples of differentiable manifolds. Tangent spaces. Jacobian map.
One parameter group of transformations. Lie derivatives. Immersion and imbeddings. Distributions.
Riemannian manifolds. Riemannian Connection. Curvature tensors. Sectional curvature. Schur’s theorem.
Geodesics. Projective curvature tensor. Conformal curvature tensor. Semi-symmetric connections.
Submanifolds and Hypersurfaces. Normals. Gauss’s formula. Weingarten equations. Lines of curvature.
Generalized Gauss and Mainardi-Codazzi equations.
References:
1. R. S. Mishra, A Course in Tensors with Applications to Riemanian Geometry, Pothishala, Allahabad,
1965.
2. Y. Matsushima, Differentiable Manifolds, Marcel Dekker, 1972.
3. B. B. Sinha, An Introduction to Modern Differential Geometry, Kalyani Prakashan, New Delhi, 1982.
4. Y. Talpiert, Differential Geometry with applications to Mechanics and Physics, Marcel Dekkar Inc.
2001.
5. N.J. Hicks, Notes on Differential Geometry, D. Van Nostrand Inc. , 1965.
MTM 105 Set Theory & Complex Analysis Credits : 4
Set Theory: Countable and uncountable sets, Cardinal numbers, Schroeder- Bernstien theorem, POSET, Zorn`s
lemma and its application.
Complex Analysis: Complex Integration. Cauchy-Goursat Theorem. Cauchy’s integral formula. Higher order
derivatives. Morera’s theorem. Cauchy’s inequality and Liouville’s theorem. The fundamental theorem of
algebra. Taylor’s Theorem. Maximum modulus Principle, Schwarz lemma.
Laurent’s Series. Isolated singularities. Casporati-Weierstress theorem. Meromorphic functions. The argument
principle. Rouche’s theorem.
Residues. Cauchy’s residue theorem. Evaluation of integrals. Branches of many valued functions with special
reference to arg Z, Log Z, and Za.
Analytic continuation.

References:
1. K. Knopp, Theory of Functions, Vol. 1.
2. E. C. Titchmarsh, The Theory of Functions, Oxford University Press.
3. J. B. Conway, Functions of One Complex Variable, Narosa Publishing House, 1980.
4. E. T. Copson, Complex Variables, Oxford University Press.
5. L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1977.
6. D. Sarason, Complex Function Theory, Hindustan Book Agency, Delhi, 1994.
7. P. Suppes, Axiomatic Set Theory, Van Nostrand, 1960.
8. P.R. Halmos, Naive Set Theory, Van Nostrand, 1960.
9. K.K. Jha, Advanced Set Theory & Fundamentals of Mathematics, P.C. Dwadesh Shreni & Co.,
Aligarh, 1993.
MTM 106M Mathematical Methods Credits : 3
The objective of the course is to introduce the mathematical methods to the PG students of Physical sciences for
the possible requirements in the modeling of the problems in their respective discipline of studies.
Integral Transforms.
Z-Transforms.
Fourier Series.
Matrix Coputations.
Chebyshev Polynomials
Complex Integration.
References:
1. G.B. Thomas, R.L.Finney, M.D.Weir, Calculus and Analytic Geometry, Pearson Education Ltd, 2003.
2. Erwin Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1999.

