{"id":96872,"date":"2021-03-20T18:06:30","date_gmt":"2021-03-20T12:36:30","guid":{"rendered":"https:\/\/blog.forumias.com\/?page_id=96872"},"modified":"2023-10-26T17:36:10","modified_gmt":"2023-10-26T12:06:10","slug":"mathematics-syllabus","status":"publish","type":"page","link":"https:\/\/forumias.com\/blog\/upsc-syllabus\/mathematics-syllabus\/","title":{"rendered":"Mathematics Optional UPSC Syllabus"},"content":{"rendered":"

Mathematics Optional-Syllabus\u00a0<\/strong><\/p>\n

PAPER- I<\/h5>\n

(1) Linear Algebra :<\/strong>
\nVector spaces over R and C, linear dependence and independence, subspaces, bases,
\ndimensions, Linear transformations, rank and nullity, matrix of a linear transformation.
\nAlgebra of Matrices; Row and column reduction, Echelon form, congruence\u2019s and similarity;
\nRankof a matrix; Inverse of a matrix; Solution of system of linear equations; Eigenvalues and
\neigenvectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew-symmetric,
\nHermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues.<\/p>\n

(2) Calculus :<\/strong>
\nReal numbers, functions of a real variable, limits, continuity, differentiability, mean-value
\ntheorem, Taylor\u2019s theorem with remainders, indeterminate forms, maxima and minima,
\nasymptotes; Curve tracing; Functions of two or three variables; Limits, continuity, partial
\nderivatives, maxima and minima, Lagrange\u2019s method of multipliers, Jacobian.
\nRiemann\u2019s definition of definite integrals; Indefinite integrals; Infinite and improper
\nintegral; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.<\/p>\n

(3) Analytic Geometry :<\/strong>
\nCartesian and polar coordinates in three dimensions, second degree equations in three
\nvariables, reduction to Canonical forms; straight lines, shortest distance between two skew
\nlines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and
\ntheir properties.<\/p>\n

(4) Ordinary Differential Equations :<\/strong>
\nFormulation of differential equations; Equations of first order and first degree, integrating
\nfactor; Orthogonal trajectory; Equations of first order but not of first degree, Clairaut\u2019s equation,
\nsingular solution.
\nSecond and higher order liner equations with constant coefficients, complementary function,
\nparticular integral and general solution.
\nSection order linear equations with variable coefficients, Euler-Cauchy equation;
\nDetermination of complete solution when one solution is known using method of variation of
\nparameters.
\nLaplace and Inverse Laplace transforms and their properties, Laplace transforms of
\nelementary functions. Application to initial value problems for 2nd order linear equations with
\nconstant coefficients.<\/p>\n

(5) Dynamics and Statics :<\/strong>
\nRectilinear motion, simple harmonic motion, motion in a plane, projectiles; Constrained
\nmotion; Work and energy, conservation of energy; Kepler\u2019s laws, orbits under central forces.
\nEquilibrium of a system of particles; Work and potential energy, friction, Common catenary;
\nPrinciple of virtual work; Stability of equilibrium, equilibrium of forces in three dimensions.<\/p>\n

(6) Vector Analysis :<\/strong>
\nScalar and vector fields, differentiation of vector field of a scalar variable; Gradient,
\ndivergence and curl in cartesian and cylindrical coordinates; Higher order derivatives; Vector
\nidentities and vector equation.
\nApplication to geometry : Curves in space, curvature and torsion; Serret-Furenet’s
\nformulae.
\nGauss and Stokes\u2019 theorems, Green’s indentities.<\/p>\n

PAPER-II<\/h5>\n

(1) Algebra :<\/strong>
\nGroups, subgroups, cyclic groups, cosets, Lagrange\u2019s Theorem, normal subgroups, quotient
\ngroups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley\u2019s
\ntheorem.
\nRings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal
\ndomains, Euclidean domains and unique factorization domains; Fields, quotient fields.<\/p>\n

