Optional
Mathematics
UPSC Civil Services Mains, optional subject
Syllabus
Paper I
- Linear Algebra
- Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions, Linear transformations, rank and nullity, matrix of a linear transformation.
- Algebra of Matrices
- Row and column reduction, Echelon form, congruence’s and similarity
- Rankof 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.
- 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 integral
- Double and triple integrals (evaluation techniques only)
- Areas, surface and volumes.
- 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.
- 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 liner equations with constant coefficients, complementary function, particular integral and general solution.
- Section 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.
- 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
- 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 equation.
- Application to geometry : Curves in space, curvature and torsion
- Serret-Furenet's formulae.
- Gauss and Stokes’ theorems, Green's identities.
Paper II
- 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.
- 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 function, 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.
- 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 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,heatequation,Laplaceequation and their solutions.
- 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-Jorden (direct), Gauss-Seidel (iterative) methods.
- Newton’s (forward and backward) and interpolation, Lagrange’s interpolation.
- Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula.
- Numerical solution of ordinary differential equations : Eular 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.
- 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
- 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.
Syllabus as published by UPSC in the Civil Services Examination 2026 notification.