UPPSC Assistant Professor Online Program

UNIT– 1

Analysis: Elementary set theory, finite, countable, and uncountable sets, the Real number system as a complete ordered field, Archimedean property, supremum, and 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 examples, 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 φ-function, Fermat's theorem, Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation 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, systems of first-order ODEs. General theory of homogeneous 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 Gauss Seidel 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-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, Correlation 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, Two-dimensional motion of rigid bodies, Generalized coordinates, generalized momentum, Lagrange's equations, Hamilton's equations, Hamilton's principle of least action, Contact transformations, Poisson bracket. Fluid Dynamics: Equation of Continuity, 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.