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Group Axioms
Subgroups
Group Homomorphisms
Normal subgroups
Qutient Groups
Examples of Automorphism Groups
Cyclic Group
Permutation Groups
Group action
Sylow Theorem I
Sylow Theorem II
Sylow Theorem III
Definition of a Ring
Divisibility and Ideals
Quotient Rings
Factorization and the Noetherian Condition
Euclidean Domains
Polynomial rings 1
Eisenstein criterion and Problems
Lecture 20 : Field
Field extensions 1
Field extensions 2
Algebraic elements form a field
Functions of several variables
Limits for multivariable functions - I
Limits for multivariable functions - II
Continuty of a multivariable functions
Partial derivatives -I
Partial derivatives -II
Differentiability -I
Chain rule - I
Chain rule - II
Extream values - I
Extream values - II
Lagrange multipliers
Multiple integrals
Applications of Multiple Integrals
Vector differentiation
Gradient of a Scalar Field and Directional Derivative
Gradient(Identities), Divergence and Curl (Identities)
Some Identities on Divergence and Curl
Line Integral (I)
Green's theorem
Surface Area
Surface integrals
Divergence Theorem of Gauss
Stoke's Theorem
Introduction to Complex Functions
Limits and Continuity
Differentiation
Cauchy-Riemann Equations and Differentiability
Sine, Cosine and Harmonic functions
Introduction to Cauchy's Theorem
Cauchy's Integral Formula and its Consequences
Winding Number and Maximum Modulus Principle
Morera's Theorem and Higher Order Derivatives of Analytic Functions
Zeroes of Analytic Functions
Removable Singularities
Introduction to Complex Power Series
Taylor's Theorem
Essential Singularity & Introduction to Laurent Series
Residue Theorem and Applications
The Argument (Counting) Principle, Rouche's Theorem and The Fundamental Theorem of Algebra
Proofs of Maximum Principles and Introduction to Schwarz Lemma
Bilinear Transformations
Conformal Mapping-I
Normed Linear Spaces
Hahn-Banach Theorems
Open mapping and closed graph theorems
Baire's Theorem and Applications
Hilbert spaces - Part 1
Orthonormal bases - Part 2
Projection Theorem, Orthonormal Sets and Sequences
Representation of Functionals on a Hilbert Spaces
Spectrum of a compact self-adjoint operator
Vector spaces
Elementary Properties in Vector Spaces. Subspaces
Subspaces (continued), Spanning Sets, Linear Independence, Dependence
Basis for a vector space
Dimension of a vector space
Linear transformations
The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces
The Matrix of a Linear Transformation
Linear Equations
Eigenvalues and Eigenvectors of Linear Operators
Diagonalization of Linear Operators. A Characterization
The Minimal Polynomial
The Cayley-Hamilton Theorem
Inner Product Spaces
Norms on Vector spaces. The Gram-Schmidt Procedure I
The Gram-Schmidt Procedure II. The QR Decomposition.
Unitary Operators
Unitary operators II. Self-Adjoint Operators I.
Normal Operators - Spectral Theorem
Quadratic Forms
Jordan canonical form
Introduction to Linear Programming Formulations
Linear Programming Solutions- Graphical Methods
Linear Programming Solutions - Simplex Algorithm
Two Phase
Revised Simplex Method
Unbounded solution of LPP
Infeasible solution of LPP
Hungarian algorithm; Alternate optimum
Primal Dual Construction
Weak Duality Theorem
Weak and Strong duality
Transportation Problem
Stepping stone method and Modified Distribution method
Assignment Problem
Solution Of Linear Systems Of Equations - 1
Solution Of Linear Systems Of Equations - 3
Solutions Of Linear Systems Of Equations - 8 Iterative Method - 1
Solutions Of Linear Systems Of Equations - 8 Iterative Method - 2
Root Finding Methods - 1 The Bisection Method - 1
Root Finding Methods - 5 Secant Method, Method Of false Position
Root Finding Methods - 3 Newton-Raphson Method - 1
Root Finding Methods - 6 Fixed Point Methods - 1
Lagrange Interpolation Polynomial , Error In Interpolation - 1
Divide Difference Interpolation Polynomial
Lagrange Interpolation Polynomial ; Error In Interpolation - 1
Numerical Differentiation - 3 Operator Method Numerical Integration - 1
Numerical Integration - 2 Error in Trapezoidal Rule Simpson's Rule
Numerical Integration - 3 Error in Simpson's Rule Composite in Trapezoidal Rule , Error
Numerical Integration - 4 Composite Simpsons Rule , Error Method of Undetermined Coefficients
Numerical Solution Of ODE - 3 Examples of Taylor Series Method Euler's Method
Numerical Solution Of ODE-4 Runge-Kutta Methods
Introduction to Ordinary Differential Equations
Picard's Existence and Uniqueness Theorem
First Order ODE's - Homogeneous Equations
First order ODE's - Exact Equations
First order ODEs - Non-Exact Equations - Finding Integrating Factors
Linear First Order ODE and Bernoulli's Equation
Second order ODE's
Properties of solutions of second order homogeneous ODE's
Abel's formula to find the other solution
Second Order ODE's with constant coefficients
Cauchy- Euler equation
Non homogeneous ODEs Variation of Parameters
Method of undetermined coefficients
Power Series Solutions to Second Order ODE's
Legendre Differential Equation
Properties of Legendre Polynomials
Power series solutions around a regular singular point
Frobenius method of solutions
Bessel differential equation
Properties of Bessel functions
Introduction to Sturm-Liouville theory
Sturm-Liouville Problems
Regular Sturm-Liouville problem
Sturm -Liouville problem and its properties
Solving Linear ODE's with polynomial coefficients
General System and Diagonalizability
2 by 2 systems and Phase Plane Analysis
Stability Equilibrium Points
Lyapunov Function
Relation between Characteristic curves and Integral surfaces for Quasilinear equations
Second Order Partial Differential Equations - Classification
SOPDE's - Canonical form for an equation of Hyperbolic type
SOPDE's - Canonical form for an equation of Parabolic type
SOPDE's - Canonical form for an equation of Elliptic type
Laplace Equation-I
IBVP for Heat equation Subtitle: Method of Separation of Variables
IBVP for Wave Equation - Separation of Variables Method
Wellposedness of Cauchy problem for Wave Equation
Wave Equation in one space dimension - d'Alembert formula
Qualitative analysis of Wave equation - Domain of dependence, domain of influence
Nonhomogeneous Wave Equation - Duhamel principle
Cauchy Problem for Heat Equation - 1
Solution of PDEs using Laplace transform
Solution of PDEs using Fourier transform
Metric Spaces: Definition and Examples
Completeness
Connectedness
Compactness
Sequences and Series of Functions
Uniform Convergence
Equicontinuous family of Functions: Arzela - Ascoli Theorem
Approximation of a Continuous Function by Polynomials: Weierstrass Theorem
Existence using Fixed Point Theorem
Power Series
Derivatives: Possible Definition
Implicit Function Theorem
Application of IFT: Inverse Function Theorem
Lebesgue measure and its properties
Measurable functions
Lebesgue Integral and its properties
Monotone convergence theorem & Fatou's Lemma
Properties of Integral functions & Dominated Convergence Theorem
Definition of aTopology
Examples of different topologies
Basis for a topology
Connectedness of the Real Line
The Product topology
Chapter II - Creating New Spaces : Lecture 28
Metric topology - Part 2
First and second countable topological spaces
Separation axioms
Urysohn's Lemma
