IMPORTANT TOPICS WHICH COVER MAJOR PORTION OF QUESTION PAPER

UNIT – 1,2 and 3

Analysis-

Topics- countability, supremum ,infimum, convergence of sequences and series , limsup, liminf,

Bw Theorem, continuity , uniform continuity, differentiability , mean value theorem ,

sequence and series of functions, uniform convergence , monotonocity , function of bounded variation ,

directional derivative , metric space – compactness and connectedness.

Linear Algebra

vector space and its subspace, L.D , L.I , Basis and dimension , Linear Transformation and its matrix transformation , Matrices, Rank of determinant of matrices, system of linear equations, eigen values and eigen vectors, Diagonalisation, triangularization, Jordan canonical form, orthogonality , Quadratic forms.

Complex Analysis

Power series, Analyticity, Cauchy Riemann equations, Cauchy integral formula, Maximum modulus principle, open mapping theorem, Taylor and Laurent series , residues, conformal mappings ,Mobius Transformation.

Algebra

Solution of Congruence equations, Euler phi function, primitive roots, groups , subgroups- Normal , Quotient. Homomorphisms , Isomorphisms, cyclic , permutations and dihedral groups, class equations, sylow theorems, Maximal and prime ideals , finite fields , field extension , polynomial and Quotients rings, irreducibility, Euclidean domain , PID , UFD.

ODEs and PDEs

All topics

Numerical Analysis -

Newton Raphson method, Rate of convergence, Lagrange interpolation, Numerical differentiation, Picard method , Euler and modified Euler, RK methods.

Calculus of variations

Euler Lagrange equation , Necessary and sufficient conditions for extrema.

Linear Integral Equations

All topics