Pros of Taking Maths as Optional

• High scoring potential
• High chance of getting direct questions
• No theory to study
• Increases critical thinking and problem-solving skills
• It can help to develop logical reasoning

UPSC Maths Optional Syllabus – Paper 1

1. Linear Algebra

• Vector spaces over R and C
• Linear dependence and independence, subspaces, bases, dimension
• 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 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.

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, shortest distance between two skew lines, plane, sphere, cone, cylinder, paraboloid, ellipsoid
• Hyperboloids 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 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 solution.
• Second-order linear equations with variable coefficients,
• Euler-Cauchy equation, the determination of a 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.

5. 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, the 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.

UPSC Maths Optional Syllabus – Paper 2

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, and quotient fields.

2. 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.

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

• 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 Runge 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 the 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

• 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, path of a particle, Potential flow
• Two-dimensional and axisymmetric motion, Sources and sinks, vortex motion;
• Navier-Stokes equation for a viscous fluid

Books to Refer for Maths Optional

Recommended books for Paper 1

• 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
• Ordinary and partial differential equations – M D Raisinghania
• Dynamics and Statics
• Krishna Series
• Vector Analysis
• Schaum Series – Murray R. Spiegel

Recommended books for Paper 2

• Algebra – Contemporary Abstract algebra – Joseph Gallian
• Real analysis
• Complex analysis
• Schaum series – Spiegel, Lipschitz, Schiller, Spellman
• Linear programming
• Linear programming and game theory – Lakshmi Shree Bandopadhyay
• Partial differential equations
• Advanced differential equations – MD Raisinghania
• Numerical analysis and computer programming
• Computer-based numerical and statistical techniques – M Goyal
• Numerical Methods – Jain, Iyengar and Jain
• Digital logic and Computer design – M Morris Mano
• Mechanics and Fluid Dynamics

The syllabus of maths might look vast, but with the right strategy and preparation, anyone can ace the maths optional exam.