B.Sc. (Hons) Maths
Semester V

COURSE CODECOURSE NAMECREDITS

JBMH501
AlgebraII:
UNITI
definition and examples of groups, elementary properties of groups, subgroups and examples of subgroups, centralizer, normalizer, center of a group, cyclic groups, generators of a cyclic groups, classification of subgroups of cyclic groups.
UNITII
Cycle notation for permutations, properties of permutations, even and odd permutations, alternating group, product (HK) of two subgroups.
UNITIII
definition and properties of cosets, Lagrange's theorem and consequences including Fermat Little Theorem, an application of cosets to permutations groups.
UNITIV
Normal subgroups, factor groups, applications of factor groups to the alternating group A4, commutator subgroup.
Books:
 1. Joseph A. Gallian, Contemporary Abstract Algebra(4th Edition), Narosa Publishing house, New Delhi, 1999.
04 
JBMH502
AnalysisII:
UNITI
Limits of functions, Limit theorems, sequential criterion for limits, Cauchy Critarian for finite limits divergence criteria, one sided limits.
UNITII
Continuous functions, sequential criterion for continuity & discontinuity, Dirchlet’s nowhere continuous functions, combination of Continuous functions, Compositions of Continuous functions, Continuous functions on an interval, boundedness theorem intermediate value theorem, intermediate value property, Uniform continuity, nonuniform continuity criteria, uniform continuity theorem.
UNITIII
Differentiability of a function at a point & in an interval, algebra of differentiable functions. interior extremumtheorem, Darboux’s theorem , Rolle’s theorem, Mean value theorem, Applications of mean value theoremCauchy’s mean value theorem.
UNITIV
Taylor’s theorem with Lagrange’s form of remainder, Taylor’s theorem with Cauchy’s form of remainder, application of Taylor’s theorem, Maclaurin;s Theorem, Taylor’s series &Maclaurin’s series convex functions, Application of mean value theorem and Maxima and Minima of two variables , Lagrange’s Multiplier.
 1. R. G. Bartle & D.R. Sherbert, Introduction to Real Analysis, John Wiley & Sons.
 2. K. A. Ross, Elementary Analysis: The Theory of Calculus, Springer,
 3. A. Mattuck, Introduction to Analysis, Prentice Hall.
 4. S.R.Ghorpade&B.V.Limaye, A Course in Calculus and Real Analysis  Springer.
04 
JBMH503
Elective PaperI(Integral Equation and Optimization/Mechanics/Statistics)
Integral Equation and Optimization:
UNITI
Linear integral equations, Equations of the first and second kind, Volterra and fredholm integral equations, Solution of volterra and Fredholm, Integral equations by the methods of successive approximations, Iterated and resolvent kernels, Neumann series reciprocal functions, Volterra solutions of Fredholm equations.
UNITII
Fredholm theorems, Fredholm associated equation, Solution of integral equations using Fredholm determinant and minor, Homogeneous integral equations, Integral equations with separable kernels, The Fredholm alternatives.
UNITIII
Symmetric kernals, fundamental theorems on symmetric equations, Theorem on symmetric kernels, solution of symmetric integral equations.
UNITIV
Formulation of problems, linear programming in matrix notation, Graphical solution of LPP, feasible solution, basic solution, basic feasible solution, Optimal solution, degenerate basic feasible solution, Convex set, extreme point of a convex set, Fundamental theorem of linear programming, convex and concave functions, and their properties, Simplex method.
Mechanics:
UNITI
Basic Concept of Mechanics: Fundamental laws of Newtonian mechanics. Intertial frame of reference. Particle, mass, rigid body, rest and motion and force. External and internal forces. Forces acting at a point. Triangle law of forces and polygon law of forces. Lami's theorem. Equilibrium of a system of particles. Necessary conditions for equilibrium forces.
UNITII
Moments, parallel forces, couples, moment of a force about a point and a line. Theorem of varignon. Necessary conditions for equilibrium (moment). Equilibrium of two couples. Reduction of a general plane force system, parallel force system in two and three dimensions. Friction.
