Linear Algebra, Calculus, Analytic Geometry of two and three dimensions and Differential Equations
Vector space bases, dimension of finitely generated space. Linear transformations, Rank and nullity of a linear transformation, Cayley Hamilton theorem. Eigenvalues and Eigenvectors. Matrix of a linear transformation. Row and Column reduction. Echelon form. Equivalence. Congruence and similarity. Reduction to canonical forms. Orthogonal, symmetrical, skew-symmetrical, unitary, Hermitian and Skew-Hermitian matrices – their eigenvalues, orthogonal and unitary reduction of quadric and Hermitian forms, positive definite quadratic forms. Simultaneous reduction.
Real numbers, limits, continuity, differentiability, Mean-Value theorem, Taylor's theorem, indeterminate forms, Maxima and Minima, Curve Tracing, Asymptotes, Functions of several variables, partial derivatives. Maxima and Minima, Jacobian. Definite and indefinite integrals, double and triple integrals (techniques only). Application to Beta and Gamma functions. Areas, Volumes, Centre of gravity.
Analytic Geometry of two and three dimensions:
First and second degree equations in two dimensions in Cartesian and polar coordinates. Plane, Sphere, Paraboloid, Ellipsoid. Hyperboloid of one and two sheets and their elementary properties. Curves in space, curvature and torsion. Fernet’s formula.
Order and Degree and a differential equation, differential equation of first order and degree, variables separable. Homogeneous, Linear and exact differential equations. Differential equations with constant coefficients. The complementary function and the particular integral of eax, cosax, sinax, xm, eax cosbx, eax sin bx.
Vector Analysis, Tensor Analysis, Statics, Dynamics and Hydrostatics:
(i) Vector Analysis – Vector Algebra, Differential of Vector function of a scalar variable, Gradient, Divergence and Curl in Cartesian Cylindrical and spherical co-ordinates and their physical interpretation. Higher order derivatives. Vector identities and Vector equations, Gauss and Stocks theorems.
(ii) Tensor Analysis – Definition of Tensor, transformation of co-ordinates, contravariant and covariant tensor. Addition and multiplication of tensors, contraction of tensors, Inner product, fundamental tensor, Christoffel symbols, covariant differentiation. gradient, Curl and divergence in tensor notation.
(iii) Statics – Equilibrium of a system of particles, work and potential energy. Friction, Common catenary. Principle of Virtual Work stability of equilibrium, Equilibrium of forces in three dimensions
(iv) Dynamics – Degree of freedom and constraints. Rectilinear motion. Simple harmonic motion. Motion in a plane. Projectiles. Constrained motion. Work and energy motion under impulsive forces. Kepler's laws. Orbits under central forces. Motion of varying mass. Motion under resistance.
(v) Hydrostatics – Pressure of heavy fluids. Equilibrium of fluids under given system of forces Centre of pressure. Thrust on curved surfaces, Equilibrium and pressure of gases, problems relating to atmosphere.
Algebra, Real Analysis, Complex Analysis, Partial Differential equations. Mechanics, Hydrodynamics, Numerical Analysis. Statistics including probability operational research.
Groups, sub-groups, normal sub-groups, homomorphism of groups, quotient groups. Basic isomorphism theorems. Sylow theorems. Permutation Groups, Cayley's theorem. Rings and Ideals, Principal Ideal domains, Unique factorization domains and Euclidean domains. Field Extensions. Finite fields.
Metric spaces, their topology with special reference to Rn sequence in a metric space, Cauchy sequence, Completeness, Completion Continuous functions, Uniform Continuity, Properties of Continuous function on Compact sets. Riemann-Stieltjes integral, Improper integrals and their conditions of existence. Differentiation of functions of several variable, Implicit function theorem, maxima and minima, Absolute and conditional convergence series of real and complex terms, Re-arrangement of series. Uniform convergence, Infinite products, Continuity, differentiability and integrability for series, Multiple integrals.
Analytic functions, Cauchy's theorem, Cauchy's integral formula, Power series, Taylor's, Singularities, Cauchy's Residue theorem and Contour integration.
Partial Differential Equations:
Formations of partial differential equations. Types of integrals of partial differential Equations of first order Charpit’s method. Partial differential equation with constant co-efficient.
Generalized co-ordinates, Constraints, Holonomic and Non-holonomic systems, D' Alembert's principle and Lagrange’s equations. Moment of Inertia, Motion of rigid bodies in two dimensions.
Equation of continuity, Momentum and energy. Inviscid Flow Theory – Two-dimensional motion, streaming motion, Sources and Sinks.
Transcendental and Polynomial Equations – Methods of tabulation, bisection, Regula-Falsi, secant, and Newton-Raphson and order of its convergence. Interpolation and Numerical differentiation – Polynomial interpolation with equal or unequal step size. Spline interpolation – Cubic splines. Numerical differentiation formulae with error terms. Numerical integration – Problems of approximate quadratic quadrature formula with equi-spaced arguments. Gaussian quadrature convergence. Ordinary differential equations – Euler’s method, Multistep-predictor corrector methods – Adam's and Milne's method, convergence and stability, Runge – Kutta methods.
Probability and Statistics:
1. Statistical methods – Concept of statistical population and random sample. Collection and presentation of data. Measure of location and dispersion. Moments and Sheppard's correction cumulants. Measures of Skewness and Kurtosis. Curve fitting by least squares regression, correlation and correlation ratio. Rank correlation, Partial correlation co-efficient and Multiple correlation co-efficient.
2. Probability – Discrete sample space, Events, their union and intersection, etc., Probability – Classical relative frequency and axiomatic approaches. Probability in continuum probability space conditional probability and independence, Basic laws of probability, Probability of combination of events, Bayes theorem, Random variable probability function, Probability density function. Distributions function, Mathematical expectation. Marginal and conditional distributions, Conditional expectation.
3. Probability distributions – Binomial, Poisson Normal Gamma, Beata. Cauchy, Multinomial, Hyper-geometric, Negative Binomial, Chebyshev’s Lemma. (Weak) law of large numbers, Central limit theorem for independent and identical varieties, standard errors, Sampling distribution of T.F and Chi-square and their uses in tests of significance large sample tests for mean and proportion. Operational Research.
Definition and some elementary properties of convex sets, simplex methods, degeneracy, duality, sensitivity analysis rectangular games and their solutions. Transportation and assignment problems. Kuhn Tucker conditions for nonlinear programming. Bellman's optimality principle and some elementary applications of dynamic programming.
Theory of Queues:Analysis of steady-state and transient solution for queueing system with poission arrivals and exponential service time. Deterministic replacement models, sequencing problems with two machines, n jobs, 3 machines, n jobs (special case) and n machines, 2 jobs.