(2) Real Analysis :<\/strong>
\nReal number system as an ordered field with least upper bound property; Sequences, limit of
\na sequence, Cauchy sequence, completeness of real line; Series and its convergence, absolute
\nand conditional convergence of series of real and complex terms, rearrangement of series.
\nContinuity and uniform continuity of functions, properties of continuous functions on compact
\nsets.
\nRiemann integral, improper integrals; Fundamental theorems of integral calculus.
\nUniform convergence, continuity, differentiability and integrability for sequences and series of
\nfunctions; Partial derivatives of functions of several (two or three) variables, maxima and minima.<\/p>\n

(3) Complex Analysis :<\/strong>
\nAnalytic function, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula,
\npower series, representation of an analytic function, Taylor\u2019s series; Singularities; Laurent\u2019s series;
\nCauchy\u2019s residue theorem; Contour integration.<\/p>\n

(4) Linear Programming :<\/strong>
\nLinear programming problems, basic solution, basic feasible solution and optimal solution;
\nGraphical method and simplex method of solutions; Duality.
\nTransportation and assignment problems.<\/p>\n

(5) Partial Differential Equations :<\/strong>
\nFamily of surfaces in three dimensions and formulation of partial differential equations;
\nSolution of quasilinear partial differential equations of the first order, Cauchy\u2019s method of
\ncharacteristics; Linear partial differential equations of the second order with constant coefficients,
\ncanonical form; Equation of a vibrating string, heat equation, Laplace
\nequation and their solutions.<\/p>\n

(6) Numerical Analysis and Computer Programming :<\/strong>
\nNumerical methods: Solution of algebraic and transcendental equations of one variable by
\nbisection, Regula-Falsi and Newton-Raphson methods, solution of system of linear equations by
\nGaussian Elimination and Gauss-Jorden (direct), Gauss-Seidel (iterative) methods. Newton\u2019s
\n(forward and backward) and interpolation, Lagrange\u2019s interpolation.
\nNumerical integration: Trapezoidal rule, Simpson\u2019s rule, Gaussian quadrature formula.
\nNumerical solution of ordinary differential equations : Eular and Runga Kutta methods.
\nComputer Programming : Binary system; Arithmetic and logical operations on numbers; Octal
\nand Hexadecimal Systems; Conversion to and from decimal Systems; Algebra of binary numbers.
\nElements of computer systems and concept of memory; Basic logic gates and truth tables,
\nBoolean algebra, normal forms.
\nRepresentation of unsigned integers, signed integers and reals, double precision reals and
\nlong integers.
\nAlgorithms and flow charts for solving numerical analysis problems.<\/p>\n

(7) Mechanics and Fluid Dynamics :<\/strong>
\nGeneralised coordinates; D\u2019Alembert\u2019s principle and Lagrange\u2019s equations; Hamilton
\nequations; Moment of inertia; Motion of rigid bodies in two dimensions.
\nEquation of continuity; Euler\u2019s equation of motion for inviscid flow; Stream-lines, path of a
\nparticle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex
\nmotion; Navier-Stokes equation for a viscous fluid.<\/p>\n","protected":false},"excerpt":{"rendered":"

Mathematics Optional-Syllabus\u00a0 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\u2019s and similarity; Rankof a matrix; Inverse of a matrix; Solution of system of linear… Continue reading Mathematics Optional UPSC Syllabus<\/span><\/a><\/p>\n","protected":false},"author":61,"featured_media":0,"parent":50153,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false,"footnotes":""},"class_list":["post-96872","page","type-page","status-publish","hentry","entry"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/pages\/96872","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/users\/61"}],"replies":[{"embeddable":true,"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/comments?post=96872"}],"version-history":[{"count":0,"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/pages\/96872\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/pages\/50153"}],"wp:attachment":[{"href":"https:\/\/forumias.com\/blog\/wp-json\/wp\/v2\/media?parent=96872"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}