Sequences, Series and Differential Calculus : Sequences of real numbers. Convergent sequences and series, absolute and conditional convergence. Mean value theorem. Taylor 's theorem. Maxima and minima of functions of a single variable. Functions of two and three variables. Partial derivatives, maxima and minima.
Integral Calculus : Integration, Fundamental theorem of calculus. Double and Triple, integrals, Surface areas and volumes.
Differential Equations : Ordinary differential equations of the first order of the form y'=f(x,y). Linear differential equations of second order with constant coefficients. Euler-Cauchy equation. Method of variation of parameters.
Vector Calculus : Gradient, divergence, curl and Laplacian. Green's, Stokes' and Gauss' theorems and their applications.
Algebra : Groups, subgroups and normal subgroups, Lagrange's Theorem for finite groups, group homomorphisms and basic concepts of quotient groups, rings, ideals, quotient rings and fields.
Linear Algebra : Systems of linear equations. Matrices, rank, determinant, inverse. Eigenvalues and eigenvectors. Finite Dimensional Vector Spaces over Real and Complex Numbers, Basis, Dimension, Linear Transformations.
Real Analysis : Open and closed sets, limit points, completeness of R, Uniform Continuity, Uniform convergence, Power series.
Probability: Probability spaces, Conditional Probability, Independence , Bayes Theorem, Univariate and Bivariate Random Variables, Moment Generating and Characteristic Functions, Binomial, Poisson and Normal distributions.
Statistics: Sampling Distributions of Sample Mean and Variance, Exact Sampling Distribution (Normal Population), Simple and Composite hypothesis, Best critical region of a Test, Neyman-Pearson theorem, Likelihood Ratio Testing and its Application to Normal population, comparison of normal populations, large sample theory of test of hypothesis, approximate test on the parameter of a binomial population, comparison of two binomial populations.
Complex Analysis: Analytical functions, Harmonic functions, Cauchy's theorem, Cauchy's Integral Formula, Taylor and Laurent Expansion, Poles and Residues.
Numerical Analysis: Difference table, symbolic operators, differences of a factorial, representation of a polynomial by factorials, Forward, backward and central difference approximation formulae. Simpson's one-third rule and the error in it, Gauss-Siedel method and method of elimination for numerical solution of a system of linear equations, iteration method and its convergence, Gradient and Newton-Raphson method and their convergence.
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