1. Linear
Algebra:
Vector space, linear dependence
and independence, subspaces, bases, dimensions. Finite dimensional vector
spaces. Matrices, Cayley Hamilton theorem, Eigen values and Eigenvectors.
Matrix of linear transformation, row and column reduction, Echelon form, equivalence,
congruence and similarity, reduction to canonical form, rank, orthogonal, symmetrical,
skew symmetrical, unitary, Hermitian, skew Hermitian forms their Eigen values. Orthogonal and unitary reduction of quadratic
and Hermitian forms, positive definite quadratic form.
2. Calculus:
Real numbers, limits, continuity,
differentiability, mean value theorems, Taylor’s theorem with remainders,
indeterminate forms, maxima and minima, asymptotes. 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 integrals,
Double and triple integrals (evaluation techniques only). Repeated Integrals. Beta and Gamma functions. Areas, surface and volumes, centre of
gravity.
3. Analytic
Geometry:
Cartesian and polar coordinates in
two and three dimensions, second degree equations in two and three dimensions, Reduction
to canonical forms, straight lines, plane, sphere, cone, cylinder, paraboloid,
ellipsoid, hyperboloid of one and two sheets and their properties. shortest
distance between two skew lines.
4. 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, Clairaut’s
equation, singular solution. Higher order
linear equations with constant coefficients, complementary function and particular
integral, general solution. Euler-Cauchy equations. Second order linear equations with variable
coefficients, Determination of complete solution when one solution is known, method
of variation of parameters.
5. Statics:
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.
Lemi’s theorem
6. Dynamics:
Degree of freedom and constraints,
rectilinear 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.
7.
Hydrostatics:
Pressure of heavy fluids,
equilibrium of fluids under given system of forces. centre of pressure, thrust on
curved surfaces, equilibrium of floating bodies, stability of equilibrium, metacentre,
pressure of gases. Problems relating to
atmosphere.
1. 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 equations. Application to
Geometry: Curves in space, Curvature and Torsion. Serret?Frenet's formulae,
Gauss and Stokes' theorems, Green's identities.
2. Real
Analysis:
Real number system, ordered sets,
Bounds, ordered field, real number system as an ordered field with least upper
bound. Cauchy sequence. Completeness.
Continuous Functions. Uniform continuity. Properties of continuous functions on
compact sets. Riemann integral, improper
integrals. Differentiation of functions of several variables, maxima and
minima. absolute and conditional convergence of series of real and complex
terms, rearrangement of series. Uniform convergence. Infinite Products.
Continuity, differentiability and integrability for series. Multiple integrals. Infinite and alternating
series.
3. Numerical
Analysis:
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.
Interpolations: Newton’s (Forward and
backward) and Lagrange's method
4. Mechanics:
Concepts of Particles, Lamina,
Rigid body, Displacement, Force, Mass, Weight, Motion, Velocity, Speed,
Acceleration. Parallelogram of forces. Parallelogram of velocity, acceleration,
resultant, equilibrium of coplanar forces. Moments. Couple. Friction. Centre of
mass. Gravity. Laws of motion under conservative forces. Motion under gravity.
Projectiles, Escape velocity, Motion of artificial satellites.
5. Probability:
Sample space, Events, Algebra of
Events, Probability-Classical, Statistical and Axiomatic Approaches.
Conditional Probability and Baye’s Theorem Random Variables and Probability.
Distributions Discrete and Continuous.
Mathematical Expectations. Binomial, Poisson and Normal
Distributions.
6.
Statistical Methods:
Collection, Classification,
Tabulation and Presentation of Data. Measures of Central Value. Measures of
Dispersion. Skewness, Moments and Kurtosis. Correlation and Regression.