### Syllabus for Mathematics

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.