MATHEMATICS
PAPER – II
(1) Algebra:
Groups, subgroups, cyclic groups,
cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of
groups, basic isomorphism theorems, permutation groups, Cayley’s theorem. Rings,
subrings and ideals, homomorphisms of rings; Integral domains, principal ideal
domains, Euclidean domains and unique factorization domains; Fields, quotient
fields.
(2) Real Analysis:
Real number system as an ordered
field with least upper bound property; Sequences, limit of a sequence, Cauchy
sequence, completeness of real line; Series and its convergence, absolute and
conditional convergence of series of real and complex terms, rearrangement of
series. Continuity and uniform continuity of functions, properties of
continuous functions on compact sets. Riemann integral, improper integrals;
Fundamental theorems of integral calculus. Uniform convergence, continuity,
differentiability and integrability for sequences and series of functions; Partial
derivatives of functions of several (two or three) variables, maxima and
minima.
(3) Complex Analysis:
Analytic functions, Cauchy-Riemann
equations, Cauchy’s theorem, Cauchy’s integral formula, power series representation
of an analytic function, Taylor’s series; Singularities; Laurent’s series;
Cauchy’s residue theorem; Contour integration.
(4) Linear Programming:
Linear programming problems, basic
solution, basic feasible solution and optimal solution; Graphical method and
simplex method of solutions; Duality. Transportation and assignment problems.
(5) Partial differential equations:
Family of surfaces in three
dimensions and formulation of partial differential equations; Solution of
quasilinear partial differential equations of the first order, Cauchy’s method
of characteristics; Linear partial differential equations of the second order
with constant coefficients, canonical form; Equation of a vibrating string,
heat equation, Laplace equation and their solutions.
(6) Numerical Analysis and Computer
programming:
Numerical methods: Solution of
algebraic and transcendental equations of one variable by bisection, Regula- Falsi
and Newton-Raphson methods; solution of system of linear equations by Gaussian
elimination and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods. Newton
Numerical solution of ordinary
differential equations: Euler and Runga Kutta-methods. Computer Programming:
Binary system; Arithmetic and logical operations on numbers; Octal and
Hexadecimal systems; Conversion to and from decimal systems; Algebra of binary
numbers. Elements of computer systems and concept of memory; Basic logic gates
and truth tables, Boolean algebra, normal forms. Representation of unsigned
integers, signed integers and reals, double precision reals and long integers. Algorithms
and flow charts for solving numerical analysis problems.
(7) Mechanics and Fluid Dynamics:
Generalized coordinates; D’
Alembert’s principle and Lagrange’s equations; Hamilton
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