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- UPPSC Maths Optional Syllabus
UPPSC Maths Optional Syllabus
:: UPPPSC Maths Optional Syllabus PAPER - I ::
1. Linear Algebra and Matrix: Vector spaces, Sub Spaces, basis and
dimensions, Quotient. space, co-ordinates, linear transformation, rank and
nullity of a linear transformation, matrix representation of linear
transformation, linear functionals, dual space, transpose of a linear
transformation, characteristic values, annihilating polynomials,
Cayley-Hamilton theorem, Inner product spaces, Cauchy-Schwarz inequality,
Orthogonal vectors, orthogonal complements, orthonormal sets and bases,
Bessel's inequality of finite dimensional spaces, Gram-Schmidt
orthogonalisation process. Rank of Matrix, Echelon form, Equivalence,
congruence and similarity, Reduction to canonical form, orthogonal,
symmetrical, skew-symmetrical, Hermitian and skew-Hermitian matrices, their eigen
values, orthogonal and unitary reduction of quadratic and Hermitian form,
Positive definite quadratic forms, simultaneous reduction.
2. Calculus: Limits, continuity, differentiability, mean
value theorems, Taylor's theorem, indeterminate forms, maxima and minima,
tangent and normal, Asymptotes, curvature, envelope and evolute, curve tracing,
continuity and differentiability of function of several variables
Interchangeability of partial derivatives, Implicit functions theorem, double
and triple integrals. (techniques only), application of Beta and Gamma
functions, areas, surface and volumes, centre of gravity.
3. Analytical Geometry of two and three
dimensions: General equation of second
degree, system of conics, confocal conics, polar equation of conics and its
properties. Three dimensional co-ordinates, plane, straight line, sphere, cone
and cylinder. Central conicoids, paraboloids, plane section of conicoids,
generating lines, confocal conicoids.
4. Ordinary differential equations: Order and Degree of a differential equation,
linear, and exact differential equations of first order and first degree, equations
of first order but not of first degree, Singular solutions, Orthogonal
trajectories, Higher order linear equations with constant coefficients,
Complementary functions and particular integrals.
Second order linear differential
equations with variable coefficients: use of known solution to find another,
normal form, method of undetermined coefficients method of variation of
parameters.
5. Vector and Tensor Analysis: Vector Algebra, Differentiation and integration of vector function of a scalar variable gradient, divergence and curl in cartesian, cylindrical and spherical coordinates and their physical interpretation, Higher order derivates, vector identities and, vector equations, Gauss and Stoke
:: UPPPSC 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, Rune-Kutta method.