Paper-I | Section-A
Vector, space, linear dependance and independance, subspaces, bases,
dimensions. Finite dimensional vector spaces.
Matrices, Cayley-Hamiliton theorem, eigenvalues and eigenvectors, matrix of
linear transformation, row and column reduction, Echelon form, eqivalence,
congruences and similarity, reduction to cannonical form, rank, orthogonal,
symmetrical, skew symmetrical, unitary, hermitian, skew-hermitian
formsâ€“their eigenvalues. Orthogonal and unitary reduction of quadratic and
hermitian forms, positive definite quardratic forms.
Real numbers, limits, continuity, differerentiability, mean-value theorems,
Taylor's theorem with remainders, indeterminate forms, maximas and minima,
asyptotes. Functions of several variables: continuity, differentiability,
partial derivatives, maxima and minima, Lagrange's method of multipliers,
Jacobian. Riemann's definition of definite integrals, indefinite integrals,
infinite and improper intergrals, beta and gamma functions. Double and triple
integrals (evaluation techniques only). Areas, surface and volumes, centre of
Analytic Geometry :
Cartesian and polar coordinates in two and three dimesnions, second degree
equations in two and three dimensions, reduction to cannonical forms, straight
lines, shortest distance between two skew lines, plane, sphere, cone,
cylinder., paraboloid, ellipsoid, hyperboloid of one and two sheets and their
Ordinary Differential Equations :
Formulation of differential equations, order and degree, equations of first
order and first degree, integrating factor, equations of first order but not
of first degree, Clariaut's equation, singular solution.
Higher order linear equations, with constant coefficients, complementary
function and particular integral, general solution, Euler-Cauchy equation.
Second order linear equations with variable coefficients, determination of
complete solution when one solution is known, method of variation of
Dynamics, Statics and Hydrostatics :
Degree of freedom and constraints, rectilinerar motion, simple harmonic
motion, motion in a plane, projectiles, constrained motion, work and energy,
conservation of energy, motion under impulsive forces, Kepler's laws, orbits
under central forces, motion of varying mass, motion under resistance.
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. Pressure of heavy fluids,
equilibrium of fluids under given system of forces Bernoulli's equation,
centre of pressure, thrust on curved surfaces, equilibrium of floating bodies,
stability of equilibrium, metacentre, pressure of gases.
Vector Analysis :
Scalar and vector fields, triple, products, differentiation of vector function
of a scalar variable, Gradient, divergence and curl in cartesian, cylindrical
and spherical coordinates and their physical interpretations. Higher order
derivatives, vector identities and vector quations. Application to Geometry:
Curves in space, curvature and torision. Serret-Frenet's formulae, Gauss and
Stokes' theorems, Green's identities.
Groups, subgroups, normal subgroups, homomorphism of groups quotient groups
basic isomorophism theorems, Sylow's group, permutation groups, Cayley
theorem. Rings and ideals, principal ideal domains, unique factorization
domains and Euclidean domains. Field extensions, finite fields.
Real Analysis :
Real number system, ordered sets, bounds, ordered field, real number system as
an ordered field with least upper bound property, cauchy sequence,
completeness, Continuity and uniform continuity of functions, properties of
continuous functions on compact sets. Riemann integral, improper integrals,
absolute and conditional convergence of series of real and complex terms,
rearrangement of series. Uniform convergence, continuity, differentiability
and integrability for sequences and series of functions. Differentiation of
fuctions of several variables, change in the order of partial derivatives,
implict function theorem, maxima and minima. Multiple integrals. Complex
Analysis : Analytic function, Cauchy-Riemann equations, Cauchy's theorem,
Cauchy's integral formula, power series, Taylor's series, Laurent's Series,
Singularities, Cauchy's residue theorem, contour integration. Conformal
mapping, bilinear transformations.
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. Travelling salesman problmes.
Partial differential equations:
Curves and surfaces in three dimesnions, formulation of partial differential
equations, solutions of equations of type dx/p=dy/q=dz/r; orthogonal
trajectories, pfaffian differential equations; partial differential equations
of the first order, solution by Cauchy's method of characteristics; Charpit's
method of solutions, linear partial differential equations of the second order
with constant coefficients, equations of vibrating string, heat equation,
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)
methods, Gauss-Seidel(iterative) method. Newton's (Forward and backward) and
Lagrange's method of interpolation.
Numerical integration: Simpson's one-third rule, tranpezodial rule, Gaussian
quardrature formula. Numerical solution of ordinary differential equations:
Euler and Runge Kutta-methods.
Computer Programming: Storage of numbers in Computers, bits, bytes and words,
binary system. arithmetic and logical operations on numbers. Bitwise
operations. AND, OR , XOR, NOT, and shift/rotate operators. Octal and
Hexadecimal Systems. Conversion to and form decimal Systems. Representation of
unsigned integers, signed integers and reals, double precision reals and long
Algorithms and flow charts for solving numerical analysis problems.
Developing simple programs in Basic for problems involving techniques covered
in the numerical analysis.
Mechanics and Fluid Dynamics :
Generalised coordinates, constraints, holonomic and non-holonomic ,
systems. D' Alembert's principle and Lagrange' equations, Hamilton equations,
moment of intertia, motion of rigid bodies in two dimensions.
Equation of continuity, Euler's equation of motion for inviscid flow,
stream-lines, path of a particle, potential flow, two-dimensional and
axisymetric motion, sources and sinks, vortex motion, flow past a cylinder and
a sphere, method of images. Navier-Stokes equation for a viscous fluid.