Mathematics Optional Syllabus

Here is the official UPSC Mathematics Optional syllabus for Paper I and Paper II, arranged paper by paper and unit by unit. Under each unit we give the official text first, then break it into numbered topics, so you can track your preparation and match every previous year question to its topic.

Source. UPSC Civil Services Examination 2026 notification (Examination Notice No. 05/2026-CSE, dated 04.02.2026, pp. 124–126), upsc.gov.in. The Mathematics syllabus has been unchanged in recent years.

Scheme. Two papers of 250 marks each (total 500), 3 hours each. Each paper has two sections (A and B) of four questions each; Questions 1 and 5 are compulsory, and three more are attempted with at least one from each section. The syllabus itself is not divided into sections. Paper I has six numbered units (Linear Algebra; Calculus; Analytic Geometry; Ordinary Differential Equations; Dynamics and Statics; Vector Analysis); Paper II has seven (Algebra; Real Analysis; Complex Analysis; Linear Programming; Partial Differential Equations; Numerical Analysis and Computer Programming; Mechanics and Fluid Dynamics).

Syllabus at a glance

Paper	Unit	Topics
Paper I	1. Linear Algebra	4
Paper I	2. Calculus	5
Paper I	3. Analytic Geometry	3
Paper I	4. Ordinary Differential Equations	5
Paper I	5. Dynamics and Statics	5
Paper I	6. Vector Analysis	3
Paper II	1. Algebra	5
Paper II	2. Real Analysis	6
Paper II	3. Complex Analysis	4
Paper II	4. Linear Programming	3
Paper II	5. Partial Differential Equations	4
Paper II	6. Numerical Analysis and Computer Programming	7
Paper II	7. Mechanics and Fluid Dynamics	5

Paper I

1. Linear Algebra

1.1 Vector spaces

• Vector spaces over R and C
• Linear dependence and independence
• Subspaces, bases, dimensions

1.2 Linear transformations

• Linear transformations
• Rank and nullity
• Matrix of a linear transformation

1.3 Algebra of matrices and linear equations

• Row and column reduction, echelon form
• Congruence and similarity
• Rank and inverse of a matrix
• Solution of system of linear equations

1.4 Eigenvalues and special matrices

• Eigenvalues and eigenvectors
• Characteristic polynomial, Cayley-Hamilton theorem
• Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues

2. Calculus

2.1 Limits, continuity and differentiability

• Real numbers, functions of a real variable
• Limits, continuity, differentiability
• Mean-value theorem
• Taylor’s theorem with remainders
• Indeterminate forms

2.2 Maxima, minima, asymptotes and curve tracing

• Maxima and minima
• Asymptotes
• Curve tracing

2.3 Functions of two or three variables

• Limits, continuity, partial derivatives
• Maxima and minima
• Lagrange’s method of multipliers
• Jacobian

2.4 Integration

• Riemann’s definition of definite integrals
• Indefinite integrals
• Infinite and improper integrals

2.5 Multiple integrals, areas and volumes

• Double and triple integrals (evaluation techniques only)
• Areas, surface and volumes

3. Analytic Geometry

3.1 Coordinates, lines and planes

• Cartesian and polar coordinates in three dimensions
• Straight lines
• Shortest distance between two skew lines
• Plane

3.2 Sphere, cone and cylinder

• Sphere
• Cone
• Cylinder

3.3 Conicoids and second degree equations

• Second degree equations in three variables, reduction to canonical forms
• Paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties

4. Ordinary Differential Equations

4.1 First order first degree equations

• Formulation of differential equations
• Equations of first order and first degree, integrating factor
• Orthogonal trajectory

4.2 First order higher degree equations

• Equations of first order but not of first degree
• Clairaut’s equation
• Singular solution

4.3 Higher order linear equations with constant coefficients

• Complementary function
• Particular integral
• General solution

4.4 Second order equations with variable coefficients

• Euler-Cauchy equation
• Complete solution when one solution is known
• Variation of parameters

4.5 Laplace transforms

• Laplace and inverse Laplace transforms and their properties
• Laplace transforms of elementary functions
• Initial value problems for 2nd order linear equations

5. Dynamics and Statics

5.1 Rectilinear motion and SHM

• Rectilinear motion
• Simple harmonic motion

5.2 Motion in a plane and constrained motion

• Motion in a plane
• Projectiles
• Constrained motion
• Work and energy, conservation of energy

5.3 Central orbits

• Kepler’s laws
• Orbits under central forces

5.4 Equilibrium, friction and catenary

• Equilibrium of a system of particles
• Work and potential energy
• Friction
• Common catenary

