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Based on the documents provided, here is a detailed list of topics covered in the Mathematics syllabus for the Assistant Professor Exam.

Unit 1: Calculus and Advanced Calculus
Successive differentiation and Leibnitz’s theorem.
Maclaurin’s and Taylor’s expansions.
Asymptotes, curvature, concavity, and convexity.
Point of inflexion, multiple points, and tracing of curves.
Integration, definite integrals, reduction formulae, quadrature, and rectification.
Sequences and series, including convergence tests, bounded and monotonic sequences, and Cauchy’s convergence criterion.
Alternating series, absolute and conditional convergence.
Limits and continuity of functions of single and two variables.
Change of variables, uniform continuity, mean value theorem, and chain rules.
Maxima and minima of functions of two variables.
Beta and Gamma functions.
Double and triple integrals, including change of order of integration.
Volume and surface of solids of revolution.

Unit 2: Ordinary and Partial Differential Equations
Linear and reducible to linear differential equations.
Exact differential equations.
First-order and higher-degree equations solvable for x, y, and p, including Clairaut equations and singular solutions.
Orthogonal trajectories.
Linear equations with constant coefficients and second-order linear equations.
Method of variation of parameters.
Series solutions of differential equations, Bessel’s and Legendre’s equations and functions.
Orthogonality of functions.
Laplace and inverse Laplace transformations, and the convolution theorem.
First-order partial differential equations (PDEs) and Charpit’s general method.
Second and higher-order PDEs, including classification, homogeneous and non-homogeneous equations with constant coefficients.
Heat equation, wave equation, and Laplace equation.

Unit 3: Discrete Mathematics, Logic, Algebra, and Trigonometry
Contributions of Indian mathematicians: Baudhayan, Madhavan, Bhaskaracharya II, Shridharacharya, Aryabhata, Brahmagupta, Varahmihira, Srinivasa Iyengar Ramanujan, and Bharti Krishna Tirtha.
Thematic analysis on Vedic mathematics and its importance.
Boolean functions, partial order relations, totally ordered sets, Hasse diagrams, and lattice.
Graphs, paths, circuits, trees, and matrix representation of graphs.
Rank of a matrix, Eigenvalues, Eigenvectors, and Cayley’s Hamilton theorem.
Adjoint and inverse of a matrix and their application to solve linear equations.
Logic: logical connectives, truth tables, tautology, contradiction, and logic gates.
Relationship between roots and coefficients of a polynomial equation.
Demoivre’s theorem and its applications.
Hyperbolic and inverse hyperbolic functions, and expansion of trigonometric functions.

Unit 4: Vector Analysis, Geometry, and Analytic Geometry
Product of vectors, vector differentiation, gradient, divergence, and curl.
Vector identities and vector equations.
Vector integration, Gauss’s, Green’s, and Stock’s theorems and their applications.
General equation of a second degree and tracing of conics.
Polar equation of conics.
Equation of a cone, generators, and condition for three mutually perpendicular generators.
Right circular cone and cylinder with their properties.
Central conicoid, paraboloid, ellipsoid, hyperboloid and their properties.

Unit 5: Abstract Algebra
Sets, algebra of sets, subsets, power sets, relations, and equivalence relations.
Mapping, congruence modulo-m relation, permutation, and combination.
Pigeonhole principle and inclusion-exclusion principle.
Primitive roots, divisibility in Z, and remainder theorem.
Groups, subgroups, normal subgroups, and order of an element.
Lagrange’s theorem, cyclic groups, homomorphism and isomorphism.
Kernel of homomorphism, quotient group, permutation group, and Cayley’s theorem.
Class equation, counting principle, Cauchy’s theorems, and Sylow’s theorems.
Ring, subring, ideals, and quotient ring.
Unique factorization domain, principal ideal domain, Euclidean domain, integral domain, field, and subfield.
Polynomial rings, reducibility, and irreducibility of polynomials.
Finite fields and field extension.

Unit 6: Linear Algebra
Vector spaces, vector subspaces and their algebra.
Linear dependence and independence, basis, and dimension.
Linear transformations and their matrix representation.
Change of basis, canonical forms, diagonalization, triangular forms, and Jordan form.
Algebra of matrices, rank and nullity, and Sylvester’s theorem.
Eigenvalues and Eigenvectors.
Inner product spaces and orthonormal basis.
Quadratic forms, their reduction, and classification.

Unit 7: Real Analysis
Real numbers, R as a complete ordered field, and intervals.
Bounded and unbounded sets in R, Archimedean property, and absolute value.
Open and closed sets in R and countability of sets.
Real sequences, subsequences, and infinite series, including various convergence tests.
Alternating series, absolute and conditionally convergent series, and power series.
Real functions, limit, continuity, uniform continuity, and differentiability.
Mean value theorem, higher derivatives, and intermediate forms.
Riemann integrability, Riemann-Stieltjes integrability.
Uniform convergence of sequences and series of functions, and improper integrals.
Metric spaces, subspaces, and complete metric space.
Contraction principle, Bolzano Weierstrass Property (BWP), and connected metric space.

Unit 8: Complex Analysis
Algebra of complex numbers, complex plane, polynomials, and power series.
Analytic functions and Cauchy-Riemann equations.
Contour integral, Cauchy’s theorem, and Cauchy’s Integral formula.
Liouville’s theorem and Maximum modulus principle.
Schwarz’s lemma, Taylor’s series, and Laurent Series.
Fixed points, cross ratio, and bilinear transformation.
Residue Theory: residue at a pole, residue at infinity, Cauchy’s residues theorem, and computation of residues.

Unit 9: Statistics and Numerical Analysis
Central Tendency: mean, median, mode.
Measures of dispersion: range, interquartile range, and mean deviation.
Sampling of large samples, null and alternating hypothesis.
Test of significance based on χ2, t, F, and Z-statistics.
Probability: events, sample space, addition and multiplication theorem, and Baye’s theorem.
Numerical solution of algebraic equations using iteration and Newton-Raphson methods.
Solution of linear algebraic equations using Gauss elimination and Gauss-Seidal methods.
Numerical differentiation and integration.
Numerical solutions of ordinary differential equations using Picard, Euler, Modified Euler, and Runge-Kutta methods.

Unit 10: Functional Analysis
Normed linear spaces, Banach spaces, finite dimension normed linear spaces, and normed linear subspaces.
Equivalent norms, Riesz lemma, and compactness.
Quotient spaces.
Linear operators, bounded linear operators, continuous and non-linear operators.
Linear and bounded linear functional, and dual space.
Hilbert space, orthogonal complements, and orthonormal sets.
Bessel’s inequality and complete orthonormal sets.
Hilbert adjoint operators, self-adjoint operators, and adjoint operators.
Unitary and normal operators, and positive operators.
Hahn-Banach theorem and reflexive spaces.
Category theorem, Bair’s Category theorem, and Uniform boundedness theorem.
Open mapping theorem and closed graph theorem.
Strong and weak convergence in normed spaces.