UPSC Mathematics Syllabus Math Optional Subject (IAS Mains)

UPSC Maths Syllabus

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’s 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.

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 integral
Infinite and improper integrals
Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

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.

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 2nd order linear equations with constant coefficients.

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.

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 Syllabus for

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.

UPSC Math Syllabus of 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.

UPSC Mathematics Syllabus of 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.

UPSC Mathematics Syllabus of 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.

IAS Mains Mathematics Syllabus of 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.

UPSC Math Syllabus for 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

UPSC Mathematics Syllabus of 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.