Ramanujan Classes - Course Structure

Ramanujan Classes

Module 1:- Real Analysis

Capsule 1: Elementary set theory

Capsule 2: Finite, Countable and uncountable sets

Capsule 3: Real number system as a complete ordered field, Archimedean property, supremum & infimum

Capsule 4, 5: Sequences and their convergence

Capsule 6, 7: Series and its convergence

Capsule 8: Limsup, lim inf, Bolzano Weierstrass Theorem, Heine Borel Theorem

Capsule 9: Continuity and uniform continuity, types of discontinuity

Capsule 10: Differentiability, Mean value Theorem

Capsule 11: Sequences and series of functions

Uniform convergence

Riemann sums and Riemann Integral

Improper Integrals

Monotone functions, Functions of bounded variation, Lebesgue measure and integral

Functions of several variables, Directional derivatives, Partial derivatives

Derivative as a linear transformation, Inverse function Theorem, Implicit function theorem

Metric space, compactness

Connectedness

Normed linear space, Properties of C[a, b]

Linear Algebra

Vector space

Vector subspace

Linear dependence, Basis, and dimension

Algebra of matrices, rank

Determinant of matrices

Linear equations

Eigenvalues and eigenvectors, Cayley - Hamilton Theorem

Matrix representation of linear transformation, Change of basis

Canonical forms, Diagonal forms

Triangular forms, Jordan forms

Inner product space, Orthonormal basis

Quadratic forms, Reduction & classification of quadratic forms

Complex Analysis

Algebra of Complex numbers, The complex plane

Polynomials and Power series

Transcendental functions: Exponential function, Trigonometric and hyperbolic functions

Analytic function and C-R Equations

Contour Integral

Cauchy's Theorem, Cauchy's integral formula

Liouville's theorem, Maximum modulus principle

Schwarz lemma, Open mapping theorem

Taylor series, Laurent series

Calculus of Residues

Conformal mapping

Mobius transformations

Modern Algebra

Pigeon-hole principle, Inclusion-Exclusion principle, Derangements

Fundamental Theorem of Arithmetic, Divisibility in Z

Congruences, Chinese Remainder Theorem, Euler's phi-function, primitive roots

Groups, Subgroups

Normal subgroups, Quotient groups, homomorphism

Cyclic groups, Permutation groups, Cayley's theorem

Class Equations, Sylow Theorems

Rings, Ideals, Prime and maximal ideals

Quotient Rings, UFD, PID, ED

Polynomial ring and irreducibility criteria

Field, Finite field

Field Extensions

Galois Theory

Topology

Basis, dense sets, Subspace and product topology

Separation axiom

Connectedness

Compactness

Ordinary Differential Equations (ODEs)

Existence and uniqueness of solutions of IVP for Ist order ODE

Singular solution of Ist order ODEs

System of Ist order ODEs

The general theory of homogeneous and non-homogeneous linear ODEs

Variation of parameters

Sturm-Liouville BVP

Green's function

Partial Differential Equations (PDEs)

Lagrange's and Charpit's methods for solving Ist order PDEs

Cauchy problem for Ist order PDEs

Classification of IInd order PDEs

The general solution of higher-order PDEs with constant coefficients

Method of separation of variable for Laplace, Heat, and Wave equations

Numerical Analysis

Numerical solutions of algebraic equations

Method of iteration and Newton-Raphson method, Rate of Convergence

Solution of a system of linear algebraic equations using Gauss elimination and Gauss-Siedel methods

Finite differences Lagrange, Hermite, and spline interpolation

Numerical differentiation and integration

Numerical solutions of ODEs using Picard, Euler, Modified Euler, and Runge-Kutta methods

Calculus of Variations

Variation of a functional, Euler-Lagrange equation, Necessary and sufficient conditions for extrema

Variational methods for BVPs in ODEs and PDEs

Linear Integral Equations

Linear integral equation of the first and IInd Kind of Fredholm and Volterra type

Solutions with separable kernels

Characteristic numbers and Eigenfunctions

Resolvent Kernel

Classical Mechanics

Generalized Coordinates, Lagrange's Equations

Hamilton's canonical equations, Hamilton's principle and the principle of least action

Two-dimensional motion of the rigid body

Euler's dynamical equations for the motion of a rigid body about an axis.

Theory of small oscillations

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