UPSC Civil Services Examination
Mathematics Syllabus
Complete UPSC CSE Optional Subject syllabus for Mathematics, covering Paper I and Paper II.
Optional Subject
Mathematics
The UPSC Civil Services Examination optional subject consists of two papers. Each paper covers specific topics prescribed under the Mathematics syllabus.
Optional Subject
Paper I
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.
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.
- 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 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 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 method of variation of parameters.
- Laplace and inverse Laplace transforms and their properties.
- Laplace transforms of elementary functions.
- Application to initial value problems for second order linear equations with constant coefficients.
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.
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.
Optional Subject
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 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.
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 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.
- 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
- Generalized 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.