 # Important Topics for Mathematical Science for CSIR NET

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.

Algebra

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

Classical Mechanics

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.