Topics covered for
Rapid Revision Batch- Mathematical Sciences For CSIR NET 2022
Mathematical Sciences
Common Subject
Linear Algebra
Matrices
Diagonalizability & Canonical Forms
Linear Equations
Vector Spaces
Linear Transformations
Polynomials
Determinants
Elementary Canonical Forms
Rational and Jordan Forms
Inner Product Spaces
Operators on Inner Product Spaces
Bilinear Forms
Numerical Analysis
Soln. of Algebraic & Trans Eqn
Interpolation and Approximation
Numerical Differentiation
Initial Value Problems for ODE
Boundary Value Problems-ODE & PDE
Linear System of Algebraic Eqn
Numerical Integration
Complex Analysis
Preliminaries
Bilinear Transformation & Mapping
Elementary Functions
Analytic Functions
Power Series
Complex Integration
Applications of Cauchy's Theorem
Laurent Series & Residue Theorem
Harmonic Functions
Riemann Mapping Theorem
Entire and Meromorphic Functions
Analytic Continuation
Max. & Min. Modulus Principle & Schwarz Lemma
Fundamental Concepts of Complex Analysis
Stereographic Projection & Point Set Topology
Limit, Continuity & Differentiability
Singularites of Analytic Functions
Important Theorems & Applications
Taylor and Laurent Expansion
Special Functions:The Exponential
Argument and Rouche’s Theorem
Calculus of Residues
Conformal Mapping
Descriptive Statistics
Introduction Meaning and Scope
Frequency Distributions
Measures of Central Tendency
Measures of Dispersion
Skewness and Kurtosis
Theory of Probability
Random Variables
Mathematical Expectation
Discrete Distributions
Continuous Distributions
Curve Fitting
Correlation and Regression
Theory of Attributes
Sampling and Large Sample Tests
Chi-Square Distribution
t,F and z distributions
Theory of Estimation
Statistical Inference-II
Probability
Special Univariate Distributions
Joint & Conditional Distributions
Inference - Parameter Estimation
Maximum Likelihood Estimation
Hypothesis Testing
Elements of Bayesian Inference
Markov Chains
Miscellaneous
Ordinary Differential Equation
Differential Equations
Equations of First Order & Degree
Exact Differential Equations
Equations of Special Forms
Chebyshev Polynomials
Beta and Gamma Functions
Bessel Functions
First Order First Degree Differential Equation
General Theory of Linear Differential Equation of Higher Order
Boundary Value Problems
Equtn of the 1st Order but Not of The 1st Degree
Partial Differential Equation
Origin of PDE
Linear PDE of Order One
Non-linear PDE of Order One
Linear PDE with Constant Coeff.
PDE Reducible to Equations
2nd Order Variable Coeff. PDE
PDE into Cononial/Normal Form
Monge’s Method
Classification & Formation of PDE
Heat, Wave & Laplace Equation
Classification of Second Order PDE
Analysis
Sets & Number System
Series & Convergence
Theorem's in Analysis
Notions of Continuity
Integration Theory
Multivariable Calculus
Metric Spaces & Functions
Algebra
Fundamental Theory of Arithmetic
Group Theory
Ring Theory
Fields
Permutations & Combinations
ED, PID & UFD
Algebraic Extension of Fields
Set and Relations
Basic Algebraic Structure
Groups within Groups
Some Important Groups of Finite Order
Symmetric Group or Permutation Group
Some Important Groups of Infinite Order
Conjugate Classes and Class Equation
Invariant/Normal Subgroup
Homomorphism and their Counting
Sylow Theorems
Continuity
Differentiability
Uniform Convergence (Sequence & Series of Function)
Reimann Integration & Bounded Variation and Improper Integral
Basic Concepts and Definitions
Some Important Structures
Subring and Ideals
Ring of Polynomials
Linear Integral Equations
Fredholm and Volterra Type
Sol. with Separable Kernels
Characteristic Numbers
Eigen Functions
Resolvent Kernel
Volterra’s Integral Equation
Fredholm Integral Equation
Hilbert Schmidt Theory
Integral Equation
Calculus of Variations
Variation of a Functional
Euler-Lagrange Equation
Sufficient Conditions for Extrema
Boundary Value Problems in PDE
Miscellaneous
Real Analysis
Countability
Point Set Topology
Sequence
Series
Function and its Property
General Aptitude
General Aptitude
Operation Research
Linear Programming
Classical Mechanics
Generalized Coordinates
Lagrange’s Equations
Hamilton’s Canonical Equations
Hamilton’s Principle
Principle of Least Action
2-D Motion of Rigid Bodies
Euler’s Eq. (Rigid Body)
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