VECTOR ANALYSIS AND GEOMETRY
- Mechanics : Inertial and non-inertial frames, Conservation Laws, Centre of mass and its kinematics. Rotational Motion, Coriolis force, Moment of inertia and elated Theorems. Harmonic, Damped and Forced oscillations, resonance. Wave phenomenon and its properties. Gravitational field and potential, Kepler’s laws. Galilean transformation, Lorentz transformations, Relativistic Doppler effect.
- Electricity and Magnetism : Vector Fields and Electrostatics, Stoke’s theorems, Gauss’s law and its applications, Laplace and Poisson equations. Dielectric Medium, Gauss’s law, Clausius-Mossotti relation, Langevin-Debye equation, Capacitors. Continuity equation, LCR circuts, Kirchoff’s laws, circuits theorems. Laws of magnetic Effects of current and magnetic Properties. Electromagnetic induction, self and mutual inductance. Alternating current, LCR circuits, Power in AC circuits.
- Optics and Electromagnetic Theory : Fermat’s principle and its application, cardinal points of an optical system, aberrations. Maxwell’s equations and their significance, gauge transformations and gauge condition, Poynting vector, plane e.m. waves in free space, nonconducting medium and conducting media.
- Polarization : Brewster’s law, Malus law, elliptically and circularly polarized light, polaroids. Interference of polarized light. Interference, Coherence, Fresnel’s biprism, Newton’s rings, Michelson’s interferometer, Fabry – Perot interferometer.
Fraunhoffer and Fesnel diffraction, diffraction grating, diffraction at circular aperture, Lasers and its applications, Holograpy, Optical fibre.
- Electronics : p-n junction diode, its characteristics and applications, LEDs, Zener diode and its application, capacitor filter, Power supply, BJT as an aimplifier, BJT biasing, Q-point and its temperature dependence, Hybrid parameters, Classification of amplifiers, RC coupled amplifier and its frequency response characteristics, negative feedback in amplifiers and its advantage, JFET, MOSFET and their characteristics, Basic logic gates, Boolean algebra and De Morgan’s theorem, Wein’s bridge, Hartley oscillators. Multivibrators, RC differentiator and integrator, Operational amplifier and its ideal applications, Modulation, Cathode Ray Oscilloscope.
- Mathematical Method : Cauchy – Riemann differential equations, Green’s theorem, Cauchy’s integral theorem, Cauchy’s integral formula, Taylor and Laurent series, Cauchy’s residue theorem. Curvilinear Coordinates : Gauss’s theorem, Stroke’s theorem. Basel Functions, hermite Polynomials, Legendre, Associated Legendre and Laguerre Functions, their properties, generating functions, orthogonality relation. Fourier series, its properties and applications.
- Laplace and Poission equations and their simple applications. Classification of integral equations. Forier integral, Fourier transforms, Kernels and their properties.
- Classical Mechanics and Special Relativity : Lagrangian and Hamiltonian formulation, Lagrange’s equations and Hamilton’s equations of motion and applications. Seymmetry and conservation Laws. Two body central force problem. Kepler’s laws, Virial theorem, integrals of motion. Scattering cross section. Rutherford formula. Tensors, four vectors and their transformations, Lagrangian of a relativistic particle. Equation of motion. Relativistic formulae for kinematical variables, Kinematics of two body reaction Maxwell’s equation, equation of continuity in covariant form.
- Quantum Mechanics : Necessity of new formalism for microparticles, Physical observables as mathematical operators, postulates of quantum mechanics, expectation values and Ehrenfest theorem, Heisenberg uncertainly principle. Schrodinger Equation, wave functions. Particle in one dimensional infinite and finite square well, potential step, potential barrier, linear harmonic oscillator. Orbital angular momentum operators and their commutation relations, eigenvalues and eigenfunctions of L2 and LZ, rigid rotator, central potential, hydrogen like atoms. Stern-Gerlach experiment, Zeeman splitting, Spin angular momentum operators and their algebra, Total angular momentum, its commutation relation. Idea of perturbation, time independent and dependent perturbation. Scatering amplitude and cross sections, scattering by spherically symmetric potential, scattering of identical particles.
