Mathematics is a common choice of optional subject for many. Against the common thinking that it is a generalised subject, aspirants with a specialisation stand a chance to score better than those without it. The subject has two papers, paper 1 and paper 2. Each paper carries 250 marks and is conducted for three hours.

## UPSC Maths Optional Syllabus For Paper I

**(1) Linear Algebra:**

Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; Linear transformations, rank and nullity, a 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 a system of linear equations; Eigen values and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues.

**(2) 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.

**(3) 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.

**(4) Ordinary Differential Equations:**

Formulation of differential equations; Equations of first order and first degree, integrating factor; Orthogonal trajectory; Equations of the first order but not of first degree, Clairaut’s equation, singular solution. Second and higher order linear equations with constant coefficients, complementary function, particular integral and general solution. Second 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 transform and their properties; Laplace transforms of elementary functions. Application to initial value problems for 2nd order linear equations with constant coefficients.

**(5) Dynamics & 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.

**(6) 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-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.

*Aspirants planning to take UPSC Exam 2020 may refer to the linked article.*

## UPSC Maths Optional Syllabus For Paper II

**(1) 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.

**(2) Real Analysis:**

Real number system as an ordered field with the least upper bound property; Sequences, limit of a sequence, Cauchy sequence, completeness of real line; Series and its convergence, the 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.

**(3) Complex Analysis:**

Analytic functions, 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.

**(4) 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.

**(5) Partial differential equations:**

The 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.

**(6) 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-Jordan (direct), Gauss-Seidel(iterative) methods. Newton’s (forward and backward) interpolation, Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s rules, 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.

**(7) Mechanics and Fluid Dynamics:**

Generalized coordinates; D’ Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions. The 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.

### Preparation tips for mathematics optional in IAS mains examination

- Mathematics is a subject full of quantities, number analysis, structures, etc. Formulate your way to learn through the uniqueness and difficulty of the subject.
- Try to get to the core of the subject by reasoning out every answer or solution that you arrive at while solving problems.
- The subject can either fetch you great marks, or you can score very less marks in it.
- Overall, with good preparation and accuracy in problem-solving methods, it’s a high scoring subject.
- Mathematics is one of the elementary subjects that candidates must have studied since school days, so the foundation of the subject should seem strong.
- Develop more proficiency and pay attention to little details in the problems provided in the exam.
- Minimum of 9 hours per day, at least for three-four months, is essential to crack the paper with high scores.
- Paper 1 is easier compared to paper 2. Focus more on paper 2.
- Analytical geometry needs through practice from scratch, and you need to refer to a lot of solved examples from reference books.
- Just by looking at the problem, candidates have to identify the approach to solve the problem. Once the approach is identified, the steps to solve any problem become easier.
- In the area of statistics and dynamics, solve problems that are complex at a medium level. A very high level of complex problems is hardly found ever.
- Prepare chapter wise formulae and keep them at reach for quick glances.
- Candidates are advised to finish the second part of the paper and cover the first part later on.
- Under abstract algebra, it is mandatory to memorise all important theorems. The proofs of the theorems, even if skipped, are quite alright for this exam.
- Analyse sequence of functions.
- Understand the question and data well and complete the question fully before beginning to work out the problem.
- Solve previous year papers to familiarise with patterns and trends.

#### Recommended list of reference books for study

- Book on Linear Algebra by A.R.Vasista ( Schaum Series)
- Book on Calculus and Real Analysis by authors S.C Malik, Savita Arora and Shanti Narayana
- Reference series on 3-D Geometry by author P.N. Chatterjee
- Book on Ordinary Differential Equations by authors M.D. Raisinghania and Ian Sneddon
- Book on Vector Analysis written by author A.R.Vasista (Schaum Series)
- Book on Algebra by writers Joseph A. Gallian and Shramik Sen Upadhaya
- Reference series on Complex Analysis from Schaum Series. (Book authors -J.N. Sharma, Ponnu Swami, G K Ranganath)
- Reference series on Linear Programming by authors Shanti Swarup, Kanti Swarup and S D Sharma
- Book on Numerical Analysis by writers – Jain and Iyengar, K. Shankar Rao, S. S. Sastry.
- Reference textbook on Computer Programming by Raja Raman
- Book on Dynamics & Statics by A.R.Vasista and M. Ray
- Reference series on Mechanics and Fluid Dynamics by multiple authors. ( Authors list – M.D. Raisinghania, R.K. Gupta, J.K. Goyal, and K.P. Gupta, Azaroff Leonid)

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