{"id":2347,"date":"2024-01-31T11:25:42","date_gmt":"2024-01-31T11:25:42","guid":{"rendered":"https:\/\/www.khanglobalstudies.com\/blog\/?p=2347"},"modified":"2025-01-25T12:39:07","modified_gmt":"2025-01-25T07:09:07","slug":"upsc-mathematics-optional-syllabus","status":"publish","type":"post","link":"https:\/\/www.khanglobalstudies.com\/blog\/upsc-mathematics-optional-syllabus\/","title":{"rendered":"UPSC Mathematics Optional Syllabus for Paper I and Paper II"},"content":{"rendered":"\n

The UPSC optional syllabus of Mathematics consists of two papers, Paper I and Paper II, each carrying a weightage of 250 marks. Both the papers are divided into different sections, and candidates have to choose one subject from each section. Let us look into the details of each paper.<\/p>\n\n\n\n

UPSC Mathematics Optional Syllabus<\/strong><\/h2>\n\n\n\n

Mathematics as an optional subject has the highest scoring potential in UPSC CSE, provided you have done your preparation adequately and can write your answers accurately. This subject is less unpredictable than humanities subjects because of its objective and almost consistent method of research and problem-solving. This makes it a good choice for engineers, candidates with a science background or mathematics graduates who already have a technical mindset.<\/p>\n\n\n\n

UPSC Maths Syllabus<\/strong><\/td>Topics<\/strong><\/td><\/tr>
Paper I<\/strong><\/td>Linear Algebra
Calculus
Analytic Geometry
Ordinary Differential Equations
Dynamics and Statics
Vector Analysis<\/td><\/tr>
Paper II<\/strong><\/td>Algebra
Real Analysis
Complex Analysis
Linear Programming
Partial Differential Equations
Numerical Analysis and Computer Programming
Mechanics and Fluid Dynamics<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
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UPSC Mathematics Optional Syllabus for Paper I<\/strong><\/h3>\n\n\n\n
Linear Algebra<\/strong><\/h5>\n\n\n\n

Vector spaces on R and C, linear dependence and independence, subspaces, basis, dimension, linear transformation, rank and nullity, matrix of a linear transformation. Algebra of matrices; Row and column reduction, echelon form, unity and equality; Rank of matrix; matrix inverse; Solution of systems of linear equations; Eigenvalues and eigenvectors, characteristic polynomials, Cayley-Hamilton theorem, symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues.<\/p>\n\n\n\n

Calculus<\/strong><\/h5>\n\n\n\n

Real numbers, functions of real variables, limits, continuity, variation, mean-value theorem, Taylor’s theorem with remainder, indefinite forms, maximum and minimum, asymptotes; curve tracing; Functions of two or three variables; Limits, continuity, partial derivatives, maxima and minima, Lagrange’s multiplier method, Jacobian. Riemann’s definition of definite integrals; indefinite integrals; infinite and unreasonable integrals; Double and Triple Integrals (Evaluation Techniques Only); Area, surface and volume.<\/p>\n\n\n\n

Analytic Geometry<\/strong><\/h5>\n\n\n\n

Cartesian and polar coordinates in three dimensions, equations of the second degree in three variables, reduction to canonical forms; Straight lines, minimum distance between two oblique lines, plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.<\/p>\n\n\n\n

Ordinary Differential Equations<\/strong><\/h5>\n\n\n\n

Formulation of differential equations; Equations of first order and first degree, integrating factors; orthogonal trajectory; Equations of first order but not first degree, Clairaut’s equation, singular solution. Second and higher-order linear equations with constant coefficients, complementary functions, special integrals and general solutions. Section order linear equations with variable coefficients, Euler-Cauchy equation; Determination of the complete solution when one solution is known using the method of change of parameters. Laplace and inverse Laplace transforms and their properties, Laplace transform of elementary functions. Application of initial value problems to second-order linear equations with constant coefficients.<\/p>\n\n\n\n

