# UPSC Civil Services (Mains) Examination Mathematics Syllabus

**Paper ï¿½ I:**

**1. Linear Algebra:ï¿½**

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, congruenceï¿½s and similarity; Rank of 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.

**2. Calculus**:

Real numbers, functions of a real variable, limits, continuity, differentiability, meanvalue theorem, Taylorï¿½s theorem with remainders, indeterminate forms, maxima and minima, asymptotes; Curve tracing; Functions of two or three variables: limits, continuity, partial derivatives, maxima andï¿½ minima, Lagrangeï¿½s method of multipliers, Jacobian. Riemannï¿½s definition of definite integrals; Indefinite integrals; Infinite and improperï¿½ integrals; 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, shortest distance between two skew lines; Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

**4. 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, Clairautï¿½s 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; Determination of complete solution when one solution is known using method ofï¿½ 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**:

Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; constrained motion; Work and energy, conservation of energy; Keplerï¿½s laws, orbits underï¿½ central forces. Equilibrium of a system of particles; Work and potential energy, friction; common catenary; Principle of virtual work; Stability ofï¿½ equilibrium, 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 equations. Application to geometry: Curves in space, Curvature and torsion; Serret-Frenetï¿½s formulae.ï¿½ Gauss and Stokesï¿½ theorems, Greenï¿½s identities.

**Paper ï¿½ II:**

**1. Algebra: Groups**

subgroups, cyclic groups, cosets, Lagrangeï¿½s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayleyï¿½s theorem.ï¿½ Rings, subrings and ideals, homomorphisms of rings; Integral domains, principal ideal domains, Euclidean domains and unique factorization domains; Fields, quotientï¿½ fields.

**2. 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; Seriesï¿½ and its convergence, absolute and conditionalï¿½ 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,ï¿½ improper integrals; Fundamental theorems of integral calculus. Uniform convergence, continuity, differentiability and integrability for sequences and series of functions; Partial derivatives of functionsï¿½ of several (two or three) variables, maxima and minima.

**3. Complex Analysis**:

Analytic functions, Cauchy-Riemann equations, Cauchyï¿½s theorem, Cauchyï¿½s integral formula, power series representation of an analytic function, Taylorï¿½s series; Singularities;ï¿½ Laurentï¿½s series; Cauchyï¿½s residue theorem; Contour integration.

**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:ï¿½**

Family of surfaces in three dimensions and formulation of partial differential equations; Solution of quasilinear partial differential equations of the first order, Cauchyï¿½s method of characteristics; Linear partial differential equations of the second orderï¿½ 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-ï¿½ Raphson methods; solution of system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss- Seidel(iterative) methods. Newtonï¿½s (forwardï¿½ and backward) interpolation, Lagrangeï¿½s interpolation.ï¿½ Numerical integration: Trapezoidal rule, Simpsonï¿½s rules, Gaussian quadrature formula. Numerical solution of ordinary differentialï¿½ equations: Euler and Runga Kutta-methods. Computer Programming: Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal systems;ï¿½ Conversion to and from decimal systems; Algebra of binary numbers.ï¿½ 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.ï¿½ Algorithms and flow charts for solving numerical analysis problems.

**7. Mechanics and Fluid Dynamics**:

Generalized coordinates; Dï¿½ Alembertï¿½s principle and Lagrangeï¿½s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions. Equation of continuity; Eulerï¿½s equation ofï¿½ 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.

Written by princy

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