Jkpsc lecturer Math syllabus and notes 2025

SYLLABUS FOR LECTURER 10+2 MATHEMATICS (Expected)

The Jkpsc math Syllabus for 2024 is essential for individuals preparing for papers in math.

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REAL ANALYSIS

• Resume of sequences, series and Riemann integration, continuity, uniform continuity. 
• Fundamental theorem of integral calculus, classes of R-integrable functions. 
• Functions of bounded variation. Riemann stieltjes integration.
• Cauchy's general principle of uniform convergence, uniform convergence and integration, uniform convergence and differentiation, weierstras theorem.

COMPLEX ANALYSIS

• Complex numbers and functions. 
• Cauchy integral formula, Liouvilles theorem, Taylor's and Laurent's theorems, classification of singularities.
• Removable singularities, Riemann's theorem, essential singularity, Weierstrass theorem on essential singularity.
• Calculus of residues, Cauley's residue theorem, Integration by the method of residues, evaluation of f e^(-x^2) dx by residue calculum.
• The argument Princiiple of Maximum Modulus theorem, for bounded regions.

GROUPS

• Review of the product of two subgroups. 
• Structure theorem for cyclic groups, automorphisms, inner automorphisms.
• Cauchy's and sylow's theorem for Abelian groups. Cayley's theorem. 
• Simplicity of the alternating groups. Sylow's theorem and Cauchy's theorem.
• Finite Abelian groups, Fundamental theorem on finite Abelian groups. 

• Composition series. The Jordan-Holder theorem for finite groups.

RINGS

• Definition and examples of rings. 
• Integral domains and fields Homomorphisms, Principal ideals, Prime ideals and maximal ideals,
• Fields of quotients of an integral domains, Polynomial rings, Einstien's criterion.

FIELDS

• Prime fields and their structure. 
• Extensions of fields. Algebraic numbers and algebraic extensions of a field.
• Normal extensions and fundamental theorem of Galois theory.

ADVANCED CALCULUS

• Fourier series: Expansion of function f(x) in the interval (- π, π). 
• Fourier sine series and Fourier Cosine series of f(x) in (- π, π).
• Fourier series of sine and cosine in any interval (-c, c)

DIFFERENTIAL EQUATIONS

• An initial value problem, singular solutions, P-discriminant, C-discriminant.
• Equations of the second degree with variable coefficients. 
• Total differential equation Pdx+Qdy+Rdz = O. Necessary and sufficient conditions that such a differential equation may be integrable.

• Partial differential equation of the first order, Lagrange's linear equation Pp+Qq+R. Charpits method.

DIFFERENTIAL GEOMETRY:

• Curves with torsion, curvature, Frenet formulae, spherical curvature, spherical indicatrices, Involutes and evolutes. 
• Bertrand curves. Envelopes of one and two parameter family of surfaces, Developable surfaces, Developable associated with a curve.
• Curvilinear coordinates. Fundamental magnitudes of first and second order, the two fundamental forms; curvature of normal section, Meunier's theorem, Euler's theorem, Dupin's indicatrix, Rodrigue's formulae.
• Conjugate systems. Asymptotic lines, Isometric lines, null lines. The Gauss characteristic equation, Minardi-Codazzi Relations.
• Geodesic curves in relation to Geodesies. Bonnet's theorem. Geodesic curvature.

TOPOLOGY

• Metric spaces: Definition and examples, open sets, completeness, convergence, continuous mapping, completion of a metric space, Cantor's intersection theorem. Banach's contraction Principle.

TOPOLOGICAL SPACES

• Definition and examples, Elementary properties, Kuratowski's axioms, continuous mappings and their characterisation.
• Bases and subbases, concept of first countability, second countability, separability, Tychnoffs theorem, competness, Lebasgue's covering lemma, continuous maps on compact spaces, Connectedness, local connectedness, their relationship. 
• Urysohn's lemma., Urysohn's Metrization theorem. Separation axioms, one point compactification.

BANACH SPACES

• Definition and examples, Quotient spaces, Dual of a normaed linear space. Duals of L, l, l^∞ (P ≥ 1). Hahn Banach Theorem.

HILBERT SPACES

• Definition and examples, Cauchy-Schwarx inequality Bessel's inequality, orthonormal systems. 
• Riesz representation theorem, inner product spaces, Adjoint of a Hilbert space, operators, Normal operators.

MEASURE THEORY

• Lebesque outer measure, Lebesque measurable set, Measurable functions, algebra of measurable functions. Borel Measurability.
• Convergence in measure, almost uniform convergence and Egorov's theorem. 
• Lebvesgue integration; Levesgue integral of non-negative measurable function. Fatou's lemma, Lebesgues Monotone convergence, Lebesgue's Dominated convergence theorem.
• Riemann and Lebesgue integrals. R-integrability of bounded functions. Lebesgue integrability of bounded functions. Lebesque p-integrability L^p-spaces. Fubim's theorem.

PRIME NUMBERS

• Diophantine equations, solvability of linear Diophantive equations. 
• Congruences, Fermat's theorem, Wilson's theorem, Primitive roots.

COMPLEX ANALYSIS

• The maximum, modulus theorem. Schwarz lemma Hadamard's three circle theorem, Theorem of Borel and caratheodory, Theorem of Phragman lindelof.
• Power series, Hadamard formulla for the radius of convergence, a power series represents an analytic function within the circle of convergence. 
• Handamard Pringsheim, theorem i.e. If f(z) =  ∑a_n z^n has radius of convergence equal to 1, and a_n is real with ∑a_n z^n properly divergent; then z = 1 is a regularity.
• Rouch's theorem, the fundamental theorem of algebra. Morera"s theorm is Poissons integral formula for a circled and half plane, Poisson-jenson formula.

ENTIRE FUNCTIONS

• Factorization of integral functions. The theorem of weierstrass. The order of an entire function.
• Hadareards factorization theorem, thenorder of a canonical product is equal to the exponent of convergence of its zeros. 
• Order of the derived function. Pichard's theorem.

UNIFORM SPACES

• Definition and examples, Uniform Topology; Uniformity and metrizability, complete regularity of uniform spaces, compactness in uniform spaces, uniform continuity, Homotopy theory; Brouwer's fixed point theorem.

BANACH ALGEBRA

• Preliminaries on Banach Algebras, Invertible elements, the spectrum, spectral radius and a formula for the spectral radius, Gelfand-Mazur theorem, Gelfard mapping, Maximal ideal space and its characterisation, continuity of multiplicative functions on Banach Algebra. Gelfand-Naimark theorem. Ideals in CCX and application to stone-ceeh compactification. Spectral theorem for normal operators.

MODULES

• Definitions, fundamental concepts, chain conditions, Noetherian rings, Prime and primary ideals, Sequencess theorms.

LATTICES

• Partially ordered sets, Lattices, modular lattices, complemented modular lattice.

RING THEORY

• Rings, Hilbert basis theorem for Noetherian rings, Matrix rings and their ideals. 
• Direct sums of rings, Prime radical of a ring.

• Random Variables, Mathematical Expectation, cheboyshev's Inequality. 
• Conditional probability, Baye's Theorem; The correlation, coefficient; Independence.

• Binomial, Gamma and Chi-square distributions. Bivariate Normal Distributions., The t and F distributions. 

• The moment Generating Function Technique.

• The central Limit Theorem, Point Estimation, The Rao-Blockwell Theorem.

• Further topics in point estimation Maximum likelihood Estimation statistical Hypothesis; Examples and definition, Uniformly most powerful Tests; Likelihood Ratio Tests.