The UPSC Mathematics Optional syllabus is vast and highly conceptual, making it a favoured choice for candidates with a strong mathematical background. The syllabus hasn’t changed much from 2024. While it consists of key topics like Linear Algebra, Calculus, and Numerical Methods, understanding the depth of each subject is crucial for success. One often-overlooked aspect is the need for consistent practice, especially for topics like Differential Equations and Real Analysis.
Additionally, having a clear strategy for solving complex problems and revisiting foundational concepts regularly can be game-changing. This blog provides a detailed UPSC mathematics optional syllabus, book recommendations, and much more. Keep reading.
UPSC Maths Optional Syllabus 2024: Topics
| UPSC Maths Optional Paper I | UPSC Maths Optional Paper II |
| Linear AlgebraCalculusAnalytic GeometryOrdinary Differential EquationsDynamics and StaticsVector Analysis | AlgebraReal AnalysisComplex AnalysisLinear ProgrammingPartial Differential EquationsNumerical Analysis and Computer ProgrammingMechanics and Fluid Dynamics |
UPSC Maths Optional 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; Rank of 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 integrals; 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, the 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 the 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 linear equations with constant coefficients, complementary functions, particular integral and general solutions. Section order linear equations with variable coefficients, Euler-Cauchy equation; Determination of 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; 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.
UPSC Maths Optional Syllabus: 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, 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, Regula-Falsi and Newton-Raphson methods, solution of the system of linear equations by Gaussian Elimination and Gauss-Jorden (direct), Gauss-Seidel (iterative) methods. Newton’s (forward and backwards) and interpolation, Lagrange’s interpolation.
Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula.
Numerical solution of ordinary differential equations: Euler 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, the path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.
Note: If you’re looking for a UPSC maths optional syllabus PDF download, you can visit the official UPSC website.
UPSC Mathematics Optional Books
| Maths Paper | Topic | Books/Authors |
| Paper 1 | Linear Algebra | – 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 (ODE) | – Ordinary and Partial Differential Equations – M.D. Raisinghania | |
| Dynamics and Statics | – Krishna Series | |
| Vector Analysis | – Schaum Series – Murray R. Spiegel | |
| Paper 2 | Algebra | – Contemporary Abstract Algebra – Joseph Gallian |
| Real Analysis | – Same as Calculus in Paper 1 | |
| Complex Analysis | – Schaum Series – Speigel, Lipschitz, Schiller, Spellman | |
| Linear Programming | – Linear Programming and Game Theory – Lakshmi Shree Bandopadhyay | |
| Partial Differential Equations (PDE) | – Same as ODE in Paper 1- Advanced Differential Equations – M.D. Raisinghania | |
| Numerical Analysis | – Computer-Based Numerical and Statistical Techniques – M. Goyal- Numerical Methods – Jain, Iyengar, and Jain | |
| Computer Programming | – Digital Logic and Computer Design – M. Morris Mano | |
| Mechanics and Fluid Dynamics | – Krishna Series |
Conclusion
While the UPSC Mathematics Optional syllabus is detailed and comprehensive, mastering it requires more than good books. Regular practice, time management, and strategic problem-solving are essential for success. It’s also important to stay updated with past years’ question papers and refine exam-writing skills. A well-structured study plan, with consistent revisions and mock tests, can significantly boost confidence and improve performance in the actual exam.
FAQs
What is the syllabus of maths optional UPSC?
The syllabus for UPSC Mathematics Optional includes:
Paper I: Linear Algebra, Calculus, Analytic Geometry, Ordinary Differential Equations, Dynamics and Statics, and Vector Analysis.
Paper II: Algebra, Real Analysis, Complex Analysis, Linear Programming, Partial Differential Equations, Numerical Analysis and Computer Programming, Mechanics, and Fluid Dynamics.
For a detailed syllabus, visit the official website of UPSC at upsc.gov.in.
Is a calculator allowed in IAS mains?
No, UPSC Civils doesn’t allow calculators inside the examination hall for mathematics exams.
How long does it take to finish the Maths optional for IAS?
It usually takes 6-7 months to complete Maths optional for IAS. Depending on your comfort, understanding, and effort, it could take less or more time. Those without a Maths background may need at least 7-8 months or more.
Can I crack UPSC if I am weak in maths?
Yes, you can crack UPSC even if you’re weak in maths. Focus on your strengths, choose optional subjects wisely, and work hard on general studies and other non-maths areas.
Is Mathematics a good optional for UPSC Civils?
Yes, mathematics is a good optional for UPSC Civils if you have a strong background in it. It offers scoring potential but requires consistent practice and a clear understanding of concepts.
