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## 3 December 2020

Assistant professor Mathematics Detailed Syllabus

Kerala PSC announced the exam for the post of Assistant professor Mathematics in the collegiate education Department. The main topics and detailed syllabus for the Assistant professor Mathematics exam can be read and Syllabus pdf can be downloaded in this page.

Exam details

Category Number: 296/2019

Name of Post:

Date of Test: 01.01.2021

Duration: 2 Hours

Conducted by: Kerala PSC

Mode of Exam:  Descriptive Exam (offline) Written Test (Question cum Answer Booklet)

Medium of Questions: English

Total Marks:  100

Main Topics – Assistant professor Mathematics

In the Exam Calendar, the syllabus is given as questions based on educational qualifications.

The main Topics of Assistant professor Mathematics exam are:

Module I - Linear Algebra

Module II - Real Analysis

Module III - Real Analysis(continued)

Module IV - Abstract Algebra

Module V - Abstract Algebra (continued)

Module VII - Complex Analysis

Module VIII - Functional Analysis

Module IX - Ordinary Differential & Partial Equations

Module X - Theory of Numbers

Detailed Syllabus of Assistant Professor Geography

Hope you already got to know the main topics of this exam. The detailed syllabus of Assistant professor Mathematics is given below.

Comment below if you need Books of Mathematics.

Books for preparing Mathematics.

Syllabus for Mathematics Assistant Professor Examination -Collegiate Education

Module I - Linear Algebra:

Vector spaces, subspaces, linear dependence, basis, dimension, algebra of linear transformations. Algebra of matrices, rank and determinant of matrices, linear equations. Eigenvalues and eigenvectors, Cayley-Hamilton theorem. Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms-rational forms, Jordan forms. Inner product spaces, orthonormal basis. Quadratic form.

Module II - Real Analysis:

Sequences and series, convergence, lim sup. lim inf. Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, uniform continuity, diferentiability, Rolle’s theorem, Mean value theorem. Sequences and series of functionsuniform convergence. Riemann sums and Riemann integral, Improper Integrals. Double and triple integrals,

Module III - Real Analysis(continued):

Lebesgue measure, Lebesgue integral. Functions of several variables, directional derivative, partial derivative, inverse and implicit function theorems. Special functions- Beta and Gamma functions, Fourier series.

Module IV - Abstract Algebra:

Groups, subgroups, normal subgroups, quotient groups, homomorphisms, isomorphisms, cyclic groups, permutation groups, Cayley’s theorem, Direct products, Fundamental theorem for abelian groups, class equations, Sylow theorems.

Module V - Abstract Algebra (continued):

Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain. Polynomial rings and irreducibility criteria. Fields, fnite felds, feld extensions, Galois Theory.

Module VI - Topology:

Metric spaces, continuity, Topological spaces, Base, subbase, countability properties, Separation axioms, Compact space, one point compactifcation, locally compact space, connected spaces, pathwise connectedness, Quotient spaces, Product topology.

Module VII - Complex Analysis:

Complex numbers, polar form, properties of complex numbers, Analytic functions, Cauchy Reimann equations, Conformal Mappings, Mobius transformation, Power series, Zeros of analytic functions, Liouvillis theorem, Complex integration, real integrals using complex integration, Cauchy’s theorem and Cauchy’s integral formula, Morera’s theorem, open mapping theorem, Singularities and its classifcation, residues, Laurent series, Schewarz lemma, Maximum modulus principle, Argument principle.

Module VIII - Functional Analysis:

Normed Linear spaces, Continuity of linear maps, Banach spaces, Hahn Banach spaces, Open mapping theorem, closed graph theorem, uniform boundedness principle, Inner product spaces, Hilbert spaces Projections, Bounded operators, Normal, unitary and self adjoint operators.

Module IX - Ordinary Diferential & Partial Equations :

Existence and uniqueness of solutions of initial value problems for frst order ordinary diferential equations, singular solutions of frst order ODEs, system of frst order ODEs. General theory of homogenous and non-homogeneous linear ODEs. Lagrange and Charpit methods for solving frst order PDEs, Cauchy problem for frst order PDEs. Classifcation of second order PDEs, General solution of higher order PDEs with constant coefcients, Method of separation of variables for Laplace, Heat and Wave equations.

Module X - Theory of Numbers:

Fundamental theorem of arithmetic, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ã˜-function, Fermat’s theorem, Wilson’s theorem, Euler’s theorem, primitive roots.

Hope you got the syllabus of Assistant professor Mathematics exam. You can get the previous question papers of Assistant professor Mathematics HERE.

Comment below if you need Books of Mathematics.

Books for preparing Mathematics.

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