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Mathematics Optional Syllabus for UPSC

The UPSC Mathematics optional is examined in two Mains papers of 250 marks each, 500 marks in total. Paper I has 6 topics and Paper II has 7 topics, listed below with their subtopics.

Mathematics Paper I syllabus

  1. 1. Linear Algebra

    • Vector spaces, linear dependence & independence, subspaces, bases, dimensions
    • Linear transformations, rank, nullity, matrix representation
    • Row reduction, echelon form, rank of matrix, inverse, eigenvalues & eigenvectors
    • Cayley-Hamilton theorem; symmetric, Hermitian, orthogonal & unitary matrices

  2. 2. Calculus

    • Limits, continuity, differentiability, mean-value theorem, Taylor's theorem
    • Maxima & minima, asymptotes, curve tracing; functions of two/three variables
    • Riemann's definite integrals, indefinite integrals, improper integrals
    • Double & triple integrals; areas, surface areas and volumes

  3. 3. Analytic Geometry

    • Cartesian & polar coordinates in 3D; second-degree equations in three variables
    • Straight lines, shortest distance between skew lines
    • Plane, sphere, cone, cylinder; paraboloid, ellipsoid, hyperboloid
    • Reduction to canonical forms; properties of conics

  4. 4. Ordinary Differential Equations

    • First-order equations — integrating factor, orthogonal trajectories, Clairaut's equation
    • Second & higher order linear equations — complementary function, particular integral
    • Euler-Cauchy equation; variation of parameters
    • Laplace & inverse Laplace transforms; applications to IVPs

  5. 5. Dynamics, Statics & Vector Analysis

    • Rectilinear motion, SHM, projectiles, constrained motion, work & energy
    • Kepler's laws; orbits under central forces; equilibrium of particles
    • Scalar & vector fields; gradient, divergence, curl in Cartesian & cylindrical coordinates
    • Gauss's theorem, Stokes' theorem, Green's identities

  6. 6. Dynamics and Statics

    • Common catenary, principle of virtual work, stability of equilibrium
    • Equilibrium of forces in three dimensions
    • Friction, work & potential energy
    • Motion in a plane, conservation of energy

Mathematics Paper II syllabus

  1. 1. Algebra

    • Groups, subgroups, cyclic groups, cosets, Lagrange's theorem, normal subgroups
    • Quotient groups, homomorphisms, isomorphism theorems, permutation groups, Cayley's theorem
    • Rings, subrings, ideals, integral domains, Euclidean domains, UFDs
    • Fields & quotient fields

  2. 2. Real Analysis

    • Real number system as ordered field; sequences, Cauchy sequence, completeness
    • Series convergence — absolute & conditional; rearrangement of series
    • Continuity, uniform continuity; Riemann integral, improper integrals
    • Uniform convergence; partial derivatives, maxima & minima of functions of several variables

  3. 3. Complex Analysis

    • Analytic functions, Cauchy-Riemann equations, Cauchy's theorem & integral formula
    • Power series representation, Taylor's series, singularities, Laurent's series
    • Cauchy's residue theorem, contour integration
    • Conformal mappings basics

  4. 4. Linear Programming

    • LP problems — basic solution, basic feasible solution, optimal solution
    • Graphical method and simplex method; duality theory
    • Transportation & assignment problems
    • Sensitivity analysis

  5. 5. Partial Differential Equations

    • Family of surfaces, formulation of PDEs; quasi-linear first-order PDEs
    • Cauchy's method of characteristics; second-order linear PDEs with constant coefficients
    • Canonical form; vibrating string equation (wave equation)
    • Heat equation, Laplace equation and their solutions

  6. 6. Numerical Analysis & Computer Programming

    • Root-finding — bisection, Regula-Falsi, Newton-Raphson methods
    • System of equations — Gaussian elimination, Gauss-Seidel; Newton's interpolation
    • Numerical integration — trapezoidal, Simpson's rule, Gaussian quadrature
    • Binary, octal, hexadecimal systems; Boolean algebra; basic logic gates; algorithms & flowcharts

  7. 7. Mechanics & Fluid Dynamics

    • Generalised coordinates, D'Alembert's principle, Lagrange's equations, Hamilton equations
    • Moment of inertia; rigid body motion in two dimensions
    • Equation of continuity; Euler's equation for inviscid flow; stream-lines, potential flow
    • Navier-Stokes equation for viscous fluid; sources & sinks, vortex motion

Other Sciences optionals

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