5.5 Virtual work and stability

• Principle of virtual work
• Stability of equilibrium
• Equilibrium of forces in three dimensions

6. Vector Analysis

6.1 Vector differentiation

• 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

6.2 Curves in space

• Curvature and torsion
• Serret-Frenet formulae

6.3 Integral theorems

• Gauss and Stokes’ theorems
• Green’s identities

Paper II

1. Algebra

1.1 Groups and subgroups

• Groups, subgroups, cyclic groups
• Cosets, Lagrange’s theorem

1.2 Normal subgroups and homomorphisms

• Normal subgroups, quotient groups
• Homomorphism of groups
• Basic isomorphism theorems

1.3 Permutation groups

• Permutation groups
• Cayley’s theorem

1.4 Rings and ideals

• Rings, subrings and ideals
• Homomorphisms of rings

1.5 Integral domains and fields

• Integral domains
• Principal ideal domains, Euclidean domains, unique factorization domains
• Fields, quotient fields

2. Real Analysis

2.1 Real number system and sequences

• Ordered field with least upper bound property
• Sequences, limit of a sequence
• Cauchy sequence, completeness of real line

2.2 Series

• Series and its convergence
• Absolute and conditional convergence of real and complex series
• Rearrangement of series

2.3 Continuity and uniform continuity

• Continuity and uniform continuity
• Continuous functions on compact sets

2.4 Riemann integral

• Riemann integral
• Improper integrals
• Fundamental theorems of integral calculus

2.5 Sequences and series of functions

• Uniform convergence
• Continuity, differentiability and integrability of sequences and series of functions

2.6 Functions of several variables

• Partial derivatives of functions of two or three variables
• Maxima and minima

3. Complex Analysis

3.1 Analytic functions

• Analytic function
• Cauchy-Riemann equations

3.2 Cauchy’s theorem and integral formula

• Cauchy’s theorem
• Cauchy’s integral formula

3.3 Power, Taylor and Laurent series

• Power series, representation of an analytic function
• Taylor’s series
• Singularities
• Laurent’s series

3.4 Residues and contour integration

• Cauchy’s residue theorem
• Contour integration

4. Linear Programming

4.1 LPP, graphical and simplex methods

• Linear programming problems
• Basic, basic feasible and optimal solutions
• Graphical method
• Simplex method

4.2 Duality

4.3 Transportation and assignment problems

• Transportation problems
• Assignment problems

5. Partial Differential Equations

5.1 Formulation of PDEs

• Family of surfaces in three dimensions
• Formulation of partial differential equations

5.2 First order PDEs

• Quasilinear equations of the first order
• Cauchy’s method of characteristics

5.3 Second order linear PDEs

• Linear equations with constant coefficients
• Canonical form

5.4 Wave, heat and Laplace equations

• Vibrating string
• Heat equation
• Laplace equation

6. Numerical Analysis and Computer Programming

6.1 Solution of equations

• Bisection, Regula-Falsi and Newton-Raphson methods
• Gaussian elimination and Gauss-Jordan (direct)
• Gauss-Seidel (iterative)

6.2 Interpolation

• Newton’s forward and backward interpolation
• Lagrange’s interpolation

6.3 Numerical integration

• Trapezoidal rule
• Simpson’s rule
• Gaussian quadrature formula

6.4 Numerical solution of ODEs

• Euler method
• Runge-Kutta methods

6.5 Number systems and representation

• Binary, octal and hexadecimal systems
• Conversion to and from decimal
• Arithmetic and logical operations
• Algebra of binary numbers
• Representation of unsigned and signed integers, reals, double precision reals and long integers

6.6 Computer systems, logic gates and Boolean algebra

• Elements of computer systems, concept of memory
• Logic gates and truth tables
• Boolean algebra, normal forms

6.7 Algorithms and flow charts

• Algorithms and flow charts for numerical analysis problems

7. Mechanics and Fluid Dynamics

7.1 Lagrangian and Hamiltonian mechanics

• Generalised coordinates
• D’Alembert’s principle and Lagrange’s equations
• Hamilton equations

7.2 Moment of inertia and rigid body motion

• Moment of inertia
• Motion of rigid bodies in two dimensions

7.3 Kinematics and equations of fluid motion

• Equation of continuity
• Euler’s equation of motion for inviscid flow
• Stream-lines, path of a particle
• Potential flow

7.4 Two-dimensional and axisymmetric flow

• Two-dimensional and axisymmetric motion
• Sources and sinks
• Vortex motion

7.5 Viscous flow: Navier-Stokes equation

Total: 59 numbered topics across 13 units.

Last updated: 30 October 2026