Syllabus for DU entrance exam for M.Sc Mathematics

@Elementary set theory
@Finite
@Countable and uncountable sets
@Real number system as a complete     ordered field
@Archimedean property
@Supremum
@Infimum
@Sequence and series
@Covergence limsup
@Liminf
@Bolzano Weierstrass’s theorem
@Heine Borel theorem
@Continuity, Uniform continuity, @Intermediate value theorem, @Differentiability, Mean value theorem, @Maclaurin’s theorem and series, Taylor’s series.
@Sequences and series of functions, @Uniform convergence
@Riemann sums and Riemann integral, @Improper integrals
@Monotonic functions
@Types of discontinuity
@Functions of several variables
@Directional derivative
@Partial derivative
@Metric spaces
@Completeness
@Total boundedness
@Separability, Compactness, @Connectedness
@Eigenvalues and eigenvectors of matrices
@Cayley-Hamilton theorem
@Divisibility in Z, Congruences
@Chinese remainder theorem
@Euler’s φ- function
@Groups, Subgroups, Normal subgroups, Quotient groups
@Homomorphisms
@Cyclic groups, Cayley’s theorem, Class equations, Sylow theorems
@Rings fields, Ideals, Prime and Maximal ideals, Quotient rings
@Unique factorization, Domain
Euclidean domain, Polynomial rings and irreducibility criteria
@Vector spaces
@Subspaces
@Linear dependence
@Dimension
@Algebra of linear
Transformations
@Matrix representation of linear transformations
@Change of basis
@Inner product spaces
@Orthonormal basis
@Existence and Uniqueness of solutions of initial value problems for first order ordinary
@Differential equations
@Singular solutions of first order ordinary differential equations
@System of first order ordinary differential equations
@General theory of homogeneous and non- homogeneous linear ordinary differential equations
@Variation of parameters
@Sturm Liouville boundary value problem
@Green’s function
@Lagrange and Charpit methods for solving first order PDEs
@Cauchy problem for first order PDEs
@Classification of second order PDEs
@General solution of higher order PDEs with constant coefficients
@Method of separation of variables for Laplace
@Heat and Wave equation
@Numerical solutions of algebraic equation
@Method of iteration and Newton-Raphson method
@Rate of convergence
@ Solution of systems of linear algebraic equations using Guass elimination and Guass-Seidel method
@Finite differences, Lagrange, Hermite and Spline interpolation
@Numerical integration
@Numerical solutions of ODEs using Picard
@Euler modified Euler and second order Runge- Kutta methods