B.Sc. MathematicsVictor Nandi
- Distance learning
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FIRST YEAR PAPER I CALCULUS Unit I Curvature – radius of curvature – Cartesian & polar – centre of curvature – Involute & evolute – Asymptotes in Cartesian co-ordinates – Multiple points – double points. Unit II Evaluation of double & triple integrals – Jacobians, change of variables. Unit III First order differential: equations of higher degree – solvable for p, x & y – Clairut’s form / linear differential equations of second order – particular integrals for functions of the form, Xᴺ, eᵃᵡ, eᵃᵡf(x). Second order differential equations with variable coefficients. Unit IV Laplace transform – Inverse transform – Properties – Solving differential equations. Simultaneous equations of first order using Laplace transform. Unit V Partial differential equations of first order – formation – different kinds of solution – four standard forms – Lagranges method. Books: 1. Calculus 1, 2 & 3, T.K. Manickavachagom Pillai & others. 2. Calculus 1 & 2, S. Arumugam & Issac. PAPER II CLASSICAL ALGEBRA Unit I Theory of Equations: Every equation f(x)=0 of nth degree has ‘n’ roots. Symmetric functions of the roots in terms of the coefficients – sum of the rth powers of the root – Newton’s theorem – Descartes rule of sign – Rolle’s theorem. Unit II Reciprocal Equations: Transformation of Equations – solution of cubic & biquadric equation – Cardon’s land Ferrari’s methods – Approximate solution of numerical equations – Newton’s & Horner’s methods. Unit III Sequences & series: Sequences – limits, bounded, monotonic, convergent, oscillatory & divergent sequences – algebra of limits – subsequences – Cauchy sequences in R & Cauchy’s general principle of convergence. Unit IV Series – convergence, divergence – geometric, harmonic, exponential, binomial & logarithmic series – Cauchy’s general principle of convergence – comparison test – tests of convergence of positive termed series – Kummer’s test, ratio test, Raabe’s test, Cauchy’s root test, Cauchy’s condensation test. Unit V Summation of series using exponential, binomial & logarithmic series. Books: 1. Sequences & series, S. Arumugam & others. 2. Algebra – Vol. I, T.K. Manickavachagom Pillai & others. 3. Real Analysis – Vol. I, K. Chandrasekara Rao & K.S. Narayanan. 4. Infinite series, Bromwich. PAPER III STATISTICS Unit I Correlation: Karl Pearson’s coefficient of correlations. Lines of regression – Regression coefficients – Rank correlation. Unit II Probability – Definition – application of addition & multiplication theorems – conditional. Probability – Mathematical Expectations – Moment generating function – special distributions, (Binomial distribution, Poisson distribution, Normal distribution – properties). Unit III Association of attributes – coefficient of association – consistency – Time series – Definition – components of a time series – seasonal & cyclic variations. Unit IV Sampling – definition – large samples. Smaller samples – Population with one sample & population with two samples – students – t-test – applications – chi-square test & goodness of fit – application. Unit V Index umbers – Types of Index Numbers – Tests – Unit test, Commodity reversal test, time reversal test, factor reversal test – chain index numbers – cost of living index – interpolation – finite differences operators Δ, E,Ñ, - Newton’s forward, backward interpolation formulae, Lagrange’s formula. Books: 1. Statistics: S. Arumugam & Others. 2. Statistics: D.C. Saucheti & Kapoor. 3. Statistics: Mangaladas & others. 4. Statistics: T. Sankaranarayanan & others. SECOND YEAR PAPER IV ANALYTICAL GEOMETRY 3D & VECTOR CALCULUS Unit I Rectangular Cartesian Coordinates in space – Distance formula – Direction ratio & cosines – Angle between lines – simple problems. Plane – Different forms of equation – angle between two planes – perpendicular distance from a point on a plane – projection of a line or a point on a plane. Unit II Lines – symmetrical form – plane & a straight line – The perpendicular from a point on a line – Coplanar lines – shortest distance between two skew lines & its equations. Sphere – Different forms of equations – plane section – the circle & its radius & centre – tangent plane – condition for tangency – touching spheres – common tangent plane – point of orthogonality of intersection of two spheres. Unit III Vector differentiation – Gradient, Divergence & Curl operators – solenoidal & irrotational fields – formulas involving the laplace operator. Unit IV Double & triple integrals – Jacobian – change of variables – Vector integration – single scalar variables – line, surface & volume integrals. Unit V Gauss’s, Stoke’s & Green’s theorems – statements & verification only. Books: 1. Analytical Geometry of 3D – Part II, Manickavachgom Pillai. 2. Analytical Geometry of 3D & Vector Calculus – P. Duraipandian & others. 3. Analytical Geometry of 3D & Vector Calculus – S. Arumugam & others. 