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Mathematics Optional Syllabus for UPSC 2025
Complete syllabus guide for Mathematics Optional in UPSC Civil Services Examination. Master both papers with our comprehensive study material.
PAPER—I
1. Linear Algebra
- Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimensions
- 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 integral; Double and triple integrals
- 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, 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 first order but not of first degree
- Clairaut's equation, singular solution
- Second and higher order liner equations with constant coefficients
- Second order linear equations with variable coefficients, Euler-Cauchy equation
- Method of variation of parameters
- Laplace and Inverse Laplace transforms and their properties
- Application to initial value problems for 2nd order linear equations
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; 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 equation
- Application to geometry: Curves in space, curvature and torsion
- Serret-Furenet's formulae
- Gauss and Stokes' theorems, Green's identities
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
- Unique factorization domains; Fields, 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
- Rearrangement of series of real and complex terms
- 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
- Partial derivatives of functions of several variables, maxima and minima
3. Complex Analysis
- Analytic function, 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
- 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
- Solution of algebraic and transcendental equations by various methods
- Solution of system of linear equations by direct and iterative methods
- Newton's interpolation methods
- Numerical integration: Trapezoidal rule, Simpson's rule
- Numerical solution of ordinary differential equations
- Binary system and arithmetic operations
- Octal and Hexadecimal Systems; Conversion methods
- Boolean algebra and logic gates
- Algorithms and flow charts for solving numerical 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
- Navier-Stokes equation for a viscous fluid
Recommended Study Resources
Essential Books
- • Linear Algebra by Hoffman and Kunze
- • Calculus by Thomas and Finney
- • Advanced Engineering Mathematics by Erwin Kreyszig
- • Complex Analysis by J.B. Conway
- • Real Analysis by H.L. Royden
Reference Materials
- • Abstract Algebra by Dummit and Foote
- • Numerical Analysis by Richard L. Burden
- • Linear Programming by G. Hadley
- • Mathematical Analysis by Tom M. Apostol
- • Vector Analysis by Murray R. Spiegel
Practice Materials
- • Previous Years' UPSC Mathematics Papers
- • NCERT Mathematics Books (Class 11th and 12th)
- • Mathematics Today Magazine
- • Online Practice Tests
- • Solved Problems in Higher Mathematics
Online Resources
- • NPTEL Mathematics Courses
- • MIT OpenCourseWare
- • Khan Academy Advanced Mathematics
- • Mathematical Programming Tutorials
- • Numerical Methods Online Resources