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UPPSC Assistant Professor (Mathematics) Syllabus 2026, Exam Pattern
UPPSC Assistant Professor (Mathematics) Syllabus 2026, Exam Pattern, Exam Date 2026
Detail Information about UPPSC has published notification 2026 for the recruitment of Assistant Professor (Mathematics) vacancies. Those Candidates who are Interested to the following vacancy and completed all Eligibility Criteria can read the Notification & Apply Online. In this page we provide the Complete Syllabus of this Recruitment with Latest Update Exam Pattern and the Exam Date also.
UPPSC Assistant Professor Syllabus 2026 - Overview
| Organization Name | Uttar Pradesh Public Service Commission (UPPSC) |
| Post Name | Assistant Professor (Mathematics) |
| No. of Posts | 79 |
| Mode of Application | Online |
| Category | Syllabus and Exam Pattern |
| Selection Process | Preliminary Examination, Mains Exam, Interview |
| Mains Exam Date 2026 | 15th September to 01st October 2026 |
| Job Location | Uttar Pradesh |
| Official Website | uppsc.up.nic.in |
UPPSC Assistant Professor (Mathematics) Syllabus 2026
General Studies :
1. General Science.
2. Current Events of National and International Importance.
3. History of India (Including Indian National Movement).
4. Indian Polity and Economy.
5. Geography- Indian and world.
6. Mental ability and Statistical data analysis.
Candidates are expected to have general awareness about the above topics with special reference to Uttar Pradesh.
Optional Subject (Mathematics) :
UNIT – 1
Analysis: Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf, uniform convergence. Bolzano Weierstrass theorem, Heine Borel theorem. Metric spaces, completeness, connectedness. Riemann integration, Lebesgue measure, Lebesgue integration. Normed linear Spaces, Banach spaces, Spaces of continuous functions as example, open mapping theorem, closed graph theorem, Hahn Banach theorem, Hilbert spaces.
Analysis: Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum. Sequences and series, convergence, limsup, liminf, uniform convergence. Bolzano Weierstrass theorem, Heine Borel theorem. Metric spaces, completeness, connectedness. Riemann integration, Lebesgue measure, Lebesgue integration. Normed linear Spaces, Banach spaces, Spaces of continuous functions as example, open mapping theorem, closed graph theorem, Hahn Banach theorem, Hilbert spaces.
UNIT – 2
Calculus: Continuity, Types of discontinuity, uniform continuity, differentiability, Monotonic functions, Functions of bounded variation, Mean value theorems. Sequences and series of functions, Functions of two or more variables, directional derivative, partial derivative, total derivative, maxima and minima, saddle points, Method of Lagrange's multipliers, Double and triple integrals and their applications, Improper integrals and their convergence. Vector Calculus: Gradient, divergence and curl, Green's Theorem, Stokes Theorem, Gauss Divergence Theorem
UNIT – 3
Algebra: Divisibility in Z, Fundamental theorem of arithmetic, Congruences and residue classes, Chinese Remainder Theorem, Euler s -- function, Fermats theorem, Groups, subgroups, normal subgroups, quotient groups Cayley's theorem, Fundamental theorem of group homomorphism, group action, Class equation, Sylow's theorems. Rings, ideals, prime and maximal ideals, quotient rings, unique factorization domain, principal ideal domain, Euclidean domain, Polynomial rings and irreducibility criteria. Fields, finite fields, field extensions, Galois Theory, Modules, Submodules, Cyclic modules, free modules, Noetherian and Artinian modules, Hilbert basis theorem.
UNIT – 4
Linear Algebra: Vector spaces, subspaces, linear dependence and independence, basis, dimension, algebra of linear transformations. Rank-Nullity theorem, Matrix representation of linear transformations. Change of basis, Solution of system of linear equations, Eigenvalues and eigenvectors, Cayley Hamilton theorem, Reduction to diagonal form, triangular form, rational and Jordan canonical form. Inner product spaces, orthonormal basis. Quadratic forms, reduction and classification of quadratic forms.
UNIT – 5
Complex Analysis: Limit, continuity and differentiability of complex functions, Analytic functions, Cauchy-Riemann equations. Complex integration, Cauchy's theorem, Cauchy's integral formula, Liouville's theorem, Maximum modulus principle, Schwarz lemma, Taylor series, Laurent series, calculus of residues, Contour integral, Conformal mappings, Mobius transformations. Topology: Basic concepts of topology, basis, dense sets, topological subspaces, First countable & second countable spaces, Separation axioms, Connected spaces and their basic properties, components, locally connected, spaces, Compactness, basic properties, Sequential and countable compactness.
