- Groups, subgroups, Languages Theorem, cyclic groups, normal sub groups, quotient groups, Fundamental theorem of homomorphism, Isomorphism theorems of groups, Inner automorphisms, Conjugate elements, conjugate subgroups, Commutator subgroups.
- Rings, Subrings, Integral domains, Quotient fields, Isomorphism theorems, Field extension and finite fields.
- Vector spaces, Linear independence, Bases, Dimension of a finitely generated space, Linear transformations, Matrices and their algebra, Reduction of matrices to their echelon form, Rank and nullity of a linear transformation.
- Solution of a system of homogeneous and non-homogeneous linear equations, Properties of determinants, Cayley-Hamilton theorem, Eigenvalues and eigenvectors, Reduction to canonical forms, specially diagonalisation.
- Conic sections in Cartesian coordinates, Plane polar coordinates and their use to represent the straight line and conic sections, (artesian and spherical polar coordinates in three dimensions, The plane, the sphere, the ellipsoid, the paraboloid and the hyperbiloid in Cartesian and spherical polar coordinates.
- Vector equations for Plane and for space-curves. The arc length. The osculating plane. The tangent, normal and binormal, Curvature and torsion, Serre-Frenet’s formulae, Vector equations for surfaces, The first and second fundamental forms, Normal, principal, Gaussian and mean curvatures,
Calculus and Real Analysis
- Real Numbers, Limits, Continuity, Differentiabiliry, Indefinite integration, Mean value theorems, Taylor’s theorem, Indeterminate forms, Asymptotes. Curve tracing, Definite integrals, Functions of several variables, Partial derivatives. Maxima and minima Jacobians, Double and triple integration (techniques only). Applications of Beta and Gamma func tions. Areas and Volumes. Riemann-Stieltje’s integral, Improper integrals and their conditions of existences, Implicit function theorem, Absolute and conditional convergence of series of real terms, Rearrangement of series, Uniform convergence of series,
- Metric spaces, Open and closed spheres, Closure, Interior and Exterior of a set. Sequences in metric space, Cauchy sequence convergence of sequences, Examples, Complete metric spaces, Continuity in metric spaces, Properties of continuous functions,
SECTION – D
Function of a complex variable: Demoiver’s theorem and its applications, Analytic functions, Cauchy’s theorem, Cauchy’s integral formula, Taylor’s and Laurent’s series, Singularities, Cauchy residue theorem and contour integration, Fourier series and Fourier transforms, Analytic continuation.
- Vector Analysis
Vector algebra, scalar and vector product of two or more vectors, Function of a scalar variable, Gradient, divergence and curl, Expansion formulae, curvilinear coordinates, Expansions for gradient, divergence and curl in orthogonal curvilinear coordinates, Line, surface and volume integrals, Green’s, Stoke’s and Gauss’s theorms
Composition and resolution of forces, Parallel forces, and couples, Equilibrium of a system of coplanar forces, Centre of mass and centre of gravity of a system of particles and rigid bodies, Friction, Principle of virtual work and its applications, equilibrium of forces in three dimensions.
Tangential, normal, radial and transverse components of velocity and acceleration, Rectilinear motion with constant and variable acceleration, Simple harmonic motion, Work, Power and Energy, Conservative forces and principles of energy, Principles of linear and angular momentum, Motion of a projectile, Ranges on horizontal and inclined planes, Parabola of’ safety. Motion under central forces, Apse and apsidal distances, Planetary orbits, Kepler’s laws, Moments and products of inertia of particles and rigid bodies, Kinetic energy and angular momentum of a rigid body, Motion of rigid bodies, Compound pendulum, Impulsive motion, collision of two spheres and coefficient of restitution.
SECTION – C
- Differential Equations
Linear differential equations with constant and variable coefficients, the power series method.
Formation of partial differential equations. Types of integrals of partial differential equations. Partial differential equations of first order Partial differential equations with constant coefficients. Monge’s method. Classification of partial differential equations of second order, Laplace’s equation and its boundary value problems. Standard solutions of wave equation and equation of heat induction.
SECTION – D
Definition of tensors as invariant quantities. Coordinate transformations. Contravariant and covariant laws of transformation of the components of tensors. Addition and multiplication of tensors, Contraction and inner product of tensors The Kronecker delta and Levi-Civita symbol. The metric tensor in Cartesian, polar and other coordinates, covariant derivatives and the Christoffel symbols. The gradient. divergence and curl operators in tensor notation.
- Elements of Numerical Analysis
Solution of non-linear equations, Use of x = g (x) form, Newton Raphson method, Solution of system of linear equations, Jacobi and Gauss Seidel Method, Numerical Integration, Trapezoidal and Simpson’s rule. Regula falsi and interactive method for solving non-linear equation with convergence. Linear and Lagrange interpolation. Graphical solution of linear programming problems.