The beauty of mathematics as a subject in the
main examination is that you can be very selective, yet completely safe.
(more content follows the advertisement below) A D V E R T I S E M E N T
Your
efforts should be aimed at developing quality of approach rather than a broad
coverage of the course. The following sections are especially important for the
aspirants taking IAS Main 2005 with mathematics as an optional subject. The
candidates must practise a lot on the indicated sections and they should take
care to give derivation in all the cases if the result is a subsidiary one. In
case of standard results, there is no need to give derivation of an equation,
until specifically asked to.
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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