SEMESTER - II
MTM 201 Algebra–II Credits : 4
Modules, submodules, Quotient Modules, Isomorphism theorems. Cyclic modules, simple modules and semisimple
modules and rings Schur’s lemma. Free modules. Noetherian and Artinian modules and rings. Hilbert
basis theorem .
Solution of polynomial equations by radicals. Insolvability of the general equation of degree ³ 5 by radicals.
Finite fields.
Canonical forms: Similarity of linear transformations. Invariant subspaces. Reduction to triangular forms.
Nilpotent transformations. Index of nilpotency. Invariants of a nilpotent transformation. The primary
decomposition theorem. Jordan blocks and Jordan form.
References:
1. I. N. Herstein, Topics in Algebra, Wiley Eastern, 1975.
2. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra (2nd Edition), Cambridge
University Press, 1997.
3. K. Hoffman and R. Kunze, Linear Algebra, 2nd Edition, Prentice Hall of India, 1971.
4. D. S. Malik, J. N. Mordeson, and M. K. Sen, Fundamentals of Abstract Algebra, McGraw-Hill
International Edition, 1997.
5. Vivek Sahai and Vikas Bist, Algebra, Narosa Publishing House, 1999.
6. Ramji Lal, Fundamentals in Abstract Algebra, Chakra Prakashan, Allahabad, 1985.
7. J.S. Golan, Modules & the Structures of Rings, Marcl Dekkar. Inc.
MTM 202 Real Analysis-II Credits : 4
Functions of several variables. Derivative of functions in an open subset of Ân into Âm as a linear
transformation. Chain rule. Partial derivatives. Taylor’s theorem. Inverse function theorem. Implicit function
theorem. Jacobians.
Measures and outer measures. Measure induced by an outer measure, Extension of a measure. Uniqueness of
Extension, Completion of a measure. Lebesgue outer measure. Measurable sets. Non-Leesgue measurable sets.
Regularity. Measurable functions. Borel and Lebesgue measurability.
Integration of non-negative functions. The general integral. Convergence theorems. Riemann and Lebesgue
integrals.
References:
1. Walter Rudin, Principle of Mathematical Analysis (3rd edition) McGraw-Hill Kogakusha, International
Student Edition, 1976.
2. H. L., Royden, Real Analysis, 4th Edition, Macmillan, 1993.
3. P. R. Halmos, Measure Theory, Van Nostrand, 1950.
4. G. de Barra, Measure Theory and Integration, Wiley Eastern, 1981.
5. E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, 1969.
6. P. K. Jain and V. P. Gupta, Lebesgue Measure and Integration, New Age International, New Delhi,
2000.
7. R. G. Bartle, The Elements of Integration, John Wiley, 1966.
MTM 203 Analytic Dynamics Credits : 4
Rotation of a vector in two and three dimensional fixed frame of references. Kinetic energy and angular
momentum of rigid body rotating about its fixed point.
Euler dynamic and geometrical equations of motion.
Generalized coordinates, momentum and force components. Lagrange equations of motion under finite forces,
cyclic coordinates and conservation of energy.
Lagrangian approach to some known problems-motions of simple, double, spherical and cycloidal pendulums,
motion of a particle in polar system, motion of a particle in a rotating plane, motion of a particle inside a
paraboloid, motion of an insect crawling on a rod rotating about its one end, motion of masses hung by light
strings passing over pulleys, motion of a sphere on the top of a fixed sphere and Euler dynamic equations.
Lagrange equations for constrained motion under finite forces. Lagrange equations of motion under impulses,
motion of parallelogram about its centre and some of its particular cases.
Small oscillations for longitudinal and transverse vibrations.
Equations of motion in Hamiltonian approach and its applications on known problems as given above.
Conservation of energy. Legendre dual transformations.

Hamilton principle and principle of least action. Hamilton-Jacobi equation of motion, Hamilton-Jacobi theorem
and its verification on the motions of a projectile under gravity in two dimensions and motion of a particle
describing a central orbit.
Phase space, canonical transformations, conditions of canonicality, cyclic relations, generating functions,
invariance of elementary phase space, canonical transformations form a group and Liouville theorem.
Poisson brackets, Poisson first and second theorems, Poisson. Jacobi identity and invariance of Poisson bracket.
References:
1. A. S. Ramsay, Dynamic –Part II.
2. N. C. Rana and P.S. Joag, Classical Mechanics, Tata McGraw-Hill, 1991.
3. H. Goldstein, Classical Mechanics, Narosa, 1990.
4. J. L. Synge and B. A. Griffith, Principles of Mechanics, McGraw-Hill, 1991.
5. L. N. Hand and J. D. Finch, Analytical Mechanics, Cambridge University Press, 1998.
6. Naveen Kumar, Generalized Motion of Rigid Body, Narosa, 2004.
MTM 204 Differential Geometry of Manifolds-II Credits : 4
Topological groups. Lie groups and Lie algebras. Product of two Lie groups. One parameter subgroups and
exponential maps. Examples of Lie groups. Homomorphism and isomorphism. Lie transformation groups.
General linear groups.
Principal fiber bundle. Linear frame bundle. Associated fiber bundle. Vector bundle. Tangent bundle. Induced
bundle. Bundle homomorphisms. Exterior Algebra. Exterior derivative.
Almost complex and Almost contact structures. Nijenhuis tensor. Contravariant and covariant almost analytic
vector fields in almost complex manifold. F-Connexion.
Almost complex and almost contact submanifolds and hypersurfaces.
References:
1. B. B. Sinha, An Introduction to Modern Differential Geometry, Kalyani Prakashan, New Delhi, 1982.
2. K. Yano and M. Kon, Structure of Manifolds, World Scientific, 1984.
3. Y. Matsushima , Differential manifolds, Marcel Dekkar, 1972.
4. K. Yano, Differential Geometry of Complex and almost Complex Spaces, Pergaman Press, 1965.
5. R. S. Mishra, Structures on a Differentiable Manifold and Their Applications, Chandrama Prakashan,
Allahabad, 1984.
MTM 205 Theory of Optimization Credits : 4
Unconstrained Optimization: Introduction, Gradient methods, Conjugate Direction Methods, Newton’s Method,
Quasi Newton Method.
Linear Programming: Simplex Method, Duality and Non- simplex Methods.
Non- Linear Constrained Optimization: Introduction, Lagrange’s multipliers, Kuhn- Tucker conditions, Convex
Optimization.
Evolutionary Algorithms: Neural Networks: Introduction, Basic Hopfield Model, Delta Rule, Single Neuron
Training, Backpropagation algorithm. Genetic Algorithm: Basic description, Simple real number algorithm.
References:
1. Edwin K. P. P. Chong, Stanislaw H. Zak, An Introduction to Optimization, Johan Welly & Sons Inc
2001.
2. M. C. Joshi & K.M. Moudgalya, Optimization Theory & Practice, Narosa Publ. New Delhi, 2004.
3. S.S.Rao, Engg. Optimization: Theory & Practice, New Age Intl. Pub. New Delhi, 2003.
4. Laurence, Fausett, Fundamentals of Neural Networks, Pearson education Ltd, 2005.
5. D.E. Goldberg, Genetic Algorithms in neural optimization and machine learning, Pearson Education.
Ltd. 2004.
MTM 206M Mathematical Modeling Credits : 3
Simple situations requiring mathematical modeling, techniques of mathematical modeling, Classifications,
Characteristics and limitations of mathematical models, Some simple illustrations.
Mathematical modeling through differential equations, linear growth and decay models, Non linear growth and
decay models, Compartment models, Mathematical modeling in dynamics through ordinary differential
equations of first order.
Mathematical models through difference equations, some simple models, Basic theory of linear difference
equations with constant coefficients, Mathematical modeling through difference equations in economic and
finance, Mathematical modeling through difference equations in population dynamic and genetics.
Situations that can be modeled through graphs. Mathematical models in terms of Directed graphs, Mathematical
models in terms of signed graphs, Mathematical models in terms of weighted digraphs.

