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UPPSC 2010 Syllabus for Mathematics (Subject Code 05):

Paper - 1

Linear Algebra: Vector space, bases, dimensions of a finitely generated space, linear transformation: Rank and nullity of a linear transformation. Cayley Hamlllton theorem, Eigenvalues and Eigen vectors.

Matrix of linear transformation, Row and column reduction. Echelon form, Equivalence, Congruence and similarly, Reduction to canonical form. Orthogonal, symmetrical, skew-symmetrical, unitary, Hemitian and skew - Hermitian matrices their eigen values, orthogonal and unitary reduction of quadratic and Hermitian form. Positive definite quadratic form. Simultaneous reduction.

Calculus: Real numbers, limits, continuity, differentiability. Mean value theorems, Taylor's indeterminate forms, Maxima and minima. Curve Tracing Asymptotes. Functions of several variables, partial derivatives, maxima and minima, Jacobian Definite and indefinite integrals.

Double and 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 two dimensions in cartesian and polar coordinates. Plane, sphere, paraboloid, Ellipsoid, hyperboloid of one and two 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 and first degree, variables separable. Homogeneous. linear, and exact differential equations, differential equation with constant coefficients. The complementary function and the particular integral of e", cosax, sinax, xm, e, cosdx, e*, sinbx.

Vector Analysis: Vector Algebra, Differentiation of vector function of a scalar variable Gradient, divergence and curl in cartestian, cylindrical and spherical coordinates and their physical interpretation, Higher order derivates. vector identities and vector, equations, Gauss and stokes Theorems.

Tensor Analysis: Definition of Tensor, Transformation of coordinates, contravariant and contravariant tensors, Addition and multiplication of tensors, contraction of tensors. Inner product, fundamental tensors. Christoffel symbols, contravariant differentiation. Gradient, 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: Degree of freedom and constraints. Rectilinear motion Simple harmonic motion in a plane. Projectiles, Constrained motion, work and energy. Motion under impulsive forces, Kepler's laws. Orbits under central forces. Motion of varying mass. Motion under resisting medium.

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

Paper - II

Algebra: Groups, subgroups, normal subgroup, homomorphlsm of groups, quotient groups Baisc Isomorphism theorems, sylow theorems. Permutation Groups. Cayley's Theorem. Rings and ideals. Principal ideal domains, unique factorization domains and Euclidean domains, Field Extensions, Finite fields.

Real Analysis: Metric spaces, their topology with special reference to 'R' sequence in metric space Catchy sequence completeness. Completion, contineous functions. Uniform continuity. Properties of continuous function of Compact sets. Riemann Steltjes integral.

Improper integral and their condition's of existence. Differentiation of 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's series.

Singularities. Cachley's Residue theorem and Contour integration. Partial Differential Equations : Formation of partial differential equation. Types of integrals of partial differential equations of first order, Charphs method. Partial differential equation with constant coeffcients.

Mechanics: Generalised constraints, constraints, holonomic and non-holonomic systems, D' Alemberts' Principle and Language's equations, Moment of inertia. Motion of rigid bodies in two dimensions. Hydrodynamics : Equation of continuity. momentum and energy, mviscid flow theory. Two dimensional motion, streaming motion sources and Sinks.

Numerical Analysis: Transcendental and polynomial Equations-Methods of tabulation, bisection, reaula-false secants and Newton-Renhsoand order of its convergence. Interpolation and Numerical differentiation formulae with error terms. Numerical Integration of Ordinary differential Equations': Euler's method, mulistepperdictors Corrector methods.

Adams 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.

UPPSC Main Exam 2010 Syllabus for Mathematics