Stratergy for preparing Mathematics

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.

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.