1. Algebra : Elements of Set Theory; Algebra of Real and Complex
Paper-I
Section-A
Linear Algebra
Vector, space, linear dependance and independance, subspaces, bases, dimensions.
Finite dimensional vector spaces.
Matrices, Cayley-Hamiliton theorem, eigenvalues and eigenvectors, matrix of
linear transformation, row and column reducti
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A D V E R T I S E M E N T
on, Echelon form, eqivalence,
congruences and similarity, reduction to cannonical form, rank, orthogonal,
symmetrical, skew symmetrical, unitary, hermitian, skew-hermitian forms�their
eigenvalues. Orthogonal and unitary reduction of quadratic and hermitian forms,
positive definite quardratic forms.
Calculus
Real numbers, limits, continuity, differerentiability, mean-value theorems,
Taylors theorem with remainders, indeterminate forms, maximas and minima,
asymptotes. Functions of several variables: continuity, differentiability,
partial derivatives, maxima and minima, Lagranges method of multipliers,
Jacobian. Riemanns definition of definite integrals, indefinite integrals,
infinite and improper intergrals, beta and gamma functions. Double and triple
integrals (evaluation techniques only). Areas, surface and volumes, centre of
gravity.
Analytic Geometry :
Cartesian and polar coordinates in two and three dimesnions, second degree
equations in two and three dimensions, reduction to cannonical forms, straight
lines, shortest distance between two skew lines, plane, sphere, cone, cylinder.,
paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.
Section-B
Ordinary Differential Equations :
Formulation of differential equations, order and degree, equations of first
order and first degree, integrating factor, equations of first order but not of
first degree, Clariauts 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 :
Degree of freedom and constraints, rectilinerar motion, simple harmonic motion,
motion in a plane, projectiles, constrained motion, work and energy,
conservation of energy, motion under impulsive forces, Keplers laws, orbits
under central forces, motion of varying mass, motion under resistance.
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.
Pressure of heavy fluids, equilibrium of fluids under given system of forces
Bernoullis equation, centre of pressure, thrust on curved surfaces, equilibrium
of floating bodies, stability of equilibrium, metacentre, pressure of gases.
Vector Analysis :
Scalar and vector fields, triple, products, differentiation of vector function
of a scalar variable, Gradient, divergence and curl in cartesian, cylindrical
and spherical coordinates and their physical interpretations. Higher order
derivatives, vector identities and vector quations.
Application to Geometry: Curves in space, curvature and torision.
Serret-Frenets formulae, Gauss and Stokes theorems, Greens identities.
Paper-II
Section-A
Algebra:
Groups, subgroups, normal subgroups, homomorphism of groups quotient groups
basic isomorophism theorems, Sylows group, permutation groups, Cayley theorem.
Rings and ideals, principal ideal domains, unique factorization domains and
Euclidean domains. Field extensions, finite fields.
Real Analysis :
Real number system, ordered sets, bounds, ordered field, real number system as
an ordered field with least upper bound property, cauchy sequence, completeness,
Continuity and uniform continuity of functions, properties of continuous
functions on compact sets. 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 fuctions of several
variables, change in the order of partial derivatives, implict function theorem,
maxima and minima. Multiple integrals.
Complex Analysis :
Analytic function, Cauchy-Riemann equations, Cauchys
theorem, Cauchys integral formula, power series, Taylors series, Laurents
Series, Singularities, Cauchys residue theorem, contour integration. Conformal
mapping, bilinear transformations.
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. Travelling salesman problmes.
Section-B
Partial differential equations:
Curves and surfaces in three dimesnions, formulation of 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 Cauchys method of characteristics; Charpits
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: 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)
methods, Gauss-Seidel(iterative) method. Newtons (Forward and backward) and
Lagranges method of interpolation.
Numerical integration:
Simpsons one-third rule, tranpezodial rule, Gaussian quardrature formula.
Numerical solution of ordinary differential equations: Euler and Runge Kutta-methods.
Computer Programming:
Storage of numbers in Computers, bits, bytes and words,
binary system. arithmetic and logical operations on numbers. Bitwise operations.
AND, OR, XOR, NOT, and shift/rotate operators. Octal and Hexadecimal Systems.
Conversion to and form decimal Systems.
Representation of unsigned integers, signed integers and reals, double
precision reals and long integrers.
Algorithms and flow charts for solving numerical analysis problems.
Developing simple programs in Basic for problems involving techniques covered
in the numerical analysis.
Mechanics and Fluid Dynamics :
Generalised coordinates, constraints, holonomic and non-holonomic, systems. D
Alemberts principle and Lagrange equations, Hamilton equations, moment of
intertia, motion of rigid bodies in two dimensions.
Equation of continuity, Eulers equation of motion for inviscid flow,
stream-lines, path of a particle, potential flow, two-dimensional and
axisymetric motion, sources and sinks, vortex motion, flow past a cylinder and a
sphere, method of images. Navier-Stokes equation for a viscous fluid.
numbers including Demovires 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, Cramers 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, Rolles 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. |
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AKHAND PRATAP SINGH
