**:: ****UPPSC Maths Optional Syllabus ****PAPER - II ::**

**1. Algebra:** Groups, Cyclic groups, subgroups, Cosets of a subgroup, Lagrange's theorem, Normal subgroups, Homomorphism of groups, Factor groups, basic Isomorphism theorems, Permutation groups, Cayley's theorem.

Rings, Subrings, Ideals, Integral domains, Fields of quotients of an integral domain, Euclidean domains, Principal ideal domains, Polynomial rings over a field, Unique factorization domains.

**2. Real Analysis:** Metric spaces and their topology with special reference to sequence, Convergent sequence, Cauchy sequences, Cauchy's criterion of convergence, infinite series and their convergence, nth term test, series of positive terms, Ratio and root tests, limit comparison tests, logarithmic ratio test, condensation test, Absolute and conditional convergence of general series in R, Abel's Dirichlet's theorems. Uniform convergence of sequences and series of functions over an interval, Weierstrass M-test, Abel's and Dirichlet's tests, continuity of limit function. Term by term integrability and differentiability.

Riemann's theory of integration for bounded functions, integrability of continuous functions. Fundamental theorem of calculus. Improper integrals and conditions for their existence, tests.

**3. Complex Analysis:** Analytic functions, Cauchy-Riemann equations, Cauchy's theorem, Cauchy's integral formula, Power series representation of an analytic function. Taylor's series. Laurent's series, Classification of singularities, Cauchy's Residue theorem, Contour integration.

**4. Partial Differential Equations:** Formation of partial differential equations. Integrals of partial differential equations of first order, Solutions of quasi linear partial differential equations of first order, Charpit's method for non-linear partial differential equations of first order, Linear Partial differential equations of the second order with constant coefficients and their canonical forms, Equation of vibrating string. Heat equation. Laplace equation and their solutions.

**5. Mechanics:** Generalized co-ordinates, generalized velocities, Holonomic and nonholonomic systems, D'Alembert's principle and Lagrange's equations of motion for holonomic systems in a conservative field, generalized momenta, Hamilton's equations. Moments and products of inertia, Principal axes, Moment of inertia about a line with direction cosines (l,m,n), Momental ellipsoid, Motion of rigid bodies in two dimensions.

**6. Hydrodynamics:** Equation of continuity, Velocity Potential, Stream lines, Path Lines, Momentum and energy. Inviscid flow theory: Euler's and Bernoulli's equations of motion. Two-dimensional fluid motion, Complex potential, Momentum and energy, Sources and Sinks, Doublets and their images with respect line and circle.

**7. Numerical Analysis:** Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi and Newton-Raphson methods and order of their convergence. Interpolation (Newton's and Lagrange's) and Numerical differentiation formula with error terms. Numerical Integration: Trapezoidal and Simpson's rules. Numerical solutions of Ordinary differential Equations: Euler's method, Runge-Kutta method.