The beauty of mathematics as a
subject in the main examination is that you can be very selective, yet
completely safe. 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.
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A D V E R T I S E M E N T
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.
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.
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.
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,
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