UPSC Mains Syllabus for Mathematics Optional – A Complete Guide (Paper I & Paper II)

Mathematics is one of the most rewarding optional subjects in the UPSC Civil Services Examination. With a fixed syllabus, objective nature, and scoring potential, it is a popular choice among candidates with a strong math background. The syllabus is divided into two papers – Paper I and Paper II, each carrying 250 marks, making a total of 500 marks.

Let’s dive deep into the complete syllabus for Mathematics Optional in the UPSC Mains, categorized and described in detail.

Join WhatsApp community for Free Notifications, Updates, Study Material, Mock Tests, Internship Updates, and Current Affairs - CLICK HERE TO JOIN

Paper I – Pure and Applied Mathematics

1. Linear Algebra

• Vector spaces: Definition, examples, subspaces, basis, and dimension. Linear dependence and independence.
• Linear transformations: Matrix representation of linear transformations, rank and nullity, isomorphism.
• Matrices: Types of matrices, operations, determinant, adjoint, inverse.
• Eigenvalues and eigenvectors: Diagonalization, Cayley-Hamilton theorem.
• System of linear equations: Consistency, solution methods using matrices.

This section forms the foundation of abstract algebra and has practical applications in data science, economics, and physics.

2. Calculus

• Limits and continuity: Definitions, theorems, and problem-solving.
• Differentiation: Rules, mean value theorems, Taylor’s and Maclaurin’s series.
• Maxima and minima: Local and absolute extrema, functions of several variables, Lagrange multipliers.
• Indeterminate forms and L’Hôpital’s Rule.
• Integral calculus: Definite and indefinite integrals, methods of integration.
• Improper integrals: Convergence, Beta and Gamma functions.

This section is highly application-based, especially useful in optimization problems, mechanics, and engineering physics.

To Enroll in FIRST IAS INSTITUTE - Click Here

3. Analytical Geometry

• Cartesian and polar coordinates in two dimensions.
• Second degree equations in two variables: Conics, classification, reduction to standard forms.
• Three-dimensional geometry:
• Direction cosines and ratios.
• Plane, straight line, shortest distance between lines.
• Sphere, cone, and cylinder.
• Central conicoids: Ellipsoid, hyperboloid, paraboloid; their properties.

Emphasizes spatial visualization and geometric understanding critical in architecture and physics.

4. Ordinary Differential Equations (ODEs)

• First-order equations: Linear, exact, homogeneous, Bernoulli’s equations.
• Higher-order linear ODEs: Constant coefficients, method of variation of parameters.
• System of linear differential equations.
• Applications to physical problems: Mechanics, population models, electrical circuits.

Essential for understanding dynamics in real-world systems.

5. Dynamics & Statics

• Forces and equilibrium: Equilibrium of a particle, forces in two and three dimensions.
• Centre of mass, Work and energy, and Virtual work.
• Motion of a particle: Laws of motion, rectilinear and curvilinear motion, projectiles, constrained motion.
• Circular and simple harmonic motion.
• Central forces and Kepler’s laws.

Bridges physics and mathematics, useful in understanding planetary motion and mechanics.

6. Vector Analysis

• Vector differentiation and integration: Gradient, divergence, curl, Laplacian.
• Vector identities and applications.
• Line, surface, and volume integrals.
• Gauss’s, Green’s, and Stokes’ theorems: With physical interpretation and applications.

Join WhatsApp community for Free Notifications, Updates, Study Material, Mock Tests, Internship Updates, and Current Affairs - CLICK HERE TO JOIN

Key topic for fields such as fluid dynamics and electromagnetism.

Paper II – Advanced and Applied Mathematics

1. Algebra

• Group theory:
• Group definitions and examples, cyclic groups, permutation groups.
• Cosets, Lagrange’s theorem, normal subgroups, quotient groups.
• Homomorphism and isomorphism theorems.
• Ring theory:
• Rings, subrings, ideals, quotient rings.
• Integral domains and fields.
• Ring homomorphisms, polynomial rings.

Abstract algebra plays a vital role in cryptography, coding theory, and theoretical computer science.

2. Real Analysis

• Sequences and series: Convergence, tests for convergence, absolute and conditional convergence.
• Continuity and uniform continuity.
• Differentiability of real-valued functions.
• Riemann integration: Properties, improper integrals.
• Sequences and series of functions: Pointwise and uniform convergence, power series, Fourier series.

Important for deeper theoretical foundations of calculus and mathematical modeling.

To Enroll in FIRST IAS INSTITUTE - Click Here

3. Complex Analysis

• Functions of a complex variable: Analytic functions, Cauchy-Riemann equations.
• Complex integration: Cauchy’s integral theorem and formula.
• Taylor and Laurent series.
• Singularities, residues, and residue theorem.
• Conformal mappings and their applications.

Widely used in fluid mechanics, electromagnetic theory, and complex simulations.

4. Linear Programming

• Linear programming problems (LPP): Formulation and graphical method.
• Simplex method: Basic feasible solutions, optimality, degeneracy.
• Duality in LPP, dual simplex method.
• Transportation and assignment problems.

Highly practical, applied in economics, logistics, and operations research.

5. Partial Differential Equations (PDEs)

• First-order PDEs: Lagrange’s and Charpit’s methods.
• Second-order linear PDEs: Classification, canonical forms, methods of solution.
• Wave equation, heat equation, Laplace’s equation: Standard solutions and boundary value problems.

Integral to weather modeling, heat transfer, and quantum mechanics.

6. Numerical Analysis & Computer Programming

• Numerical methods:
• Root-finding: Bisection, Newton-Raphson.
• Interpolation: Lagrange and Newton’s.
• Numerical integration: Trapezoidal and Simpson’s rules.
• Numerical solution of ODEs: Euler and Runge-Kutta methods.
• Computer programming:
• Algorithms, flowcharts.
• Basics of C or pseudocode: Loops, conditionals, arrays.

Equips aspirants with skills for computational modeling and data analysis.

7. Mechanics and Fluid Dynamics

• Mechanics:
• Generalized coordinates, D’Alembert’s principle, Lagrange’s equations.
• Hamiltonian formulation, cyclic coordinates, conservation laws.
• Fluid dynamics:
• Kinematics of fluids, streamlines, velocity potential.
• Euler’s and Navier-Stokes equations (conceptual), Bernoulli’s theorem.
• Vortex motion, circulation.

Very useful for civil and mechanical engineering students; links physics and math deeply.

Join WhatsApp community for Free Notifications, Updates, Study Material, Mock Tests, Internship Updates, and Current Affairs - CLICK HERE TO JOIN

Final Tips on Mathematics Optional

• The syllabus is vast but well-defined. There are no current affairs or opinion-based questions, which makes it ideal for those who love problem-solving.
• Practice is the key. Solve previous year questions extensively.
• Ensure conceptual clarity in both proofs and applications.

Recommended Books

• Linear Algebra – K.C. Prasad and K.B. Datta
• Calculus and Real Analysis – Malik & Arora, Rudin
• ODE & PDE – M.D. Raisinghania
• Complex Analysis – Churchill
• Algebra – I.N. Herstein
• Mechanics & Fluid Dynamics – F. Chorlton
• Numerical Analysis – S.S. Sastry