LECTURER IN MATHEMATICS SYLLABYS

History of Development of Mathematics. 

Module-I 

Mensuration, length of arcs, area of sectors of cir cles, tangents to circles, circumcircle and incircl e of polygons, area of polygons, solids-volume and su rface area. Fundamentals of mumber theory. Continued fractions. Boolean Algebra Fundamentals of graph theory.

Module -II 

Sets and binary operations, groups, Sylon's Theorem s, Rings and ideals, Fields, extension fields, rings of polynomials, finite fields, Galois Theory, constructible numbers. System of Linear Equations - Vector spaces, linear transformations, characterist ic values, characteristic polynomial, Minimal polynomial, Cayley-Hamilton theorem, triangulation and diagonalization of matrics. Hyperspaces and linear functionals.

Module -III 

Normed spaces, Banach spaces and related theorems, Linear Maps, inner product spaces, Hilbert spaces and related theorems, finite dimensional and infinite dimensional normed spaces, bounded operators, spectrum, duals and transposes. Adjoint s, normal, unitary and self adjoint operators. Polynomial Equations, Trigonometry, Analytical geom etry of two dimension and three dimension, similarity of triangles, vectors, matrices. Calculus, applications of differentiation and integ ration, elemetary functions (logarithms, exponential, hyperbolic, trigonometric etc), Fundam ental theorem of calculus, mean value theorems, maxima and minima-functions of more than one independent variables, derivatives, partial derivatives, saddle point, critical point.

Module –IV 

Real numbers, rational, irrational numbers, algebra ic and order properties of Real numbers, supremum property, countable and uncountable sets, completeness property, sequences and series of red numbers, relations and functions, limits and co ntinuity of functions, uniform continuity, differentiability and integrability of functions, R iemann integral, Riemann-Stieltges integral, sequences and series of functions. Term by term di fferentiation and integration of series of functions. Lebesgue measure, lebesgue integral, convergence th eorems and applications

Module -V 

Complex numbers, De Moirre's Theorem, Algebraic pro perties of complex numbers, regions in the complex plane. Complex functions, analytic functions, harmonic fun ctions, conformal mapping, elemetary functions, derivatives and integrals of complex fun ctions and related theorems, sigularities, residue theorem and its applications, Power series, Taylor series, Laurent series. Metric spaces, topological spaces, basis, subbasis, closed set, closure, interior, boundary, neighbourhood. Connectedness and compactness, loca lly connected, path connected, locally compact spaces. Functions, continuous functions, homeomorphism, quo tient space. Seperation axioms and related theorems.

Module -VI 

First order ordinary differential equations-formati on, properties and various methods of solving. Picards method of approximation. Numerical methods Second order ordinary differential equations – form aiton, properties and various methods of solving. Equidimensional equations. Existence and uniqueness of solutions. Systems of first order equations. Series solutions of first order and second order or dinary differential equation at ordinary adn regula r singular points. Hypergeometric functions and equations, legendre eq uations and polynomials. Chebyshev's Equations and polynomials. Bessels equations and Fu nctions. Laplace transform, fourier series, beta and Gamma f unctions. Formation and solution of first order partial diffe rential equation in two independent variables. Functional dependence, analytic functions. Second o rder partial differential equation, formation, classification. Wave equation, heat diffusion equation, laplace equ ation. Numerical solutions of algebraic equations, finite differences, interpolation.

Module –VII 

Fundamentals of Theory of Wavelets, Fuzzy set theor y, Fractal geometry, Modular functions Jordan forms, elliptic functions, Riemann Zets Function, A utomate and formal languages, Block Designs.
Monodromy theorem, Reimann mapping theorem, product topology and Tychnoff theorem.
Solutions at infinity of Differential Equations, In tegral Equations, calculus of Variations.
Fundamentals of differential geometry, contractions , inverse function theorem, implicit function theorem.
Fundamentals of Mechanics and Fundamentals of Fluid Dynamics.

Module VIII Research Methodology/Teaching Aptitude 

I. TEACHING APTITUDE
• Teaching: Nature, objectives, characteristics and b asic requirements;
• Learner's characteristics; • Factors affecting teaching;
• Methods of teaching;
• Teaching aids;
• Evaluation systems.
II. RESEARCH APTITUDE
• Research: Meaning, Characteristics and types;
• Steps of research;
• Methods of research;
• Research Ethics;
• Paper, article, workshop, seminar, conference and s ymposium;
• Thesis writing: its characteristics and format.