UNITIII
Work and Energy: Conservative field and potential energy. Principle of conservation of energy for a particle and principle of virtual work for a system of particles. Catenary (cartesian, intrinsic forms). Geometrical properties of catenary. Centre of gravity and centre of parallel forces. Centre of gravity of some simple bodies: rod, triangle, arc, plane area, surface of revolution, sum of difference of two bodies, segment of a sphere and some simple curves.
UNITIV
Components of velocity and acceleration (cartesian, radial and transverse, tangential and normal). Uniformly accelerated motion. Simple harmonic motion. Resisted motion. Harmonic oscillators. Damped and forced vibration. Elastic strings. Hooke's law. Vertical and horizontal vibrations of a particle attached to an elastic string.
UNITV
Projectile and motion in a nonresisting medium. Constrained motion on a smooth vertical circle. Simple pendulum. Collisions (direct).
UNITVI
Motion of a Particle under a central force. Derential equation of a central orbit in both reciprocal polar and pedal coordinates, Newton's law of gravitation and planetary orbits. Kepler's laws of motion deducted from Newton's laws of gravitation and viceversa. Motion of the mass centre and motion relative to mass centre. Principle of linear momentum. Angular momentum and energy for a particle and for a system of particles. D'Alembert's principle. General theory of plane impulse.
 1. Synge and Griffith : Principle of Mechanics
 2. S.L. Loney : Dynamics of particles and rigid bodies
 3. A.S. Ramsey : Statics
 4. F. Chorlton : A Text book of Dynamics
 5. R.S. Verma : Statics
Statistics:
UNITI
Methods of least squares, and its use for Curve Fitting and fitting of straight lines and parabola, Normal equations.
UNITII
Bivariate distribution, Karl’s Pearson’s coefficient of Correlation, Rank Correlation and Line of Regression, Proof of 1< r < 1.
UNITIII
Consistency and Association of attributes, Theory of Attributes and their combination, class frequency. Association of datas, dependent and independent attributes
UNITIV
Finite difference and interpolation, various methods of interpolations Newton’s Gregory formula, finite difference and factorial Notation.
UNITV
Properties of χ 2 distribution, calculation of theortical freequences, problem of χ 2 distribution at significant level.
 1. "Statistics" by M. Ray and H. S. Sharma, Ram prashad & Sons
 2. "Statistics" by J. N. Kapoor and H. C. Saxena, S.Chand & Company
 3. "Statistics" by B. D. Gupta and O. P. Gupta, Krishana Prakashan Mandir
 4. "Statistics" by O. P. Gupta, Kedar Nath Ram Nath
 5. "Statistics" by J.K. Goyal and J. N. Sharma, Krishana Prakashan Mandir
 6. "Statistics" by V. K. Kapur and S. C. Gupta, Sultan Chand & Sons
04 
JBMH504
Numerical Analysis:
UNITI
Solution of algebraic and transcendental equations, The Bisection and Regula –Falsi method, Iteration methods namely, NewtonRaphson method, Solution of system o linear equations using Direct metho such as Matrix inversion method, Gauss elimination method, GaussSeidol method.
UNITII
Finite difference Operators and their relations, NewtonGregory Forward and Backward difference formulae, Gauss’s, Stirling’s and Bessel’s formulae, Lagranges Formulae, Divided difference and their properties, Newton’s General Interpolation formula, Inverse interpolation formula.
UNITIII
Numerical differentiation and integration, Numerical differentiation of tabular and nontabular functions, Numrical integration using Gauss Quadrature formula, Trapezoidal, Simpsons ‘1/3’ and ‘3/8’ rule , Weddle rule and Newto Cotes formula.
UNITIV
Ordinary differential equation, Euler’s method, Modified Euler’s method, Taylor’s series method, RungeKutta method of 4th order, MilneThomson method, Boundary value problem using finite difference method.
Books:
 1. S.S. Sastry, Introductory method of Numerical Analysis(IIIrd Edition), Prentice Hall of India (Ltd.)New Delhi110001, 1999.
