Linear Algebra, Calculus, Analytic Geometry of two and three dimensions and Differential Equations
Linear Algebra:
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.
Calculus:
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.
Differential Equations:
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.
Algebra:
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.
Real Analysis:
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.
Complex Analysis:
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.
Mechanics:
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.
Hydro Dynamics:
Equation of continuity, Momentum and energy. Inviscid Flow
Theory – Two-dimensional motion, streaming motion, Sources and Sinks.
Numerical Analysis:
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.
Mathematical Programming:
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: