# B.Sc. Mathematics

Victor Nandi
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## Important information

Typology | Bachelor |

Methodology | Distance learning |

Start | Raiganj |

Duration | 3 Years |

- Bachelor
- Distance learning
- Raiganj
- Duration:

3 Years

Suitable for: Any student having interest in the subject.

Where and when

Starts | Location |
---|---|

On request |
RaiganjLine Bazar, Opp. Coronation School, Raignaj, 733134, West Bengal, India See map |

Starts | On request |

Location |
RaiganjLine Bazar, Opp. Coronation School, Raignaj, 733134, West Bengal, India See map |

## Frequent Asked Questions

**· Requirements**

Any 10+2 candidates pass candidate or equivalent to 10+2 certificate.

## Course programme

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).