__UPSC Math’s Optional Syllabus __

**PAPER – I**

**(1) Linear
Algebra:** Vector spaces over R and C, linear dependence and
independence, subspaces, bases, dimension; linear transformations, rank and
nullity, matrix of a linear transformation. Algebra of Matrices; row and column
reduction, echelon form, congruence, and similarity; the rank of a matrix; the inverse
of a matrix; solution of a system of linear equations; eigenvalues and
eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, symmetric,
skew-symmetric, Hermitian, Skew-Hermitian, orthogonal and unitary matrices, and
their eigenvalues.

**(2)
Calculus:** Real numbers, functions of a real variable, limits,
continuity, differentiability, mean value theorem, Taylor’s theorem with
remainders, indeterminate forms, maxima and minima, asymptotes; curve tracing;
functions of two or three variables: limits, continuity, partial derivatives,
maxima, and minima, Lagrange’s method of multipliers, Jacobian. Riemann’s
definition of definite integrals; indefinite integrals; infinite and improper
integrals; double and triple integrals (evaluation techniques only); Areas,
surface, and volumes.

**(3)
Analytic Geometry:** Cartesian and polar coordinates in three
dimensions, second-degree equations in three variables, reduction to canonical
forms, straight lines, the shortest distance between two skew lines; Plane, sphere,
cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and
their properties.

**(4)
Ordinary Differential Equations:** Formulation of differential
equations; equations of the first order and first degree, integrating factor;
orthogonal trajectory; equations of first order but not of the first degree, Claimant’s
equation, singular solution. Second and higher-order linear equations with
constant coefficients, complementary function, particular integral, and general
solution. Second-order linear equations with variable coefficients,
Euler-Cauchy equation; determination of complete solution when one solution is
known using a method of variation of parameters. Laplace and inverse Laplace
transforms and their properties; Laplace transforms elementary functions.
Application to initial value problems for 2nd order linear equations with
constant coefficients.

**(5)
Dynamics & Statics:** Rectilinear motion, simple harmonic motion,
motion in a plane, projectiles; constrained motion; work and energy, conservation
of energy; Kepler’s laws, orbits under central forces. 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.

**(6) Vector
Analysis:** Scalar and vector fields, differentiation of vector
field of a scalar variable; gradient, divergence and curl in Cartesian and
cylindrical coordinates; higher-order derivatives; Vector identities and vector
equations. Application to geometry: curves in space, curvature, and torsion; Serret
-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.

__ __

__UPSC Math’s Optional Syllabus for Paper
II__

__ __

**PAPER-II**

**(1)
Algebra:** Groups, subgroups, cyclic groups, costs, Lagrange’s
Theorem, normal subgroups, quotient groups, homomorphism of groups, basic
isomorphism theorems, permutation groups, Cayley’s theorem. Rings, subrings and
ideals, homomorphism’s of rings; Integral domains, principal ideal domains,
Euclidean domains, and unique factorization domains; Fields, quotient fields.

**(2) Real
Analysis:** Real number system as an ordered field with the least upper
bound property; Sequences, the limit of a sequence, Cauchy sequence, completeness
of real line; Series and its convergence, absolute and conditional convergence
of series of real and complex terms, rearrangement of series. Continuity and
uniform continuity of functions, properties of continuous functions on compact
sets. Riemann integral, improper integrals; Fundamental theorems of integral
calculus. Uniform convergence, continuity, differentiability, and inerrability
for sequences and series of functions; partial derivatives of functions of
several (two or three) variables, maxima, and minima.

**(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; singularities; Laurent’s series; Cauchy’s residue
theorem; contour integration.

**(4) Linear
Programming:** Linear programming problems, basic solution, basic
feasible solution, and optimal solution; Graphical method and simplex method of
solutions; duality. Transportation and assignment problems.

**(5) Partial
differential equations:** Family of surfaces in three dimensions and
formulation of partial differential equations; solution of quasilinear partial
differential equations of the first order, Cauchy’s method of characteristics;
Linear partial differential equations of the second order with constant
coefficients, canonical form; equation of a vibrating string, heat equation,
Laplace equation, and their solutions.

**(6)
Numerical Analysis and Computer programming:** Numerical methods:
solution of algebraic and transcendental equations of one variable by
bisection, Regula-False, and Newton-Raphson methods; solution of a system of
linear equations by Gaussian elimination and Gauss-Jordan (direct),
Gauss-Seidel(iterative) methods. Newton’s (forward and backward) interpolation,
Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s
rules, Gaussian quadrature formula. Numerical solution of ordinary differential
equations: Euler and Runge Kutta-methods. Computer Programming: Binary system;
Arithmetic and logical operations on numbers; Octal and Hexadecimal systems;
Conversion to and from decimal systems; Algebra of binary numbers. Elements of
computer systems and concept of memory; Basic logic gates and truth tables,
Boolean algebra, normal forms. Representation of unsigned integers, signed
integers and reals, double precision reals, and long integers. Algorithms and
flow charts for solving numerical analysis problems.

**(7)
Mechanics and Fluid Dynamics:** Generalized coordinates; D’
Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of
inertia; Motion of rigid bodies in two dimensions. Equation of continuity;
Euler’s equation of motion for inviscid flow; Stream-lines, a path of a particle;
Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks,
vortex motion; Naiver-Stokes equation for a viscous fluid.

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