4. Vector Analysis, K. Viswanathan. PAPER V MODERN ALGEBRA Unit I Sets – functions – relations – partitions – compositions of functions – groups – subgroups – cyclic groups. Unit II Normal subgroups – cosets – lagrange’s theorem – Quotient groups – Homomorphism – Kernel – Cayley’s theorem – Fundamental theorem of homomorphism. Unit III Rings – types – subring – ordered integral domain – ideals – Quotient rings – P.I.D. – Homomorphism of rings – fundamental theorem of homomorphism – Euclidean rings. Unit IV Definition & example of vector spaces – subspaces – sum & direct sum of subspaces – linear span, linear dependence, independence & their basic properties – Basis – finite dimensional vector spaces – dimension of sums of subspaces – Quotient space & its dimension. Unit V Linear transformation & their representation as matrices – Algebra of linear transformations & dual spaces – Eigen values & eigen vectors of a linear transformation – Inner product spaces – Schwarz inequality – orthogonal sets & basis – Gram Schmidt orthogonalization process. Reference Books: 1. Modern Algebra, S. Arumugam & Issac. 2. Modern Algebra, Vasistha. 3. Topics in Algebra, I.N. Herstien, Vikas Publishers. PAPER VI ALLIED II NUMERICAL ANALYSIS Unit I Finite differences – difference table – operators E, Δ & Ñ - Relations between these operatous – Factorial notation – Expressing a given polynomial in factorial notation – Difference equation – Linear difference equations – Homogeneans linear difference equation with constant coefficients. Unit II Interpolation using finite differences – Newton – Gregory formula for forward interpolation – Dividend differences – Properties – Newton’s formula for unequal intervals – Lagrange’s formula – Relation between ordinary differences & divided differences – Inverse interpolation. Unit III Numerical differentiation & integration – General Quadratue formula for equidistant ordinates – Trapezoidal Rule – Simpson’s one third rule – Simpson’s three eight rule – Waddle’s rule – Cote’s method. Unit IV Numerical solution of ordinary differential equations of first & second orders – Piccards method. Eulers method & modified Euleis method – Taylor’s series method – Milne’s method – Runge-Kutta method of order 2 & 4 – Solution of algebraic & transcendent equations. Finding the initial approximate value of the root – Iteration method – Newton Raphson’s method. Unit V Simultaneous linear algebraic equations – Different methods of obtaining the solution – The elimination method by Gauss – Jordan method – Grout’s method – Method of factorization. Books: 1. Calculus of finite differences & Numerical Analysis, P.P. Gupta & G.S. Malik, Krishna Prakashan Mardin, Mecrutt. THIRD YEAR PAPER VII OPERATIONS RESEARCH Unit I Linear programming problem – Mathematical formulation – Graphical method of solution – Simplex method – The big M method (Charges method of penalties) – Two phase simplex method – Duality – Dual simplex method – integer programming. Unit II Transportation problem – mathematical formulation – North-west corner rule – Vogel’s approximation method (unit penalty method) – method of matrix minima – optimality test – maximization – Assignment problem – mathematical formulation – method of solution – maximization of the effective matrix. Unit III Sequencinhg problem – introduction – n jobs & two machines – n jobs & three machines – two jobs & n machines – graphical method – Inventory models: types of inventory models: Deterministic: 1) Uniform rate of demand, infinite rate of production & no shortage - 2) Uniform rate of demand, finite rate of replenishment & no shortage – 3) Uniform rate of demand, instantaneous production with shortages – 4) Uniform rate of demand, instantaneous production with shortage & fixed time. Unit IV Probabilistic Models: Newspaper problem - discrete & continuous type cases – Inventory models with one price break. Queuing theory: General concept & definitions – classification of queues – Poisson process, properties of poisson process – models: 1) (M/M/1) : (¥/FCFS), 2) (M/M/1) : (N/FCFS), 3) (m/M/S) : (¥/FCFS). Unit V Network Analysis: Drawing network diagram – Critical Path Method – labeling method – concept of slack & floats on network – PERT – Algorithm for PERT – Differences in PERT & CPM. Resource Analysis in Network Scheduling: Project cost – Crashing cost – Time-cost optimization algorithm – Resource allocation & scheduling. Books for Reference: 1. Operation Research: Kantiswarup, P.K. Gupta & Man Mohan. 2. Operation Research: P.K. Gupta, D.S. Hira. 3. Operation Research: V.K. Kapoor. 4. Operation Research: S.D. Sharma. 