UNIT – 6
Differential Equations: Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations, singular solutions of first order ODEs, system of first order ODEs. General theory of homogenous and non-homogeneous linear ODEs, Sturm-Liouville boundary value problem, Green's function. Linear differential equations of second order- Method of changing of dependent/independent variables, variation of parameters. Partial Differential
Equations (PDEs): Linear PDE of first order, Lagrange's method, Non-linear PDE of first order Charpit's method, General solution of higher order PDEs with constant coefficients, Classification of second order PDEs, Method of separation of variables, Laplace equation, Wave equation and Heat equation.
UNIT – 7
Numerical Analysis: Numerical solutions of algebraic equations, Method of iteration and Newton-Raphson method, Rate of convergence, Solution of systems of linear algebraic equations using Gauss elimination and GaussSeidel methods, Finite differences, Gregory-Newton, Lagrange interpolation formulae, Newton's divided difference formula, Numerical differentiation and integration, Newton Cote's formulae, Numerical solutions of ODEs using Picard, Euler, modified Euler and Runge-Kutta methods.
Calculus of Variations: Variation of a functional, Euler-Lagrange equation, Fixed end-point problem, variable end-point problem, Variational problems with subsidiary conditions. Linear Integral Equations: Linear integral equation of the first and second kind of Fredholm and Volterra type, Solution by the method of successive approximation, conversion of differential equation with initial condition, separable kernels. Eigenvalues and eigenfunctions, resolvent kernel.
UNIT – 8
Geometry: Polar equation of a conic, Cartesian and polar coordinates in three dimensions, Plane, straight lines, shortest distance between two skew lines; sphere, cone, cylinder, central conicoids, paraboloid.
Tensors: Contravariant and covariant tensors, transformation formulae, Tensor of (r, s)-type, symmetric and skew symmetric properties, contraction of tensors, inner product of tensors, quotient law.
Differential Geometry: Curves in space, curvature and torsion of curves, Serret-Frenet's formulae, Helix, first and second fundamental forms of a surface
UNIT – 9
Operations Research: Linear programming problem, basic feasible solution, Graphical method, simplex method, duality, transportation problem, assignment problem, travelling salesman problem, convex optimization, gradient descent, stochastic gradient descent.
Statistics and probability: Variance and standard deviation, Curve fitting by least squares, Corelation and regression, logistic regression, support vector regression, linear discriminant analysis, Sample space, Basic laws of probability, Independent events, Expectation, Bayes theorem. Random variables, discrete and continuous probability distribution functions-Binomial, Poisson and Normal.
Graph Theory: Graphs, isomorphism, subgraphs, matrix representations, operations on graphs, degree of a vertex, Connected
Graphs and shortest paths: Walks, trails, connected graphs, shortest path algorithms. Trees: minimum spanning trees. Bipartite graphs, Hamilton graphs, Planar graphs, Euler's formula, Eulerian directed graphs
UNIT – 10
Mechanics: Moment of inertia, Motion of a rigid body about an axis, Twodimensional motion of rigid bodies, Generalized coordinates, generalized momentum, Lagrange's equations, Hamilton's canonical equations, Hamilton's principle of least action, Contact transformations, Possion bracket.
Fluid Dynamics: Equation of continuity, Euler's equation of motion for inviscid flow, stream lines, boundary surface, Motion in two dimensions: Sources and sinks, method of images, Flow past a cylinder and sphere.
UPPSC Assistant Professor (Mathematics) Exam Pattern 2026
Preliminary Examination
Duration : 120 Minutes
Negative Marking : 0.33
Mains Examination
Duration : 180 Minutes
Mains Exam Date 2026 : 15th September to 01st October 2026
Starting Date of Application Form : 04th September 2025
Last Date of Application Form : 06th October 2025
Total Post : 79
Duration : 120 Minutes
Negative Marking : 0.33
| S.No | Subject | No.of uestion | Marks |
| 1 | General Studies | 30 | 30 |
| 2 | Optional Subject (Mathematics) | 70 | 70 |
| Total | 100 | 100 |
Mains Examination
Duration : 180 Minutes
| S.No | Subject | No.of Question | Marks |
| 1 | Optional Subject (Mathematics) | (10+10) 20 Section A: 10 Short Answer Questions (125 words each) Section B: 10 Long Answer Questions (200 words each) |
(80+120) 200 Section A: 10 questions x 8 marks = 80 marks Section B: 10 questions x 12 marks = 120 marks |
| Total | 120 | 150 |
Mains Exam Date 2026 : 15th September to 01st October 2026
Starting Date of Application Form : 04th September 2025
Last Date of Application Form : 06th October 2025
Total Post : 79
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