# UPPSC Mathematics Syllabus

## Paper-1

Linear Algebra: Vector space, bases, dimensions of a finitely generated space, linear transformation: Rank and nullity of a linear transformation. Cayley Hamiliton theorem, Eigenvalues and Eigen vectors. Matrixof linear transformation, Row and column reduction. Echelon form, Equivalence, Congruence and similarty, Reduction to canonical form. Orthogonal, symmetrical, skew-symmetrical, unitary, Hemitian and skew-Hermi-tian matrices their eigen values, orthogonal and unitary reduction of quadratic and Hermitian form. Positivedefinite quadratic form. Simultaneous reduction.

Calculus: Real numbers, limits, continuity, differentiability. Mean value theorems, Taylor's indeterminate forms, Maxima and minima. Curve Tracing Asymptotes. Functionsof several variables, partial derivatives, maxima and minima, Jacobian Definite and indefinite integrals. Doubleand tripple integrals (techniques only), application to Beta and Gamma Functions, Areas, Volumes, Centre of gravity.

Analytical Geometry of two and three dimensions: First and second degree equations in twodimensions in cartesian and polar coordinates. Plane, sphere, parabofoid, Ellipsoid, hyperboloid of one andtwo sheets and their elementary properties. Curves in space. Curvature and torsion. Frenet's formulze. Differential Equations: Order and Degree of a differential equation, differential equation of first order andfirst degree, variables separable. Homogeneous, linear, and exact differential equations, differential equa-tion with constant coefficients. The complementary function and the particular integral of eax, cosax, sinax, xm, e ax, cosdx, e ax, sinbx. Vector Analysis: Vector Algebra, Differentiation of vector function of a scalarvariable Gradient, divergence and curl in cartestian, cylindrical and spherical coordinates and their physicalinterpretation, Higher order derivates. Vector identities and vector, equations, Gauss and stokes Theorems. Tensor Analysis: Definition of Tensor, Transformation of coordinates, contravariant and contravariant ten-sors. Addition and multiplication of tensors, contraction of tensors. Inner product, fundamental tensors, Christoffel symbols, contravariant differentiation, Gradiant, curl and divergence in tensor notation. Statics: Equilibrium of a system of particles, work and potential energy. Friction. Common catenary. Principle of Virtual work Stability of equilibrium. Equilibrium of forces in three dimensions.

Dynamics: Degreeof freedom and constraints. Rectilinear motion Simple harmonic motion in a plane. Projectiles, Constrainedmotion, work and energy. Motion under impulsive forces, Kepler's laws. Orbits under central forces. Motionof varying mass. Motion under resisting medium.

Hydrostatics: Pressure of heavy fluids. Equilibrium offluids under given system of forces. Centre of pressure. Thrust on curved surfaces. Equilibrium of floatingbodies, stability of equilibrium and pressure and gases, problems relating to atmosphere.

## Paper-II

Algebra: Groups, subgroups, normal subgroup, homomorphism of groups, quotient groups Baisc isomor-phism theorems, sylow theorems. Permutation Groups. Cayley's Theorem. Rings and ideals. Principal idealdomains, unique ractorization domains and Euciiden domains, Field Extensions, Finite fields. Real Analysis: Metric spaces, their topology with special reference to ‘R’ sequence in metric space Cauchy sequencecompleteness. Completion, continuous functions. Uniform continuity. Properties of continuous function ofCompact sets. Riemann Steltjes Integral. Improper integral and their condition's of existence. Differentiationof function of several variables. Implicit function theorem, maxima and minima.

Absolute and conditional Convergence of series of real Complex terms, Rearrangement of series, Uniform-convergence, infinite products. Continuity, differentiability and integrabillity of series, Multiple integrals. Complex Analysis: Analytic functions, Cauchy's theorem, Cauchy's integral formula, power series, TayloR'sseries, Singularities, Cachley's Residue theorem and Contour integration. Partial Differential

Equations: Formation of partial differential equation. Types of integrals of partial differential equaltions of first order, Charphs method, Partial differential equation with constant coeffcients. Mechanis: Generalised constraints, constraints, holonomic and non-holonomic systems, D ‘Alemberts’ Principle and Langrange's equations, Mo-ment of intertia. Motion of rigid bodies in two dimensions. Hydrodynamics: Equation of continuity. Momentum and energy, inviscid flow theory. Two dimensional motion, streaming motion sources and Sinks. NumericalAnalysis: Transcendental and ploynomial Equations-Methods of tabulation, bisection, reaula-false secantsand Newton-Renhso and order of its converagence. Interpolation and Numerical differentiation formulae witherror terms. Numercial Integration of Ordinary differential Equations: Euler's method, mulistepperdictorsCorrector methods. Adam's and Milne's method convergence and stability, Runge Kutta Method.

Operational Research: Mathematical Programming, Definition and some elementary properties of convex sets, simplex methods, rectangular games and their solutions.