Dynamics and Statics<\/strong><\/h5>\n\n\n\n

Rectilinear motion, simple harmonic motion, motion in a plane, projectile; hindered movement; Work and energy, conservation of energy; Kepler’s laws, orbits are subject to central forces. Equilibrium of a system of particles; Work and potential energy, friction, general catenary; Theory of Virtual Work; Stability of equilibrium, balance of forces in three dimensions.<\/p>\n\n\n\n

Vector Analysis<\/strong><\/h5>\n\n\n\n

Scalar and vector fields, differentiation of vector fields of scalar variables; Slope, divergence and curl in Cartesian and cylindrical coordinates; Higher order derivatives; Vector identities and vector equations. Applications in Geometry: Curves, curvature and torsion in space; Serret-Fernet’s formula. Gauss’s and Stokes’ theorems, Green’s identity.<\/p>\n\n\n\n

UPSC Mathematics Optional Syllabus for Paper II<\/strong><\/h3>\n\n\n\n
Algebra<\/strong><\/h5>\n\n\n\n

Groups, subgroups, cyclic groups, cosets, Lagrange’s theorem, normal subgroups, quotient groups, isomorphism of groups, basic isomorphism theorem, permutation groups, Cayley’s theorem. Rings, subrings and ideals, isomorphism of rings; Integral domain, principal ideal domain, Euclidean domain and unique factorization domain; Fields, quotient fields.<\/p>\n\n\n\n

Real-Analysis<\/strong><\/h5>\n\n\n\n

The real number system is an ordered field with the minimum upper bound property; Sequences, limits of sequences, Cauchy sequences, completeness of the real line; Series and its convergence, complete and conditional convergence of series of real and complex words, rearrangement of series. Continuity of functions and uniform continuity, Properties of continuous functions on compact sets. Riemann integral, improper integral; Fundamental theorem of integral calculus. Uniform convergence, continuity, divergence and completeness for sequences and series of functions; Partial derivatives of functions of many (two or three) variables, maxima and minima.<\/p>\n\n\n\n

Complex Analysis<\/strong><\/h5>\n\n\n\n

Analytical functions, Cauchy-Riemann equation, Cauchy’s theorem, Cauchy’s integral formula, power series, representation of an analytical function, Taylor’s series; Singularities; Laurent’s series; Cauchy’s residue theorem; contour integration.<\/p>\n\n\n\n

Linear Programming<\/strong><\/h5>\n\n\n\n

Linear programming problems, basic solutions, basic feasible solutions and optimal solutions; Graphical method and simplex method of solution; Duality. Transportation and assignment problems.<\/p>\n\n\n\n

Partial Differential Equations<\/strong><\/h5>\n\n\n\n

Representation of families of surfaces and partial differential equations in three dimensions; Solution of first-order quasilinear partial differential equations, Cauchy’s method of characteristics; Linear partial differential equations of the second order with constant coefficients, canonical form; Equation of vibrating string, heat equation, Laplace equation and their solutions.<\/p>\n\n\n\n

Numerical Analysis and Computer Programming<\/strong><\/h5>\n\n\n\n

Numerical methods: Solution of algebraic and transcendental equations of one variable by bifurcation, Regula-Falcy and Newton-Raphson methods, solution of systems of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel (iterative) methods. Newton’s (forward and reverse) and interpolation, Lagrange’s interpolation. Numerical Integration: Trapezoidal rule, Simpson’s law, 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; Conversions to and from the decimal system; Algebra of binary numbers. Elements of computer systems and the concept of memory; Basic logic gates and truth tables, Boolean algebra, general forms. Representations of unsigned integers signed integers, and real, double-precision real and long integers. Algorithms and flow charts to solve numerical analysis problems.<\/p>\n\n\n\n

Mechanics and Fluid Dynamics<\/strong><\/h5>\n\n\n\n

Generalized coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton equation; moment of inertia; Motion of rigid bodies in two dimensions. continuity equation; Euler’s equation of motion for inviscid flow; Streamline, the path of a particle; potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier\u2013Stokes equations for viscous fluids.<\/p>\n\n\n\n

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Tips to Prepare UPSC Maths Syllabus<\/strong><\/h4>\n\n\n\n