## (Download) Uttar Pradesh Combined State / Upper Subordinate Services (UPPCS) Mains Optional Subject Exam Syllabus "Mathematics"

## :: 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 tripple 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's theorems, Curves in Space,
curvature and torsion, Serret-Frenet's formulae.

Definition of Tensor, Transformation of coordinates, contravariant and covariant tensors, addition and outer product of tensors. Contraction of tensors, inner product tensor, fundamental tensors, Christoffel symbols, covariant differentiation, gradiant, divergence and curl in tensor notation.

**6. Statics and Dynamics:** Virtual work, stability of
equilibrium. Catenary, Catenary of uniform strength, equilibrium of forces in
three dimensions.

Rectilinear motion, simple harmonic motion, velocities and accelerations along radial and transverse directions and along tangential and normal directions, Motion in resisting Medium, constrained motion, motion under impulsive forces, Kepler's laws, orbits under central forces, motion of varying mass.

## :: 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, test. ע – 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, Pricipal 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.