UPHESC Assistant Professor Mathematics Syllabus 2026

UPHESC Assistant Professor Mathematics Syllabus

The written exam has two parts: General Knowledge (30 questions, the same for all subjects) and Mathematics (90 questions). Both syllabi below are the official ones of the Uttar Pradesh Education Service Selection Commission, Prayagraj.

Part 1: General Knowledge (30 questions, 90 marks)

Compulsory for all disciplines. Download PDF (0.3 MB)

Unit 1: Current Affairs

• Major events and current affairs: national and international
• Personalities in news
• Sports news
• Current research in the field of science

Unit 2: Teaching and Research Aptitude

• Teaching: nature, objectives, need, methods, evaluation and factors affecting teaching
• Research: meaning and methods
• Data: source, collection and analysis
• Role of UGC in qualitative development in higher education

Unit 3: Information and Communication Technology (ICT)

• ICT: meaning, advantages and disadvantages
• Basics of internet and e-mailing
• General abbreviations and terminology

Unit 4: People and Environment

• People and environment interaction
• Environmental degradation: causes and solutions
• National and international efforts for a better environment
• Environmental control

Unit 5: Indian History and Geography

• Salient features of Indian culture
• Indian National Movement (1857–1950)
• Origin of the universe and the solar system
• Indian geography (general)

Unit 6: Indian Constitution and Economy

• Preamble, Fundamental Rights and Directive Principles
• Indian political system: legislature, executive and judiciary
• Election Commission and Public Service Commission
• Population, poverty, unemployment; planning and development; income tax
• Agriculture, industry, trade; money, currency, banking and capital market

Part 2: Mathematics (subject code 75) – 90 questions, 270 marks

The Mathematics paper is at post-graduate (Master's) level. The complete official syllabus is below. Download Mathematics syllabus PDF (0.8 MB)

Unit 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.

Unit 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, system of linear equations, eigen values and eigen vectors, Cayley – Hamilton Theorem, diagonalization, Hermitian, Skew- Hermitian and Unitary matrices, Finite dimensional inner product spaces, Orthonormal basis, Gram-Schmidt orthogonalization process, Quadratic forms, reduction and classification of quadratic forms. Rational and Jordan canonical forms.

Unit 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 and 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.

Unit 4: COMPLEX ANALYSIS

Analytic functions, Cauchy - Riemann equations, Cauchy's theorem, Morera's theorem, Liouville's theorem, Cauchy's Integral formula, Zeros of analytic functions, Taylor series, Laurent series, Calculus of residues, Contour integration, Conformal mappings, Mobius transformations.

Unit 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.

Unit 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, Sturm-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.

Unit 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.

Unit 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, Velocity potential, Stream lines, Vorticity, Euler's equation of motion for perfect fluid, Two- dimensional fluid motion, complex potential, source and sink, doublets, motion of sphere in perfect fluid.

Unit 9: OPERATION RESEARCH & NUMERICAL ANALYSIS

Linear programming problem, solution of linear programming problem by graphical and simplex methods, Big M-technique, Two-phase method, Dual problem and duality theorem, convex set, 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.

Unit 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.

How to prepare

• Match the official Mathematics syllabus unit by unit with your Master's notes and mark weak units first.
• Solve previous UPHESC/UPESSC Assistant Professor Mathematics papers (Advt 51 re-exam of April 2026 is the most recent).
• Do not leave GK for the end: Teaching and Research Aptitude and ICT are scoring units of the GK paper.
• Practise MCQs under time limits because of negative marking (-1 per wrong answer).

Syllabus PDFs of all 43 subjects are on UPHESC Assistant Professor.