# OPTIONAL : MATHEMATICS ### Syllabus

Mathematics Syllabus Paper – I

(1)Linear Algebra:

Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; Linear transformations, rank and nullity, a 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 a 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, 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:

Cartesian and polar coordinates in three dimensions, second-degree equations in three variables, reduction to canonical forms, straight lines, the 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 the 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; Determination of complete solution when one solution is known using the method of variation of parameters. Laplace and Inverse Laplace transform 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; 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:

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.

### Reference Books:

• Analytic Geometry – Shantinarayan, HC Sinha, DK Jha and Sharma
• Calculus – Santhi Narayan
• Dynamics, Statics and Hydrostatics – M.Ray
• Linear Algebra – K C Prasad, K B Datta
• Ordinary Differential eqs: MD Raising Lumina, Golden seris-NP Bali
• Vector analysis – Shantinarayan

Mathematics Syllabus Paper – II

(1) 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.

(2) Real Analysis:

Real number system as an ordered field with the least upper bound property; Sequences, limit of a sequence, Cauchy sequence, completeness of real line; Series and its convergence, the 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:

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:

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:

The 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.

(6) 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-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. 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:

Generalized coordinates; D’ Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions. The equation of continuity; Euler’s equation of motion for inviscid flow; Stream-lines, the path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.

### Reference Books:

• Algebra – K C Prasad, KB Datta
• Complex Analysis – GK Ranganath
• Linear Programming – SD sharma
• Mechanics & Fluid dynamics – AP Mathur, Azaroff leonid
• Numerical analysis and Computer Progg. – V. Rajaraman, SS Shasri
• Partial Diff.eqs. – Singhania
• Real Analysis – Shantinarayan,Royden

### Strategy:

Merits:

• Mathematics is the most advantageous and the highest scoring optional.
• Students with very strong mathematics basics and interest in maths can opt for this paper.
• Since its a aptitude paper scoring becomes easy.

Demerits:

• Very strong basics are needed in mathematics.

How to prepare

• Memorise the Syllabus.
• Stick to limited reference books
• Before starting preparation, go through previous year qps

Paper 1:

• For Analytical Geometry, read all the solved examples given in above mentioned books.
• Regularly revise particularly skew lines, sphere, cone and conicoids. In many problems you would have to remember how to start the problem i.e. you would have to mug the approach to solve specific problems.
1. For Calculus, focus more on Calculus of many variables. This is well covered in Malik and Arora. Also many topics of Paper I and Paper II overlap, which can be prepared simultaneously from the above mentioned book.
2. In Statics & Dynamics, try to solve all the problems. You can leave very complex problems which are usually given at the end of every chapter.
• Make formula sheet for every chapter and revise it regularly.
• Usually Paper II is tough for many. Hence if you are able to master it, then you will able to score very high compared to others

Paper 2:

• Abstract Algebra is a unique topic.do it from 10 markers point of view.
• Memorize all the theorems. Skip proofs of theorems which are big, particularly in Permutation groups, Cayley’s theorem, PID, Euclidean Domain and UFDs.
• For Real Analysis, Malik and Arora is the best. You can supplement it by MD Raisinghania.
• Focus more on Riemann Integral, Improper Integrals and Series and Sequences of functions.
• Linear Programming: refere books for MBA like JK Sharma
• PDE: Even though not mentioned in syllabus, Charpit’s method has to be covered as questions are regularly asked. For Boundary Value problems (heat equation etc.) first read from Grewal.
• Mechanics and Fluid Dynamics: From last year UPSC has started mixing questions from PDE, Numerical Analysis and Fluid & Rigid Dynamics.
• Therefore to score high it has become imperative to cover this topic.
• In Fluid dynamics cover Kinematics of Fluids in Motion, Equations of Motions of Inviscid Fluids, Sources and Sinks, Vortex Motion. No need to see proof of any theorems. From Navier Stokes equations, try to see only solved examples.

### Previous Year Question Papers:

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