5. Operation Research: Mangaladoss. PAPER VIII ANALYSIS Unit I Metric spaces – open sets – Interior of a set – closed sets – closure – completeness – Cantor’s intersections theorem – Baire – Category Theorem. Unit II Continuity of functions – Continuity of compositions of functions – Equivalent conditions for continuity – Algebra of continuous functions – homeomorphism – uniform continuity – discontinuities connectednon – connected subsets of R – Connectedness & continuity – continuous image of a connected set is connected – intermediate value theorem. Unit III Compactness – open cover – compact metric spaces – Herni Borel theorem. Compactness & continuity continuous image of compact metric space is compact – Continuous function on a compact metric space in uniformly continuous – Equivalent forms of compactness – Every compact metric space is totally bounded – Bolano – Weierstrass property – sequentially compact metric space. Unit IV Algebra of complex numbers - circles & straight lines – regions in the complex plane – Analytic functions Cauchy – Rienann equations – Harmonic functions – Bilinear transformation translation, rotation, inversion – Cross – ratio – Fixed points – Special bilinear transformations. Unit V Complex integration – Cauchy’s integral theorem – Its extension – Cauchy’s integral formula – Morera’s theorem – Liouville’s theorem – Fundamental theorem of algebra – Taylor’s series – Laurent’s series – Singularities. Residues – Residue theorem – Evaluation of definite integrals of the following types: 1) ò₀²ᶮ F(Cos x, Sin x)dx ¥ f(x) 2) ò ¾¾ dx ¥ g(x) Reference Books: 1. Modern Analysis – S. Arumugam & Issac. 2. Real Analysis – Vol. III – K. Chandrasekhara Rao & K.S. Narayanan, S. Viswanathan Publisher. 3. Complex Analysis – Narayanan & Manickavachagan pillai. 4. Complex Analysis – S. Arumugam & Issac. 5. Complex Analysis – P. Durai Pandian. 6. Complex Analysis – Karunakaran, Narosa Publisher. PAPER IX MECHANICS Unit I Forces acting at a point – parallelogram of forces – triangle of forces – Lami’s theorem, Parallel forces & moments – Couples – Equilibrium of three forces acting on a rigid body – Coplanar forces – Reduction of any number of Coplanar forces theorems. General conditions of equilibrium of a system of Coplanar forces. Unit II Friction – Laws of friction – Equilibrium of a particle (i) on a rough inclined plane. (ii) under a force parallel to the plane (iii) under any force – Equilibrium of strings – Equation of the common catenary – Tension at any point – Geometrical properties of common catenary – uniform chain under the action of gravity – Suspension bridge. Unit III Dynamics: Projectiles – Equation of path, Range, etc. – Range on an inclined plane – Motion on an inclined plane. Impulsive forces – Collision of elastic bodies – Laws of impact – direct & oblique impact – Impact on a fixed plane. Unit IV Simple harmonic motion in a straight line – Geometrical representation – Composition of SHM’s of the same period in the same line & along two perpendicular directions – Particles suspended by spring – S.H.M. on a curve – Simple pendulum – Simple Equivalent pendulum – The seconds pendulum. Unit V Motion under the action of Central forces – velocity & acceleration in polar coordinates – Differential equation of central orbit – Pedal equation of central orbit – Apses – Apsidal – distances – Inverse square law. Books for Reference: 1. Statics & Dynamics: S. Narayanan. 2. Statics & Dynamics: M.K. Venkataraman. 3. Statics: Manickavachagom pillai. 4. Dynamics: Duraipandian. PAPER X ASTRONOMY Unit I Spherical Trigonometry (only formulae) celestial sphere – four systems of coordinates – Diurnal motion – Zones of the earth – Perpetual day & night – Terrestrial longitude & latitude – International date line. Unit II Dip of horizon – effects – twilight – shortest twilight. Unit III Refraction – Tangent formula – Cassini’s formula – Effects – Horizontal refraction – Geocentric parallax. Unit IV Kepler’s laws – verification – Newton’s deductions – Anomalies – Planets – Inferior & superior planet – Bode’s law – Elongation – Sidereal period – Synodic period – Phase of the planet – Stationary positions of a planet. Unit V Moon – Phase – sidereal & synodic period – elongation – Metonic cycle – golden number – Eclipses – Lunar & solar eclipses – conditions – Synodic period of the nodes – Ecliptic limits – Maximum & minimum number of eclipses near a node & in a year – Saros – Lunar & solar eclipses compared. Books: 1. Astronomy: S. Kumaravelu & Susheela Kumaravelu. 2. Astronomy: G.V. Ramachandran. 3. Astronomy: K. Subramaniam & L.V. Subramaniam. PAPER XI DISCRETE MATHEMATICS Unit I Definition & examples of graphs – degrees – subgraphs – isomorphisms – Ramsey numbers – independent sets & coverings – intersection graphs & line graphs – matrices – operations in graphs – degree sequences, graphic sequences. Unit II Walks – trails & paths – connectedness & components blocks – connectivity – Eulerian graphs – Hamiltonian graphs – trees – characterization of trees – centre of a tree. Unit III Planas graph & their properties – characteristics of planas graphs – thickness – crossing & outerplanarity – chromatic number – chromatic index – five colour theorem – four colour problem – chromatic polynomials – Directed graphs & basic properties – paths & connections in digraphs – digraphs & matrices – tournaments. Unit IV Permutations – ordered selections – unordered selections – further remarks on binomial theorem – Pairings within a set – pairings between sets an optimal assignment problem. Unit V Recurrence relations – Fibonacci type relations – Using generating functions – miscellaneous methods – The inclusion exclusion principle & rook polynomials. Text Books: 1. Invitation to graph theory, S. Arumugam & S. Ramachandran, Scitech Publication. 2. A first course in combinational mathematics, Ian Anderson (Oxford Applied Math series).