PAPER  I
(1) Linear
Algebra:
Vector spaces over R and C, linear dependence and
independence, subspaces, bases, dimension; Linear
transformations, rank and nullity, matrix of a linear
transformation.
Algebra of
Matrices; Row and column reduction, Echelon form,
congruence’s and similarity; Rank of a matrix; Inverse of a
matrix; Solution of system of linear equations; Eigenvalues
and eigenvectors, characteristic polynomial, CayleyHamilton
theorem, Symmetric, skewsymmetric, Hermitian, skewHermitian,
orthogonal and unitary matrices and their eigenvalues.
(2)
Calculus:
Real numbers, functions of a real variable, limits, continuity,
differentiability, meanvalue theorem, Taylor's theorem with
remainders, indeterminate forms, maxima and minima,
asymptotes; Curve tracing; Functions of two or three
variables: limits, continuity, partial derivatives, maxima
and minima, Lagrange's method of multipliers, Jacobian.
Riemann's
definition of definite integrals; Indefinite integrals;
Infinite and improper integrals; Double and triple integrals
(evaluation techniques only); Areas, surface and volumes.
(3) Analytic
Geometry:
Cartesian and polar coordinates in three dimensions, second
degree equations in three variables, reduction to canonical
forms, straight lines, shortest distance between two skew
lines; Plane, sphere, cone, cylinder, paraboloid, ellipsoid,
hyperboloid of one and two sheets and their properties.
(4) Ordinary
Differential Equations:
Formulation
of differential equations; Equations of first order and
first degree, integrating factor; Orthogonal trajectory;
Equations of first order but not of first degree, Clairaut's
equation, singular solution.
Second and
higher order linear equations with constant coefficients,
complementary function, particular integral and general
solution.
Second order
linear equations with variable coefficients, EulerCauchy
equation; Determination of complete solution when one
solution is known using method of variation of parameters.
Laplace and
Inverse Laplace transforms and their properties; Laplace
transforms of elementary functions. Application to initial
value problems for 2^{nd} order linear equations
with constant coefficients.
(5) Dynamics
& Statics:
Rectilinear motion, simple harmonic motion, motion in a
plane, projectiles; constrained motion; Work and energy,
conservation of energy; Kepler's laws, orbits under central
forces.
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.
(6) Vector
Analysis:
Scalar and vector fields, differentiation of vector field of
a scalar variable; Gradient, divergence and curl in
cartesian and cylindrical coordinates; Higher order
derivatives; Vector identities and vector equations.
Application
to geometry: Curves in space, Curvature and torsion;
SerretFrenet’s formulae.
Gauss and
Stokes’ theorems, Green’s identities.
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, CauchyRiemann 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, RegulaFalsi and
NewtonRaphson methods; solution of system of linear
equations by Gaussian elimination and GaussJordan (direct),
GaussSeidel(iterative) methods. Newton's (forward and
backward) interpolation, Lagrange's interpolation.
Numerical
integration: Trapezoidal rule, Simpson's rules, Gaussian
quadrature formula.
Numerical
solution of ordinary differential equations: Euler and Runga
Kuttamethods.
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 equations; Moment of inertia;
Motion of rigid bodies in two dimensions.
Equation of
continuity; Euler's equation of motion for inviscid flow;
Streamlines, path of a particle; Potential flow;
Twodimensional and axisymmetric motion; Sources and sinks,
vortex motion; NavierStokes equation for a viscous fluid.