Module IX(a) Salient Features of Indian Constitution 

Salient features of the Constitution - Preamble- It s significance and its place in the interpretation of the Constitution.
Fundamental Rights - Directive Principles of State Policy - Relation between Fundamental Rights and Directive Principles - Fundamental Duties.
Executive - Legislature - Judiciary - Both at Union and State Level. - Other Constitutional Authorities.
Centre-State Relations - Legislative - Administrati ve and Financial.
 Services under the Union and the States.
Emergency Provisions.
Amendment Provisions of the Constitution.

Module IX (b) Social Welfare Legislations and Programmes 

Social Service Legislations like Right to Informati on Act, Prevention of atrocities against Women & Children, Food Security Act, Environmental Acts e tc. and Social Welfare Programmes like Employment Guarantee Programme, Organ and Blood Don ation etc.

Module X (a) Renaissance in Kerala 

TOWARDS A NEW SOCIETY
Introduction to English education - various mission ary organisations and their functioning- founding of educational institutions, factories, pr inting press etc.
EFFORTS TO REFORM THE SOCIETY 
(A) Socio-Religious reform Movements 
SNDP Yogam, Nair Service Society, Yogakshema Sabha, Sadhu Jana Paripalana Sangham, Vaala Samudaya Parishkarani Sabha, Samathwa Samajam, Isla m Dharma Paripalana Sangham, Prathyaksha Raksha Daiva Sabha, Sahodara Prasthanam etc.
(B) Struggles and Social Revolts 
Upper cloth revolts.Channar agitation, Vaikom Sathy agraha, Guruvayoor Sathyagraha, Paliyam Sathyagraha. Kuttamkulam Sathyagraha, Temple Entry Proclamation, Temple Entry Act .Malyalee Memorial, Ezhava Memorial etc. Malabar riots, Civil Disobedience Movement, Abstent ion movement etc.
ROLE OF PRESS IN RENAISSANCE 
Malayalee, Swadeshabhimani, Vivekodayam, Mithavadi, Swaraj, Malayala Manorama, Bhashaposhini, Mathnubhoomi, Kerala Kaumudi, Samada rsi, Kesari, AI-Ameen, Prabhatham, Yukthivadi, etc
AWAKENING THROUGH LITERATURE 
Novel, Drama, Poetry, Purogamana Sahithya Prasthanam, Nataka Prashtanam, Library movement etc
WOMEN AND SOCIAL CHANGE 
Parvathi Nenmenimangalam, Arya Pallam, A V Kuttimal u Amma, Lalitha Prabhu.Akkamma Cheriyan, Anna Chandi, Lalithambika Antharjanam and others
LEADERS OF RENAISSANCE
Thycaud Ayya Vaikundar, Sree Narayana Guru, Ayyan K ali.Chattampi Swamikal, Brahmananda Sivayogi, Vagbhadananda, Poikayil Yohannan(Kumara G uru) Dr Palpu, Palakkunnath Abraham Malpan, Mampuram Thangal, Sahodaran Ayyappan, Pandi t K P Karuppan, Pampadi John Joseph, Mannathu Padmanabhan, V T Bhattathirippad, Vakkom A bdul Khadar Maulavi, Makthi Thangal, Blessed Elias Kuriakose Chaavra, Barrister G P Pill ai, TK Madhavan, Moorkoth Kumaran, C. Krishnan, K P Kesava Menon, Dr.Ayyathan Gopalan, C V Kunjuraman, Kuroor Neelakantan Namboothiripad, Velukkutty Arayan, K P Vellon, P K Chathan Master, K Kelappan, P. Krishna Pillai, A K Gopalan, T R Krishnaswami Iyer, C Kesav an. Swami Ananda Theerthan , M C Joseph, Kuttippuzha Krishnapillai and others
LITERARY FIGURES 
Kodungallur Kunhikkuttan Thampuran, KeralaVarma Val iyakoyi Thampuran, Kandathil Varghesc Mappila. Kumaran Asan, Vallathol Narayana Menon, Ul loor S Parameswara Iyer, G Sankara Kurup, Changampuzha Krishna Pillai, Chandu Menon, V aikom Muhammad Basheer. Kesav Dev, Thakazhi Sivasankara Pillai, Ponkunnam Varky, S K P ottakkad and others

Module X (b) General Knowledge and Current Affairs 

General Knowledge and Current Affairs

NOTE: - It may be noted that apart from the topics detailed above, questions from other topics prescribed for the educational qualifi cation of the post may also appear in the question paper. There is no undertaking that all the topics above may be covered in the question paper
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