Algebra:

• Groups, subgroups, Abelian groups, non-abelian groups, cyclic groups, permutation groups
• Normal subgroups, Lagrange’s Theorem for finite groups
• Group homomorphism and quotient groups
• Rings, Subrings, Ideal, Prime ideal
• Maximal ideals; Fields, quotient field

Vector Spaces:

• Linear dependence and Independence of vectors
• Basis, dimension, linear transformations
• Matrix representation with respect to an ordered basis
• Range space and null space, rank-nullity theorem
• Rank and inverse of a matrix, determinant
• Solutions of systems of linear equations, consistency conditions
• Eigenvalues and eigenvectors
• Cayley-Hamilton theorem
• Symmetric, Skew symmetric, Hermitian, Skew-Hermitian, Orthogonal and Unitary matrices

Real Analysis:

• Sequences and series of real numbers
• Convergent and divergent sequences
• Bounded and monotone sequences
• Convergence criteria for sequences of real numbers
• Cauchy sequences
• Absolute and conditional convergence
• Tests of convergence for series of positive terms:
  • Comparison test
  • Ratio test
  • Root test
  • Leibnitz test for convergence of alternating series

Functions of One Variable:

• Limit, continuity, differentiation
• Rolle’s Theorem, Cauchy’s Taylor’s theorem
• Interior points, limit points, open sets, closed sets
• Bounded sets, connected sets, compact sets
• Completeness of R
• Power series including Taylor’s and Maclaurin’s
• Domain of convergence
• Term-wise differentiation and integration of power series

Functions of Two Real Variables:

• Limit, continuity, partial derivatives
• Differentiability, maxima and minima
• Method of Lagrange multipliers
• Homogeneous functions including Euler’s theorem

Complex Analysis:

• Functions of a complex Variable
• Differentiability and analyticity
• Cauchy Riemann Equations
• Power series as an analytic function
• Properties of line integrals
• Goursat Theorem, Cauchy theorem
• Consequence of simply connectivity
• Index of closed curves
• Cauchy’s integral formula
• Morera’s theorem
• Liouville’s theorem
• Fundamental theorem of Algebra
• Harmonic functions

Integral Calculus:

• Integration as inverse process of differentiation
• Definite integrals and their properties
• Fundamental theorem of integral calculus
• Double and triple integrals
• Change of order of integration
• Calculating surface areas and volumes using double integrals
• Calculating volumes using triple integrals

Differential Equations:

• Ordinary differential equations of first order
• Bernoulli’s equation
• Exact differential equations
• Integrating factor
• Orthogonal trajectories
• Homogeneous differential equations-separable solutions
• Linear differential equations of second and higher order with constant coefficients
• Method of variation of parameters
• Cauchy-Euler equation

Vector Calculus:

• Scalar and vector fields
• Gradient, divergence, curl and Laplacian
• Scalar and vector line integrals
• Scalar and vector surface integrals
• Green’s, Stokes and Gauss theorems and their applications

Linear Programming:

• Convex sets, extreme points, convex hull
• Hyper plane & polyhedral Sets
• Convex function and concave functions
• Concept of basis, basic feasible solutions
• Formulation of Linear Programming Problem (LPP)
• Graphical Method of LPP
• Simplex Method