 2. Numerical Methods by Jain and Iyengar , New Age International Publishers.
 3. M.K. Jain , S.R.K. Iyenger and R.K. Jain, Numerical Methods for scientic and Engineering computation, New Age publication.
 4. H.C. Saxena , Finite Difference and Numerical Analysis, S. Chand and Company Ltd. New Delhi, 1998.
 5. Numerical Analysis by S. Rajan, S.J. Publication , Meerut.
04 
JBMH505
Elective paper II (Differential Geometry and Tensor/Discrete Mathematics/Number Theory)
Differential Geometry and Tensor:
UNITI
Curves in space, space curves, are lengths, tangent plane lines, osculating plane, normal plane, unit vectors t, n, b, serretfernet formula, curvature and torsion of curves helix, osculating circle and osculation sphere.
UNITII
Fundamentals of surfaces, definition of surface, class of a surface, regular and singular point, tangent and normal planes, fundamental form and relation between E, F, G, Fundamental magnitude of slandered surface.
UNITIII
Envelopes and Developable surfaces, characteristics envelop, edge of regression, developable surface, envelops of a plane etc.
UNITIV
Contra variant & Covariant Vectors & Tensors, Contraction, Tensor algebra, Associated Vectors and Tensors.
 1. "Differential Geometry" by A. R. Vasistha and J. N. Sharma, KedarnathRamnath
 . "Tensor Calculus" by G. C. Sharma and S.K. Singh Laxmi Narayan Publisher Agra
 3. "Differential Geometry" by A.B. Chandra Moule and J. B. Chauhan, Siksha Sahitya Prakashan
 4. "Differential Geometry" by P. P. Gupta and G. S. Malik, PragatiPrakashan
 5. "Differential Geometry" by S. C. Mittal and D. C. Agarwal, Krishna Pracashan
 6. "Differential Geometry" by T. J. Willmore Oxford University Press, New Delhi
Discrete Mathematics:
UNITI
Definition of set, countable and uncountable sets, venn diagrams, proof of some general identity of sets, relation, types of relation , composition of relation, pictorial representation of relation, equivalence relation, function , types of function, one to one, into and onto function, inverse function, composition of function, mathematical induction(simple and strong).
UNITII
Posets, hass diagram of posets, isomorphism of ordered sets, well ordered sets, properties of lattice, Boolean algebra, SOP and POS form, logic gates, Kmaps.
UNITIII
Propositional logic, basic logic operator, truth tables, tautology, contradiction, algebra of proposition, logical implications, logical equivalence, predicates.
UNITIV
Recurrence relations, generating function, graph definition, types of graphs, representation of graphs, graph coloring, chromatic number, isomorphism and homomorphism of graphs.
 1. Discrete mathematics by Vinaya Rawol and bhakti Raul.(Techmax Publications)
 2. Discrete mathematics and its applications by Kenneth H Rose
 3. Discrete mathematics by S. Rajan, S.J. Publication, Meerut
Number Theory:
UNITI
Division Algorithm, greatest common divisor, least common multiple, unique factorization theorem, Relatively prime integers, Euler's function, Congruences, Complete set of residues (mod m), Euler's theorem, order of an element, (mod m).
UNITII
Linear congruences, Chinese remainder theorem, algebraic congruences (mod p), Lagrange's theorem, Wilson theorem, Algebraic congruences with composite modules.
UNITIII
Number theoretic functions, sum and number of divisors, totally multiplicative functions, definition and properties of the Dirchilet product, the mobius inverse formula, the greatest integer function, Euler's phi function, Properties of Euler phy function.
UNITIV
Order of an integer modulo n, primitive roots for primes, composite numbers having primitive roots, Euler's criterion, the Legendre symbol and its properties, quadratic reciprocity, quadratic congruences with composite moduli, Fermat's last theorem.
 1. David M. Burton, Elementary number theory, Tata McGrewHill, Indian reprint.
 2. Neville Robinns, Beginning Number Theory, Narosa publishing house Pvt. Limited, Delhi
04 
Total Credits20