STRATEGY
The beauty of mathematics as a subject in the main examination is that you can be very selective, yet completely safe. Your efforts should be aimed at developing quality of approach rather than a broad coverage of the course. The following sections are especially important for the aspirants taking IAS Main 2005 with mathematics as an optional subject. The candidates must practise a lot on the indicated sections and they should take care to give derivation in all the cases if the result is a subsidiary one. In case of standard results, there is no need to give derivation of an equation, until specifically asked to.
Paper I
Section A
Linear Algebra: Vector, space, linear dependance and independance, subspaces, bases, dimensions. Finite dimensional vector spaces. Eigenvalues and eigenvectors, eqivalence, congruences and similarity, reduction to canonical form, rank, orthogonal, symmetrical, skew symmetrical, unitary, hermitian, skew-hermitian formstheir eigenvalues.
Calculus : Lagrange's method of multipliers, Jacobian. Riemann's definition of definite integrals, indefinite integrals, infinite and improper integrals, beta and gamma functions. Double and triple integrals (evaluation techniques only). Areas, surface and volumes and centre of gravity.
Analytic Geometry: Sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.
Analytic Geometry: Sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.
Section B
Ordinary Differential Equations : Clariaut's equation, singular solution. Higher order linear equations, with constant coefficients, complementary function and particular integral, general solution, Euler-Cauchy equation. Second order linear equations with variable coefficients, determination of complete solution when one solution is known, method of variation of parameters.
Dynamics, Statics and Hydrostatics: You canskip this entire section, if you have prepared other sections well.
Vector Analysis: Triple products, vector identities and vector equations. Application to Geometry: Curves in space, curvature and torision. Serret-Frenet's formulae, Gauss and Stokes' theorems, Green's identities.
Dynamics, Statics and Hydrostatics: You can
Vector Analysis: Triple products, vector identities and vector equations. Application to Geometry: Curves in space, curvature and torision. Serret-Frenet's formulae, Gauss and Stokes' theorems, Green's identities.
Paper II
Section A
Algebra: Normal subgroups, homomorphism of groups quotient groups basic isomorophism theorems, Sylow's group, principal ideal domains, unique factorisation domains and Euclidean domains. Field extensions, finite fields.
Real Analysis: Riemann integral, improper integrals, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Uniform convergence, continuity, differentiability and integrability for sequences and series of functions. Differentiation of functions of several variables, change in the order of partial derivatives, implicit function theorem, maxima and minima. Multiple integrals.
Complex Analysis: You can skip this entire section, if you have prepared other sections well.
Linear Programming: Basic solution, basic feasible solution and optimal solution, Simplex method of solutions. Duality. Transportation and assignment problems. Travelling salesman problems.
Real Analysis: Riemann integral, improper integrals, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Uniform convergence, continuity, differentiability and integrability for sequences and series of functions. Differentiation of functions of several variables, change in the order of partial derivatives, implicit function theorem, maxima and minima. Multiple integrals.
Complex Analysis: You can skip this entire section, if you have prepared other sections well.
Linear Programming: Basic solution, basic feasible solution and optimal solution, Simplex method of solutions. Duality. Transportation and assignment problems. Travelling salesman problems.
Section B
Partial differential equations: Solutions of equations of type dx/p=dy/q=dz/r; orthogonal trajectories, pfaffian differential equations; partial differential equations of the first order, solution by Cauchy's method of characteristics; Char-pit's method of solutions, linear partial differential equations of the second order with constant coefficients, equations of vibrating string, heat equation, laplace equation.
Numerical Analysis and Computer programming: Numerical methods, Regula-Falsi and Newton-Raphson methods Numerical integration: Simpson's one-third rule, tranpesodial rule, Gaussian quardrature formula. Numerical solution of ordinary differential equations: Euler and Runge Kutta-methods.
Computer Programming: Binary system. Arithmetic and logical operations on numbers. Bitwise operations. Octal and Hexadecimal Systems. Convers-ion to and from decimal Systems.
Mechanics and Fluid Dynamics: D'Alembert's principle and Lagrange' equations, Hamilton equations, moment of intertia, motion of rigid bodies in two dimensions.
Numerical Analysis and Computer programming: Numerical methods, Regula-Falsi and Newton-Raphson methods Numerical integration: Simpson's one-third rule, tranpesodial rule, Gaussian quardrature formula. Numerical solution of ordinary differential equations: Euler and Runge Kutta-methods.
Computer Programming: Binary system. Arithmetic and logical operations on numbers. Bitwise operations. Octal and Hexadecimal Systems. Convers-ion to and from decimal Systems.
Mechanics and Fluid Dynamics: D'Alembert's principle and Lagrange' equations, Hamilton equations, moment of intertia, motion of rigid bodies in two dimensions.
1. Algebra : Elements of Set Theory; Algebra of Real and Complex numbers including Demovire's theorem; Polynomials and Polynomial equations, relation between Coefficients and Roots, symmetric functions of roots; Elements of Group Theory; Sub-Group, Cyclic groups, Permutation, Groups and their elementary properties.
Rings, Integral Domains and Fields and their elementary properties.
2. Vector Spaces and Matrices : Vector Space, Linear Dependence and Independence. Sub-spaces. Basis and Dimensions, Finite Dimensional Vector Spaces. Linear Transformation of a Finite Dimensional Vector Space, Matrix Representation. Singular and Nonsingular Transformations. Rank and Nullity.
Matrices : Addition, Multiplication, Determinants of a Matrix, Properties of Determinants of order, Inverse of a Matrix, Cramer's rule.
3. Geometry and Vectors : Analytic Geometry of straight lines and conics in Cartesian and Polar coordinates; Three Dimensional geometry for planes, straight lines, sphere, cone and cylinder. Addition, Subtraction and Products of Vectors and Simple applications to Geometry.
4. Calculus : Functions, Sequences, Series, Limits, Continuity, Derivatives.
Application of Derivatives : Rates of change, Tangents, Normals, Maxima, Minima, Rolle's Theorem, Mean Value Theorems of Lagrange and Cauchy, Asymptotes, Curvature. Methods of finding indefinite integrals, Definite Integrals, Fundamental Theorem of integrals Calculus. Application of definite integrals to area, Length of a plane curve, Volume and Surfaces of revolution.
5. Ordinary Differential Equations : Order and Degree of a Differential Equation, First order differential Equations, Singular solution, Geometrical interpretation, Second order equations with constant coefficients.
6. Mechanics : Concepts of particles-Lamina; Rigid Body; Displacements; force; Mass; weight; Motion; Velocity; Speed; Acceleration; Parallelogram of forces; Parallelogram of velocity, acceleration; resultant; equilibrium of coplanar forces; Moments; Couples; Friction; Centre of mass, Gravity; Laws of motion; Motion of a particle in a straight line; simple Harmonic Motion; Motion under conservative forces; Motion under gravity; Projectile; Escape velocity; Motion of artificial satellites.
7. Elements of Computer Programming : Binary system, Octal and Hexadecimal systems. Conversion to and from Decimal systems. Codes, Bits, Bytes and Words. Memory of a computer, Arithmetic and Logical operations on numbers. Precisions. AND, OR, XOR, NOT and Shit/Rotate operators, Algorithms and Flow Charts
PAPER - I
(1) Linear Algebra:
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; 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, 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, 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 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:
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.
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 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:
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:
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.
Equation of continuity; Euler's equation of 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.
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