Maths is one of the optional subjects in the UPSC mains exam stage. Candidates who are well versed in Mathematics can choose Maths as their optional subject. Choosing Maths optional will require candidates to give two papers, namely Maths Optional Paper 1 & 2, each for 250 marks. So, to prepare well for the mains exam, candidates need to have clarity of the UPSC Maths Syllabus. 

In this article on UPSC Maths Syllabus, we have provided the detailed UPSC Maths Optional Syllabus as per the latest UPSC Notification.

Check out UPSC Maths Optional Syllabus in Hindi here.

Note: The UPSC Maths Optional Syllabus is for the Maths Optional Subject in the Mains Exam, while the UPSC CSAT Maths Syllabus or the maths syllabus for upsc prelims is for the Prelims CSAT Paper of the IAS Exam. So, don't get confused between the two syllabi.

Check out the complete UPSC Mains Syllabus in the linked article.

Topic Wise UPSC Maths Optional Syllabus For IAS Paper 1

The IAS Maths Syllabus for Optional Paper 1 covers the topics like Linear Algebra, Calculus, Analytic Geometry, Differential Equations and more. Candidates can look at the detailed official UPSC Maths Syllabus in the section below.

Note - Answers must be written in the medium authorized in Admission Certificate

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 and similarity; Rank of a matrix; Inverse of a matrix; Solution of a 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 integrals 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 equations. Application to geometry: Curves in space, curvature and torsion; Serret-Furenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.

Also, check out UPSC Prelims Syllabus here.

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Topic Wise UPSC Maths Optional Syllabus For IAS Paper 2

The Offical Maths Optional Syllabus for optional paper 2 covers topics & sub-topics like Algebra, Real & Complex Analysis, Linear Programming, Mechanics & Fluid Dynamics and more. Candidates can get detailed clarity of the IAS Maths Syllabus for Optional Paper 2 in the section below.

Note - Answers must be written in the medium authorized in Admission Certificate

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 the least upper bound property; Sequences, the 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 a system of linear equations by Gaussian Elimination and Gauss-Jordan (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, a path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.

Take the UPSC Test Series here and ace the examination.

UPSC Maths Optional Syllabus PDF Download Here

Candidates can make use of the UPSC Maths Optional Syllabus PDF which includes the Maths topics and sub-topics in detail, study material sources and more. Download the UPSC Maths Syllabus PDF and keep a copy with yourself and refer to it while preparing for this optional subject.

Also, check out the UPSC CSAT Paper details in the linked article.

UPSC Maths Optional Paper Pattern

Candidates who have chosen maths as their optional subject in the mains exam need to give two optional papers of 250 marks each. Have a look at the Maths Optional Exam Pattern mentioned in the below table.

Check the complete UPSC Exam Pattern here.

How To Prepare For UPSC Maths Optional Paper?

For many candidates, preparation for the Maths Optional UPSC Syllabus can be a daunting task. However, with the right strategy, it is possible to cover the IAS Maths syllabus effectively and efficiently. Have a look at the below-mentioned tips to prepare for the Maths Optional UPSC Syllabus for the mains exam.

• Understand the Maths Optional Syllabus and Exam Pattern - The first step towards preparing for the maths optional syllabus is to understand the syllabus and exam pattern. The syllabus includes topics such as Linear Algebra, Calculus, Real Analysis, Ordinary Differential Equations, and Partial Differential Equations, among others. You must make sure to have a clear understanding of the topics and sub-topics included in the syllabus.
• Create a study plan - Create a UPSC study plan that covers all the topics in the maths syllabus by allocating enough time for each topic based on its weightage in the mains exam. Divide your study plan into daily, weekly, and monthly goals, and make sure you stick to them.
• Choose the right study material - You must choose books and study materials that are recommended by experts. Some of the popular Maths Optional Books for UPSC include Linear Algebra by Hoffman and Kunze, Calculus by Spivak, and Real Analysis by Royden.
• Solve previous year question papers - Solving UPSC Maths Optional Question Papers helps you to identify your strengths and weaknesses and work on them accordingly. Make sure you solve at least 6-10 previous year's papers before the exam.

Also, check out more UPSC Previous Year Papers in the linked article.

Conclusion

Candidates from a mathematical or engineering background may find it easy to prepare for the maths optional syllabus. Overall, preparing for the UPSC Mathematics Optional can be challenging, but with the right strategy and approach, it is possible to cover the syllabus effectively. Make sure you go through the complete UPSC Maths Syllabus mentioned above and also follow the tips and stay consistent in your preparation. All the best!

We hope this article on UPSC Maths Optional Syllabus helped you in understanding the important topics needed to prepare for the maths optional subject. To boost your preparations, join the Testbook community by downloading the Testbook App!