Differential and Integral Calculus

Partial Differentiation

Euler's Theorem for homogeneous functions

Total Differentiation, Maxima and Minima of two and three variables

Lagrange's Multipliers Method

Curvature

Asymptotes

Envelopes and Evolutes, Singular Points

Rectification, Multiple Integral

volume and surface of revolution of curves

Beta and Gamma functions.

Two Dimensional Coordinate Geometry (Catesian and Polar coordinates)

Polar equation of conics

Polar equation of tangent, normal, asymptotes and chord of contact

Auxiliary and Director circle

Second degree equation of General Conic

Centre, Asymptotes, eccentricity, foci, directrix axes and latus rectum of a conic, Co-ordinate of center, equation of conic referred to center as origin, lengths and position of axes of a standard conic

Three Dimensional Coordinate Geometry

Straight Line, Sphere, Cylinder, Cone and their properties (Rectangular Coordinates only)

Central Conicoids and their properties (Referred to principal axes only)

Vector Calculus

Differentiation of Vectors

Del operator, Gradient, divergent, Curl and directional derivative, their identities and related theorems

Integration of Vectors, line, Surface and Volume integration of vectors

Gauss Divergence, Stokes and Green theorem.

Ordinary Differential Equations

First order non-linear differential equation

singular solutions and extraneous Loci

Second order linear differential equation with constant and variable coefficients

imultaneous and Total Differential Equations

Partial Differential Equations

Linear and Non-linear Partial differential equation of first order

Liner Partial Differential Equations of Second Order

Solution of Partial Differential Equations by Lagrange's, Charpit's and Monge's Method

Mechanics

Equilibrium of coplanar forces, Moments, Friction, Catenary

Simple harmonic motion

Rectilinear motion under variable laws

Motion in resisting medium

Projectile

Abstract Algebra

Groups- Normal Sub-groups, Quotient groups, Homomorphism, Isomorphism of groups

Classification of finite groups

Cauchy's Theorem for finite abelian groups, Permutation groups, Solvable groups and their properties

Rings, Morphism, Principal Ideal domain

Euclidean Rings, Polynomial Rings

Irreducibility criteria, Fields, Finite fields, Field extensions

Integral domain

Linear Algebra

Vector Spaces, Linear dependence and independence

Bases, Dimensions, Linear transformations

Matrix representation of Linear transformations, Change of bases

Inner product spaces, Orthonormal basis, Quadratic forms, reduction and classification of quadratic forms

Algebra of Matrices, Eigenvalues and Eigenvectors, Cayley-Hamilton theorem

Canonical, Diagonal, Triangular and Jordan forms, Rank of Matrix

Complex Analysis

Analytic Functions, Cauchy's Theorem, Cauchy's Integral Formulae

Power Series, Laurent's Series

Singularities, Theory of Residues

Complex Transformations, Contour Integration

Special Functions

Beta and Gamma Functions

Hypergeometric Functions

Bessel Functions

Legendre Function of first kind

Hermite Polynomials

Laguerre Polynomials.

Integral Transforms

Laplace transform

Inverse Laplace transform

convolution theorem

Fourier transform,

Inverse Fourier transform

Parseval theorem

Hankel transform

Mellin transform.

Differential and Integral Equations

Classification of second order Partial Differential Equations, Green's Functions, Sturm-Liouville Boundary Value Problems, Cauchy's problems and Characteristics, Calculus of variation, Euler-Lagrange equation

Integral Equations of first and second kind of Fredholm and Volterra type

Solution by successive substitutions and successive approximations.

Metric spaces and Topology

Metric spaces, compactness, connectedness, Topological spaces, closed sets, closure, Dense set, Neighbourhood

Interior, exterior and boundary points, Accumulation points and derived sets

Bases and sub-bases

First and second countable spaces, separable spaces, Separation axioms, compactness, continuous functions and compact sets, connected spaces.

Differential Geometry

Curves in space

Osculating plane, Normal plane, rectifying plane, Serret-Frenet formulae, curvature torsion, circle of curvature, Sphere of curvature

envelopes

curves on sufaces.

Tensors

Covariant

Contravariant and mixed tensors

Invariants

Addition

Subtraction and Multiplication of tensors

Contraction of tensors

Quotient law of tensors

Fundamental tensors

Associated tensors

Christoffel symbols

Covariant differentiation of tensors

Law of covariant differentiation.

Mechanics

D'Alembert's Principle, Moment and product of inertia, Motion in two-dimensions

Lagrange's equations of motion, Euler's Equations of motion, motion of a top.

Numerical Analysis

Interpolation, Difference schemes, Lagrange interpolation, Numerical differentiation and integration, Bisection, Secant, Regula-Faisi and Newton's Methods, Roots of polynominal

Linear Equation - Direct Methods (Jacobi, Gauss and Siedal Method).

Operations Research

Simplex methods, Duality, Degeneracy, Revised Simplex method, Integer Programming Problems, Assignment problems, Transportation Problems

Game Theory - Two person zero sum game.

Mathematical Statistics

Probability, conditional Probability, Addition and multiplication theorems of probability, Baye's Theorem, Expectations, Moment Generating Function

Probability Distributions : Binomial, Poisson, Uniform and Normal, Correlation and Regression.