Mathematics
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 reduction, 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, Taylor's theorem with remainders, indeterminate forms, maximas and
minima, asyptotes. Functions of several variables: continuity,
differentiability, partial derivatives, maxima and minima, Lagrange's method of
multipliers, Jacobian. Riemann's 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, 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 :
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, Kepler's 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 Bernoulli's 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-Frenet's formulae, Gauss and Stokes' theorems, Green's identities.
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