UPHESC Mathematics 2021 syllabus has been released. UPHESC 2021 syllabus designed for UPHESC MATHEMATICS Exam comprises 10 topics, ALGEBRA, LINEAR ALGEBRA, ANALYSIS, COMPLEX ANALYSIS, TOPOLOGY & DIFFERENTIAL GEOMETRY, DIFFERENTIAL EQUATIONS, MATHEMATICAL METHODS, MECHANICS, OPERATION RESEARCH & NUMERICAL ANALYSIS, VEDIC MATHEMATICS AND NUMBER THEORY.

EXAM NAME | GENERAL KNOWLEDGE | MATHEMATICS |
---|---|---|

UPHESC ASSISTANT PROFESSOR MATHEMATICS SYLLABUS |

## 1. ALGEBRA

Cyclic groups, permutation groups, Cayley's Theorem, Fundamental Theorem of homomorphism, Group action, class equation, Sylow theorems and their applications

Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain, Polynomial rings, Fields, finite fields, field extensions, Modules, Noetherian modules, Hilbert basis theorem

## 2. LINEAR ALGEBRA

Linear transformations, Algebra of linear transformations, kernel, range, Rank-Nullity Theorem, Matrix representation of a linear transformation, change of bases, linear functionals, dual spaces, rank, system of linear equations, eigen values and eigen vectors

Cayley – Hamilton Theorem, diagonalization, Hermitian, Skew- Hermitian and Unitary matrices, Finite dimensional inner product spaces

Gram Schmidt-Quadratic forms, reduction and classification of quadratic forms

Rational and Jordan canonical forms

## 3. ANALYSIS

Sequence and series of functions, uniform convergence, Fourier series, Riemann integral, improper integrals, functions of bounded variations, Lebesgue measure, measurable functions, Lebesgue integral

Multivariable calculus- functions of several variables, directional derivative, partial derivative and total derivative, maxima and minima

Elements of metric spaces- convergence, continuity, uniform continuity, compactness, connectedness, completeness

Normed linear spaces, Banach spaces, open mapping theorem, closed graph theorem

Hahn- Banach theorem, Hilbert spaces, Orthogonal complement of a subspace in a Hilbert space, Orthogonal basis, Gram-Schmidt orthogonalization process

## 4. COMPLEX ANALYSIS

Analytic functions, Cauchy - Riemann equations, Cauchy’s theorem, Morera's theorem, Liouville's theoren

Cauchy's Integral formula, zeros of analytic functions, Taylor series, Laurent series

Calculus of residues, contour integration, conformal mappings, Mobius transformations

## 5. TOPOLOGY & DIFFERENTIAL GEOMETRY

Basic concepts of Topology, Continuity, Homeomorphism, connectedness, compactness, countability, separation axioms, subspaces, product spaces, quotient spaces, Tychonoff's theorem, Urysohn's Metrization theorem

Space curves - Their curvature and torsion

Serret - Frenet formulae, First and Second fundamental forms, Gaussian curvatures, Principal directions and principal curvatures, Geodesics, Manifolds

## 6. DIFFERENTIAL EQUATIONS

Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ordinary differential equations, General theory of homogeneous and non-homogenous linear ordinary differential equations, Strum-Liouville boundary value problem

Lagrange and Charpit methods for solving first order partial differential equations, General solution of higher order partial differential equations with constant coefficients, classification of second order partial differential equations, Method of separation of variables for Laplace, Heat and Wave equations

## 7. MATHEMATICAL METHODS

Calculus of variations- Linear functionals, Necessary and sufficient conditions for extrema

Euler-Lagrange equation, Linear integral equations of Fredholm and Volterra type, solution by successive substitutions and successive approximations, solution of integral equations with separable Kernels, Laplace transforms

## 8. MECHANICS

Generalised coordinates, Lagrange's equations, Hamilton's canonical equations, Hamilton's variational principle, Euler's dynamical equations of motion of a rigid body, theory of small oscillations, Poisson bracket, Canonical transformations

Equation of continuity in fluid motion, Euler's equation of motion for perfect fluid, Two- dimensional fluid motion, complex potential, source and sink, doublets, motion of sphere in perfect fluid and motion of liquid past a sphere, vorticity, Navier-Stokes' equations for viscous flows

## 9. OPERATION RESEARCH & NUMERICAL ANALYSIS

Linear programming problem, solution of linear programming problem by graphical and simplex methods, M-technique, Two-phase method, Dual problem and duality theorem, convex set theory, balanced and unbalanced transportation problems, Hungarian method for solving assignment problems, Game theory

Numerical solutions of algebraic equations, Fixed point iteration and Newton-Raphson methods, Solution of systems of linear algebraic equations using Gauss elimination and Gauss-Seidel methods

Finite differences, Lagrange, Hermite and spline interpolations, Numerical differentiation and integration, Numerical solutions of ordinary differential equitation using Picard, Euler, modified Euler and Runge-Kutta methods

## 10. VEDIC MATHEMATICS AND NUMBER THEORY

Contributions of ancient Indian mathematicians, Basic concepts of vedic mathematics, Contribution of Ramanujan in number theory

Fundamental theorem of arithmetic, arithmetical functions, Mobious inversion, Congruences, Chinese remainder theorem, Quadratic reciprocity law and its applications

**NOTE :** Syllabus contains 10 Units. In preparation of question paper, it is suggested that at least 06
questions should be asked from each unit.