Vector, space, linear dependance
and independance, subspaces, bases, dimensions. Finite dimensional vector
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.
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A D V E R T I S E M E N T
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.
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
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
Application to Geometry: Curves in
space, curvature and torision. Serret-Frenet's formulae, Gauss and Stokes'
theorems, Green's identities.