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
