UPSC Maths Syllabus 2020 - Check IAS Mathematics Optional Syllabus & Preparation Tips

UPSC Maths Syllabus 2020 - UPSC releases the syllabus of UPSC History 2020 along with its notification @upsc.gov.in. Candidates who are planning to appear for the UPSC IAS exam must know about the complete details of UPSC Maths 2020 syllabus. Maths is one of the optional subjects with 2 papers (Paper 1 and Paper 2). UPSC Maths syllabus 2020 is of graduate level. UPSC syllabus for mathematics comprises Algebra, Real Analysis, Complex Analysis, Mechanics of Rigid Bodies, Mechanics of Deformable Bodies and much more. To help IAS aspirants, we have shared the detailed UPSC IAS Maths syllabus 2020. Candidates must go through the details syllabus if they are eyeing qualification for the UPSC IAS 2020 exam. UPSC CSE preliminary exam will be held on October 04, 2020, while the mains will be held on January 08, 2021. Along with the syllabus, we have also shared the exam pattern of CSE along with the best books of IAS. The detailed UPSC Maths 2020 Syllabus is mentioned below.
Latest: Check the expected UPSC CSE 2020 cutoff and Unofficial UPSC IAS 2020 answer key here.

UPSC Maths Syllabus 2020 - Paper 1

(1) Linear Algebra :

Vector spaces over R and C, independence and linear dependence, subspaces, bases,

dimensions, Linear transformations, rank and nullity, a matrix of a linear transformation.

Algebra of Matrices; Echelon form, Row and column reduction, congruence and similarity;

The rank of a matrix; Inverse of a matrix; Solution of a 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, continuity, differentiability, limits, mean-value

theorem, Taylor’s theorem with remainders, indeterminate forms, maxima and minima,

asymptotes; Curve tracing; Functions of two or three variables; continuity, Limits, 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 (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, the shortest distance between two skew

lines, cone, cylinder, paraboloid, ellipsoid, Plane, sphere, hyperboloid of one and two sheets and their properties.

(4) Ordinary Differential Equations :

Formulation of differential equations; Equations of the first order and first degree, integrating

factor; Orthogonal trajectory; Equations of first order but not of the first degree, Clairaut’s equation, singular solution. Second and higher-order linear equations with constant coefficients, complementary function, particular integral and general solution. Section order linear equations with variable coefficients, Euler-Cauchy equation; Determination of 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 second order linear equations with constant coefficients.

(5) Dynamics and Statics :

Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; Constrained

motion; Work and energy, conservation of energy; orbits under central forces, Kepler’s laws.

Equilibrium of a system of particles; Work and potential energy, Common catenary, friction;

Principle of virtual work; Stability of equilibrium, the 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.

UPSC Maths Syllabus 2020 - 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 the least upper bound property; Sequences, the 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.

(3) Complex Analysis :

Analytic function, Cauchy-Riemann equations, Cauchy's theorem, Cauchy's integral formula,

power series, representation of an analytic function, Laurent’s series; Cauchy’s residue theorem; Contour integration, Taylor’s series; Singularities.

(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;

The 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 :

Numerical methods: Solution of algebraic and transcendental equations of one variable by

bisection, Regula-Falsi and Newton-Raphson methods, solution of a system of linear equations by Gaussian Elimination and Gauss-Jordan (direct), Gauss-Seidel (iterative) methods. Newton’s

(forward and backwards) and interpolation, Lagrange’s interpolation.

Numerical integration: Trapezoidal rule, Simpson’s rule, 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 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.

(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, the path of a

particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex

motion; Navier-Stokes equation for a viscous fluid.

UPSC other Optional Subjects Syllabus

Apart from checking the UPSC Maths syllabus 2020, candidates must also check the syllabus for other optional subjects. Go through the links below:

Optional subject syllabus of IAS Mathematics 2020

UPSC Maths Syllabus 2020 - Preparation Tips

UPSC IAS is one of India's biggest and most competitive exams, to ace the exam candidates must have the best UPSC IAS preparation strategy. A better strategy covering all the important sections and topics of the UPSC Maths Syllabus 2020 will help candidates to score well in the exam. Here are some quick of the best preparation tips of UPSC IAS for candidates to help them score better in the exam.

• Always refer to the NCERT Maths book.

• To score more marks it is always for aspirants to study in a planned manner and prepare a perfect IAS study plan for the exam.

• UPSC CSE Maths requires through the practising of the maths problems, deep understanding of the formulas and the concepts.

• The best way is to solve previous year questions papers of UPSC IAS for better preparations and give a mock test if available online.

• Candidates must do self-evaluation of their performance. It is better to do UPSC IAS exam analysis, to know about the difficulty level of the exam as well as the question paper trends.

UPSC Mathematics Books 2020

It is always advisable to refer to only the selected and reliable study materials for UPSC IAS. In the following table, candidates can find the best books for UPSC IAS for Maths.

UPSC Maths Syllabus 2020 for Paper 1

UPSC Maths Syllabus 2020 for Paper 2

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