UPSC CSE Mathematics Optional Syllabus 2021
By On June 5th, 2021
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UPSC CSE Mathematics Optional Syllabus for Paper-I, UPSC CSE Mathematics Optional Syllabus for Paper-II
UPSC CSE Mathematics Optional Syllabus for Paper-I
  • 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’s and similarity;
    Rankof 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
    integral; 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 liner 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 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.
  • 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.
  • 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 indentities.
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UPSC CSE Mathematics Optional Syllabus for 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 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.
  • 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-Jorden (direct), Gauss-Seidel (iterative) methods. Newton’s
    (forward and backward) and interpolation, Lagrange’s interpolation.
    Numerical integration: Trapezoidal rule, Simpson’s rule, Gaussian quadrature formula.
    Numerical solution of ordinary differential equations : Eular 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.
  • 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, vortex
    motion; Navier-Stokes equation for a viscous fluid.

Frequently Asked Questions (FAQs) On UPSC CSE Optional Syllabus 

Question- What is the UPSC’s Annual Programme (Calendar) of Examinations/RTs (Recruitment Tests)?
Answer – The UPSC publishes an Annual Programme (Calendar) of all the Structured Examinations/RTs conducted by it at least 6 months in advance (i.e. in June) for the Examinations/RTs to be conducted during the next calendar year. The Programme is uploaded on the UPSC’s website as also published in the leading news papers of the country. The date of issue of Examination Notice for each Examination is also mentioned in this Annual Programme.

Question- What happens if a candidate submits multiple online applications?
Answer – While a candidate should avoid submitting more than one online application, in case of doing so, the data provided in the last application (highest RID Number), that is successfully submitted online, is accepted by the Commission. All previous applications are ignored as these are amalgamated with the last completed & finally submitted application. If an applicant (who has already submitted an application successfully) wants to
make amendments in the application, then he has to submit a fresh application on or before the last date of submission of application of the Examination. Therefore, it must be ensured that fee is submitted against the
last online application only, which should also be complete in all respects including its final submission. Fee paid against one RID shall not be adjusted against any other RID number

Question- What action is taken by the Commission in case of submission of false information by the candidates?
Answer – A candidate found to be furnishing false information to the Commission or suppressing information, adopting various unfair means in the Examination like impersonation, cheating, etc., is liable to be disqualified
and/or debarred from writing UPSC Examinations as decided by the Commission. A detailed stipulation in this regard is incorporated in the Rules of Examination/ Examination Notices.

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