Every year, CSIR administers the NET Mathematical Scientific test at the national level to choose qualified science students for Junior research and Lecturership. One needs to qualify for CSIR NET exam to pursue a career as a mathematics professor in Indian Colleges and Universities. For those aspiring applicants seeking admission to become Junior Research Fellows (JRF) and Associate Professors in Mathematics or any other discipline, the exam is a doorway. The most crucial topics discussed here must be reviewed if you’re studying for CSIR NET Mathematical Science 2022 examination in order to prevent missing any crucial information.
UNIT I
Analysis:
Real number system as a completely ordered field
Archimedean property, supremum, infimum
Sequences and series, convergence
Bolzano Weierstrass theorem
Heine Borel theorem.
Continuity & Differentiability
Linear Algebra
Algebra of matrices
Cayley-Hamilton theorem
Matrix representation of linear transformations
Diagonal forms, triangular forms, Jordan forms. Quadratic forms, reduction and classification of quadratic forms
UNIT II: TOPOLOGY, COMPLEX ANALYSIS AND ALGEBRA
Topology: Basis, dense sets, subspace and product topology, separation axioms, connectedness and compactness.
Chinese Remainder Theorem
Cayley’s theorem
Fields, finite fields, field extensions, Galois Theory
Euler’s Ø- function
Permutations & Combinations,
Sylow theorems.
UNIT III Classical Mechanics AND ODEs, PDEs
Ordinary Differential Equations (ODEs):
Existence and uniqueness of solutions of initial value problems for first-order ordinary differential equations, singular solutions of first-order ODEs, and the system of first-order ODEs.
Partial Differential Equations (PDEs):
Lagrange and Charpit methods for solving first-order PDEs
Cauchy problem for first-order PDEs
Classification of second-order PDEs, General solution of higher-order PDEs with constant coefficients
Generalized coordinates, Lagrange’s equations, Hamilton’s canonical equations
Hamilton’s principle and the principle of least action
UNIT IV Probability:
Descriptive statistics, exploratory data analysis, Sample space, discrete probability, independent events, Bayes theorem
Random variables and distribution functions (univariate and multivariate); expectation and moments.
Independent random variables, marginal and conditional distributions.
Data Reduction Techniques:
Principle component analysis, Discriminant analysis, School Analytics, Cluster analysis, Canonical correlation.
Simple random sampling, stratified sampling and systematic sampling. Probability is proportional to size sampling.
