# IAS Mathematics Syllabus for UPSC, Civil Service Mains Examinations.

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** IAS Mathematics Syllabus for PAPER – I Section-A**

** ****(1) Linear Algebra|IAS Mathematics Syllabus**

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’s and similarity; Rank of a matrix; Inverse of a matrix; Solution of system of linear equations; Eigen values and eigen vectors, characteristic polynomial, Cayley-Hamilton theorem, Symmetric, skew-symmetric, Hermitian, Skew-Hermitian, orthogonal and unitary matrices and their eigen values.

**(2) Calculus|IAS Mathematics Syllabus**

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 integrals; Double and triple integrals (evaluation techniques only); Areas, surface and volumes.

**(3) Analytic Geometry|IAS Mathematics Syllabus**

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.

** IAS Mathematics Syllabus for ****PAPER – I Section-B**

**(4) Ordinary Differential Equations |IAS Mathematics Syllabus**

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 linear equations with constant coefficients, complementary function, particular integral and general solution. Second order linear equations with variable coefficients, Euler-Cauchy equation (Homogeneous linear equations); 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.

**(5) Dynamics & Statics |IAS Mathematics Syllabus**

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.

**(6) Vector Analysis |IAS Mathematics Syllabus**

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.** **

** IAS Mathematics Syllabus for ****PAPER – II ****Section-A**

** ****(1) Modern Algebra |IAS Mathematics Syllabus**

Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem. Rings, sub rings and ideals, homeomorphisms of rings; Integral domains, principal Ideal domains, Euclidean domains and unique factorization domains; Fields, quotient fields.

**(2) Real Analysis|IAS Mathematics Syllabus**

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.

**(3) Complex Analysis| IAS Mathematics Syllabus**

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|IAS Mathematics Syllabus**

Linear programming problems, basic solution, basic feasible solution and optimal solution; Graphical method and simplex method of solutions; Duality. Transportation and assignment problems.

** IAS Mathematics Syllabus for ****PAPER – II Section-B**

**(5) Partial differential equations|IAS Mathematics Syllabus**

Family of surfaces in three dimensions and formulation of partial differential equations; Solution of quasi linear 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 & Computer Programming|IAS Mathematics Syllabus**

**(i) Numerical Analysis**

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-Jordan (direct), Gauss-Seidel (iterative) methods. Newton’s (forward and backward) interpolation, Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s rules, Gaussian quadrature formula. Numerical solution of ordinary differential equations: Euler and Runga Kutta-methods.

** (ii) ****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|IAS Mathematics Syllabus**

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 in viscid flow; Stream-lines, path of a particle; Potential flow; Two-dimensional and asymmetric motion; Sources and sinks, vortex motion; Navier-Stokes Equation for a viscous fluid.

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