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# UPSC Mains Exam Syllabus - Mathematics Optional

**UPSC Mains Exam Syllabus - Mathematics Optional**

## 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’s and
similarity; Rank of a matrix; Inverse of a
matrix; Solution of 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, meanvalue
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, 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 first order and first degree, integrating
factor; Orthogonal trajectory; Equations
of first order but not of first degree,
Clairaut’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 method of
variation of parameters.
Laplace and Inverse Laplace transforms
and their properties; Laplace transforms of
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.

## IAS Mains General Studies Study Kit

## PAPER - II:

**(1) Algebra:**
Groups, subgroups, cyclic groups, cosets,
Lagrange’s Theorem, normal subgroups,
quotient groups, homomorphism of
groups, basic isomorphism theorems, permutation
groups, Cayley’s theorem. Rings, subrings and ideals, homomorphisms
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 least upper bound property; Sequences,
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 integrability 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-Falsi and Newton-
Raphson methods; solution of 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 Runga 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, path
of a particle; Potential flow; Two-dimensional
and axisymmetric motion; Sources
and sinks, vortex motion; Navier-Stokes
equation for a viscous fluid.

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## computers and mathematics