Mathematical modeling through linear programming, Linear programming models in forest management.
Transportation and assignment models.
References:
1. J. N. Kapur, Mathematical Modeling, Wiley Eastern.
2. D. N. Burghes, Mathematical Modeling in the Social Management and Life Science, Ellie Herwood
and John Wiley.
3. F. Charlton, Ordinary Differential and Difference Equations, Van Nostrand.
SEMESTER -III
MTM 301 Hydrodynamics Credits : 4
Equation of continuity, Boundary surfaces, streamlines, Irrotational and rotational motions, Vortex lines,
Euler’s Equation of motion, Bernoulli’s theorem, Impulsive actions. Motion in two-dimensions, Conjugate
functions, Source, sink, doublets and their images, conformal mapping, Two-dimensional irrotational motion
produced by the motion of circular cylinder in an infinite mass of liquid, Theorem of Blasius, Motion of a
sphere through a liquid at rest at infinity. Liquid streaming past a fixed sphere, Equation of motion of a sphere.
Stress components in real fluid, Equilibrium equation in stress components, Transformation of stress
components, Principal stress, Nature of strains, Transformation of rates of strain, Relationship between stress
and rate of strain, Navier-Stokes equation of motion.
References:
1. W. H. Besant and A. S. Ramsey, A Treatise on Hydrodynamics, CBS Publishers and Distributors,
Delhi, 1988.
2. S. W. Yuan, Foundations of Fluid Dynamics, Prentice-Hall of India, 1988.
MTM 302 Normed Linear Spaces and Theory of Integration Credits : 4
Normed linear spaces and Banach spaces. The LP-space. Convex functions. Jensen’s inequality. Holder and
Minkowski inequalities. Completeness of LP. Convergence in measure, Almost uniform convergence.
Signed measure. Hahn and Jordan decomposition theorems. Absolutely continuous and singular measures.
Radon Nikodyn theorem. Labesgue decomposition. Riesz representation theorem. Extension theorem
(carathedory). Lebesgue-Stieltjes integral.
Product measures. Fubini’s theorem. Baire sets. Baire measure. Continuous functions with compact support.
Regularity of measures on locally compact spaces. Integration of continuous functions with compact support.
Reisz-Markoff theorem.
References:
1. H. L. Royden, Real Analysis, Macmillan, 4th Edition, 1993.
2. P. R. Halmos, Measure Theory, Van Nostrand, 1950.
3. S. K. Berberian, Measure and Integration, Wiley Eastern, 1981.
4. A. E. Taylor, Introduction to Functional Analysis, John Wiley, 1958.
5. G. de Barra, Measure Theory and Integration, Wiley Eastern, 1981.
6. R. G. Bartle, The Elements of Integration, John Wiley, 1966.
7. Inder K. Rana, An Introduction to Measure and Integration, Narosa Publishing House, 1997.
MTM 303 Numerical Analysis Credits : 4
Integral equations :Fredholm and Volterra equations of first and second types. Conversions of initial and
boundary value problems into integral equations, numerical solutions of integral equations using Newton-Cotes,
Lagrange`s linear interpolation and Chebyshev polynomial.
Matrix Computations: System of linear equations, Conditioning of Matrices, Matrix inversion method, Matrix
factorization, Tridiagonal systems.
Numerical solutions of system of simultaneous first order differential equations and second order initial value
problems (IVP) by Euler and Runge-Kutta (IV order) explicit methods.
Numerical solutions of second order boundary value problems (BVP) of first, second and third types by
shooting method and finite difference methods.
Finite Element method: Introduction, Methods of approximation: Rayleigh-Ritz Method, Gelarkin Method and
its application for solution of ordinary BVP.

