IAS Quesion of Mathematics >> Strategy Paper I

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.