UPSC Maths Optional Syllabus: Choosing the right optional subject for the UPSC exams is one of the major decisions that a candidate has to make in his UPSC journey. Choosing the right subject can make your UPSC journey a smooth ride. With a lot of optional subjects to take from, sometimes it can be quite difficult to choose the right one.
Maths is one of the favorite optional subjects of many UPSC candidates. It is usually preferred by people with a good background in the subject.
Why should you choose maths as optional?
One of the main reasons why a large number of people opt for maths is the practical nature of the subject. Unlike subjects like geography, history, anthropology, and philosophy, maths doesn’t have any theory. Most of the questions are practical in nature and are easy for anyone with a good background to solve and ace the exams.
Maths optional consists of two papers. Paper 1 and paper 2 with 250 marks for each.
Pros of Taking Maths as Optional
 High scoring potential
 High chance of getting direct questions
 No theory to study
 Increases critical thinking and problemsolving skills
 It can help to develop logical reasoning
UPSC Maths Optional Syllabus – Paper 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
 Rank of a matrix, inverse of a matrix
 Solution of system of linear equations, eigenvalues and eigenvectors, characteristic polynomial,
 CayleyHamilton theorem, symmetric, skewsymmetric,
 Hermitian, skewHermitian,
 Orthogonal and unitary matrices and their eigenvalues.

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.

Analytic Geometry
 Cartesian and polar coordinates in three dimensions
 Seconddegree equations in three variables
 Reduction to canonical forms, straight lines, shortest distance between two skew lines, plane, sphere, cone, cylinder, paraboloid, ellipsoid
 Hyperboloids of one and two sheets and their properties

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 higherorder linear equations with constant coefficients, complementary functions, particular integrals and general solution.
 Secondorder linear equations with variable coefficients,
 EulerCauchy equation, the determination of a complete solution when one solution is known using the 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.

Dynamics and 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, the principle of virtual work
 Stability of equilibrium, equilibrium of forces in three dimensions.

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, SerretFrenet’s formulae.
 Gauss and Stokes’ theorems, Green’s identities.
UPSC Maths Optional Syllabus – Paper 2

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, and quotient fields.

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.

Complex Analysis
 Analytic functions
 CauchyRiemann 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.

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.

Partial differential equations
 Family of surfaces in three dimensions and formulation of partial differential equations
 Solution of quasilinear partial differential equations of the firstorder
 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.

Numerical Analysis and Computer programming
 Numerical methods: solution of algebraic and transcendental equations of one variable by bisection,
 RegulaFalsi and NewtonRaphson methods
 Solution of system of linear equations by Gaussian elimination and GaussJordan (direct), GaussSeidel(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 Kuttamethods.
 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 the 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.

Mechanics and Fluid Dynamics
 Generalised 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
 Streamlines, path of a particle, Potential flow
 Twodimensional and axisymmetric motion, Sources and sinks, vortex motion;
 NavierStokes equation for a viscous fluid
Books to Refer for Maths Optional
Recommended books for Paper 1
 Schaum series – Seymour Lipschutz
 Linear Algebra – Hoffman and Kunze
 Calculus
 Mathematical Analysis – S C Malik and Savita Arora
 Elements of Real Analysis – Shanti Narayan and M D Raisinghania
 Analytic Geometry
 Analytical Solid Geometry – Shanti Narayan and P K Mittal
 Solid Geometry – P N Chatterjee
 Ordinary differential equations
 Ordinary and partial differential equations – M D Raisinghania
 Dynamics and Statics
 Krishna Series
 Vector Analysis
 Schaum Series – Murray R. Spiegel
Recommended books for Paper 2
 Algebra – Contemporary Abstract algebra – Joseph Gallian
 Real analysis
 Complex analysis
 Schaum series – Spiegel, Lipschitz, Schiller, Spellman
 Linear programming
 Linear programming and game theory – Lakshmi Shree Bandopadhyay
 Partial differential equations
 Advanced differential equations – MD Raisinghania
 Numerical analysis and computer programming
 Computerbased numerical and statistical techniques – M Goyal
 Numerical Methods – Jain, Iyengar and Jain
 Digital logic and Computer design – M Morris Mano
 Mechanics and Fluid Dynamics
The syllabus of maths might look vast, but with the right strategy and preparation, anyone can ace the maths optional exam.