UPSC Maths Optional Syllabus: Choosing the right optional subject for the UPSC exams is one of the major decisions that a candidate has to make in his UPSC journey. Choosing the right subject can make your UPSC journey a smooth ride. With a lot of optional subjects to take from, sometimes it can be quite difficult to choose the right one.
Maths is one of the favorite optional subjects of many UPSC candidates. It is usually preferred by people with a good background in the subject.
Why should you choose maths as optional?
One of the main reasons why a large number of people opt for maths is the practical nature of the subject. Unlike subjects like geography, history, anthropology, and philosophy, maths doesn’t have any theory. Most of the questions are practical in nature and are easy for anyone with a good background to solve and ace the exams.
Maths optional consists of two papers. Paper 1 and paper 2 with 250 marks for each.
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
- 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.
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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.
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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
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
