Paper I
Section A
Linear Algebra: Vector, space, linear dependance and independance, subspaces,
bases, dimensions. Finite dimensional vector spaces. Eigenvalues and eigenvectors,
eqivalence, congruences and similarity, reduction to canonical form, rank,
orthogonal, symmetrical, skew symmetrical, unitary, hermitian, skew-hermitian
formstheir eigenvalues.
Calculus: Lagrange's method of multipliers, Jacobian. Riemann's definition
of definite integrals, indefinite integrals, infinite and improper integrals,
beta and gamma functions. Double and triple integrals (evaluation techniques
only). Areas, surface and volumes and centre of gravity. Analytic Geometry: Sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid
of one and two sheets and their properties. Section B
Ordinary Differential Equations: 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
parameters. Dynamics, Statics and Hydrostatics: You can skip this entire section,
if you have prepared other sections well. Vector Analysis: Triple products, vector identities and vector equations.
Application to Geometry: Curves in space, curvature and torision. Serret-Frenet's
formulae, Gauss and Stokes' theorems, Green's identities. ***************************************************************** Paper II
Section A
Algebra: Normal subgroups, homomorphism of groups quotient groups basic
isomorophism theorems, Sylow's group, principal ideal domains, unique
factorisation domains and Euclidean domains. Field extensions, finite
fields. Real Analysis: 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 functions of several
variables, change in the order of partial derivatives, implicit function
theorem, maxima and minima. Multiple integrals. Complex Analysis: You can skip this entire section, if you have prepared
other sections well. Linear Programming: Basic solution, basic feasible solution and optimal
solution, Simplex method of solutions. Duality. Transportation and assignment
problems. Travelling salesman problems. Section B
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;
Char-pit's method of solutions, linear partial differential equations
of the second order with constant coefficients, equations of vibrating
string, heat equation, laplace equation. Numerical Analysis and Computer programming: Numerical methods, Regula-Falsi
and Newton-Raphson methods Numerical integration: Simpson's one-third
rule, tranpesodial rule, Gaussian quardrature formula. Numerical solution
of ordinary differential equations: Euler and Runge Kutta-methods. Computer Programming: Binary system. Arithmetic and logical operations
on numbers. Bitwise operations. Octal and Hexadecimal Systems. Convers-ion
to and from decimal Systems. Mechanics and Fluid Dynamics: D'Alembert's principle and Lagrange' equations,
Hamilton equations, moment of intertia, motion of rigid bodies in two
dimensions. ********************************************************
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