References:
1. M. K. Jain, S. R. K. Iyenger and R. K. Jain, Numerical Methods for Scientific and Engineering
Computations, New Age Publications, 2003.

2. M. K. Jain, Numerical Solution of Differential Equations, 2nd edition, Wiley-Eastern.
3. S. S. Sastry, Introductory Methods of Numerical Analysis,
4. D.V. Griffiths and I.M. Smith, Numerical Methods for Engineers, Oxford University Press, 1993.
5. C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Addison- Wesley, 1998.
6. A. S. Gupta, Text Book on Calculas of Variation, Prentice-Hall of India, 2002.
7. Naveen Kumar, An Elementary Course on Variational Problems in Calculus,
Narosa, 2004.
MAJOR ELECTIVE
(Any two of the following courses each having 3 Credits)
MTM 304 Discrete Mathematics Credits : 4
Graph Theory: Graphs, planar graphs and their properties. Trees. Euler’s formula for connected planar graphs.
Bipartite graphs. Spanning trees, Minimal spanning trees, Kruskal’s Algorithms, Matrix representations of
graph, Directed graphs, Weighted undirected graphs, Dijkstra’s algorithm. Warshal’s algorithm, Directed trees ,
Search trees, Traversals.
Theory of Computation: Finite automata, Deterministic and non deterministic finite automata , Moore and
Mealy machines. Regular expressions.Grammars and Languages, Derivations, Language generated by a
grammar. Regular Language and regular grammar.Regular and Context free grammar, Context sensitive
grammars and Languages. Pumping Lemma, Kleene’s theorem.
Turing Machines: Basic definitions. Turing machines as language acceptors. Universal Turing machines. Turing
machine halting problem.
References:
1. F. Harary, Graph Theory, Narosa.
2. Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall
of India.
3. W. T. Tutte, Graph Theory, Cambridge University Press, 2001
4. D. Kelly, Automata and Formal Languages: An Introduction, Prentice-Hall, 1995.
5. J. E. Hopcroft, R. Motwani, and J. D. Ullman, Introduction to Automata, Languages, and Computation
(2nd edition), Pearson Edition, 2001.
6. P. Linz, An Introduction to Formal Languages and Automata, 3rd Edition,
MTM 305 Operations Research Credits : 4
Game Theory: Two person zero sum games, Games with mixed strategies, Graphical solution, Solution by
linear programming.
Basic Concept of Multi Objective and Multi Level Optimization.
Integer Programming, Mixed Integer Programming. Linear Fractional Programming. Goal Programming.
Sensitivity Analysis and System Reliability.
Geometric Programming: Constrained and Unconstrained Minimization Problems.
Dynamic Programming: Deterministic and Probabilistic dynamic programming.
Stochastic Programming: Stochastic Linear and Stochastic Non linear Programming.
Netwok Schedulig by PERT/CPM.
References:
1. F. S. Hiller and G. J. Leiberman, Introduction to Operations Research (6th Edition), McGraw-Hill
International Edition, 1995.
2. G. Hadley, Nonlinear and Dynamic Programming, Addison Wesley.
3. H. A. Taha, Operations Research –An Introduction, Macmillan.
4. Kanti Swarup, P. K. Gupta and Man Mohan, Operations Research, Sultan Chand & Sons, New Delhi.
5. S. S. Rao, Optimization Theory and Applications, Wiley Eastern.
6. N. S. Kambo, Mathematical Programming Techniques, Affiliated East-West Press Pvt. Ltd., New
Delhi.
MTM 306 Gravitation Credits : 4
Newtonian theory : Attraction and potential of rod, disc, spherical shell and sphere. Surface integral of normal
attractions-Gauss theorem, Laplace and Poission equations. Work done by self attracting systems. Distribution
for given potentials. Equipotential surfaces.
Einstein’s Theory : Principles of equivalence and general covariance, Geodesic postulate. Newtonian
approximation of general relativistic equations of motion. Heuristic derivation of Einstein’s field equations,

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Banaras Hindu University
Ramnagar Rd, Opposite Madan Mohan Malviya Road, Near sir Sundar Hospital, Saket Nagar Colony, Lanka, Banaras, Uttar Pradesh 221005 ‎
0542 236 8558 ‎

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