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UPSC Mathematics Syllabus 2024

The UPSC Mathematics syllabus 2024 covers a wide range of topics including abstract algebra, real analysis, complex analysis, linear algebra, analytical geometry, differential equations, probability theory, numerical analysis, mechanics, and statistics. Candidates will need to have a strong grasp of mathematical concepts and their applications to real-world problems.

UPSC Mathematics Syllabus 2024 Overview

There is a list of optional subjects for mains in which there are 48 subjects, from which candidates can choose according to their choice. Mathematics is one of the optional subjects for the Civil Services Exam conducted by UPSC. The optional paper is conducted for 250 marks. The UPSC CSE optional subject Mathematics Syllabus 2024 has two papers, Paper 1 and Paper 2.

Optional Mathematics Syllabus for UPSC CSE

The UPSC Mathematics optional tests analytical thinking and problem-solving aptitude. The syllabus covers topics like algebra, analysis, geometry, statistics. Thorough preparation on the entire syllabus is essential to maximize scores in this highly competitive exam that demands strong mathematical reasoning abilities.

If you opt for Mathematics optional in UPSC mains, prepare it thoroughly. Go through the UPSC CSE Syllabus and previous years’ Mathematics papers. Also study relevant Mathematics books for UPSC preparation.

UPSC CSE Mathematics Syllabus 2024 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
• Partial derivatives
• 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, 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
• Clairaut’s equation
• Singular solution
• Second and higher-order linear equations with constant coefficients
• Second-order linear equations with variable coefficients
• Euler-Cauchy equation
• Laplace and inverse Laplace transforms and their properties
• Application to initial value problems for 2nd order linear equations with constant coefficients.

5. 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
• 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 equations
• Application to geometry: curves in space, curvature, and torsion
• Serret-Frenet’s formulae
• Gauss and Stokes’ theorems
• Green’s identities.

UPSC CSE Mathematics Syllabus 2024 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, 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
• 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 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:

• Solution of algebraic and transcendental equations of one variable
• Solution of system of linear equations
• Interpolation methods
• Numerical integration methods
• Numerical solution of ordinary differential equations
• Computer Programming: Binary system, Arithmetic and logical operations, Conversion systems, Algorithms, and flow charts

7. 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
• Potential flow
• Navier-Stokes equation for a viscous fluid

The UPSC Mathematics optional paper tests mathematical reasoning ability, comprehension skills, and the capacity to articulate mathematical ideas clearly and concisely. Thorough preparation covering the entire syllabus is essential for aspirants to maximize scores in this optional paper known for its high difficulty level. Focus, practice and conceptual clarity will help aspirants successfully tackle the challenges of this paper.

UPSC CSE Mathematics Syllabus 2024 FAQs

1. What is the syllabus of Mathematics for UPSC CSE?

The UPSC Mathematics syllabus 2024 covers a wide range of topics including abstract algebra, real analysis, complex analysis, linear algebra, analytical geometry, differential equations, probability theory, numerical analysis, mechanics, and statistics. Candidates will need to have a strong grasp of mathematical concepts and their applications to real-world problems.
Get Syllabus PDF here: UPSC CSE Mathematics Syllabus

2. How much marks is allotted for UPSC Mains Optional subject Mathematics?

There is 250 marks is for one optional paper.