- Thermal and Statistics Physics : Kinetic Theory of Gases, principle of equipartition of energy, specific hearts of gases, Andrew’s experiment, collision cross section and mean free path, transport phenomena. Laws of thermodynamics, Carnot’s cycle, entropy, T-S diagrams, Gibb’s paradox. TdS equations, porous plug experiment, regenerative cooling, helmboltz and Gibb’s functions, maxwell’s equations. First and second order phase transitions. Phase space, microand macro-states, definition of entropy. Fermi-Dirac and Bosc Einstein distributions. Maxwell-Boltzmann distribution, partition function, Planck’s formula and other related laws. Monatomic ideal gas, Maxwell’s formula for the distribution of velocities, the law of equipartition of energy, specific heat. Atomic, Molecular, Laswer and Solid State Physics.
- Atomic Physics : Spin – orbit interaction, Relativistic correction and Lamb Shift, fine structure of hydrogen, excitation and ionization potentials. L.S. and jj coupling scheme, Pauli’s exclusion principle, Breit’s scheme, Zeeman effect for sodium doublets. Molecular Physics : Rotation and vibration of diatomic molecules, vibrating rotator and its spectrum, infrared spectrum of diatomic molecules, classical theory of Raman effect, rotational Raman spectra. Electronic spectra of diatomic molecules, Frank-Condon principle, electronic structure of N2 and O2, modes of vibration HCN, CO2 and H2O.
- Laser Physics : Einstein’s A and B coefficients, population inversion, ammonia maser, vibtational modes of a resonator, Quality factor, Schawlow – Townes conditions, Properties of Laser beams, Ruby, He-Ne and N2 lasers. Crystalline and Amorphous Structures, Symmetry Operations, Index System, Defects and Dislocations, Quasi – crystals, Crystal Binding, Cohesive energy of Ionic and Noble gas crystals, Bragg and Laue diffraction, Reciprocal lattice, Normal modes of one dimensional monatomic and diatomic lattices, Dispersion curves and their features, Free electron theory, Drude model of electrical conductivity, Mathiessen’s rule, Hall effect. Variation of Fermi-Dirac distribution with temperature, Sommerfeld model of free electron gas, Band theory of solids, Periodicity, Bloch theorem, Kronnig Penny model, Band gap for a linear monatomic crystal in nearly free electron model. Simple properties of superconductors.
- Nuclear Physics, Particle and Astrophysics : General Properties of Atomic Nuclei, Nucler angular momentum, magnetic dipote moment, g-factor, Schmidt lines, electric quardrupole moments, Nuclear force and its properties. Bainbridge and Aston man spectrograph, mass defect and binding energy, fission and fusion. Radioactive Series, bateman equations and their application to activation analysis, Alpha, beta and gamma decays. Nuclear Reactions : Energy balance, Q-value and threshold, reaction cross section. Interaction of radiations with matter and radiation detectors.
- Particle Physics : Basic interactions and their mediating quanta, classification of particles; conservation rules, determination of spin and parity of pions, strange particles quark model. Cosmic Rays Primary cosmic rays, its energy and charge spectrum, secondary cosmic rays, its composition, intensity variation, Rossi transition curve, electromagnetic cascade showers and extensive air showers.
- Astrophysics : Sun and its atmosphere, energy source, solar activities, stars and their temperatures and magnitudes, H-R diagram, Birth and death of stars, Chandrashekhar mas limit, pulsars, Schwarzschild radius.
Note : For details refer to Physics Syllabi taught to B. Sc. (Hons.) Physics students of AMU.
- General Equation of Second Degree
Intersection of a straight line and a conic, Equation of a tangent to a conic. Condition of tangency, Pair of tangents from a point, Chord of contact of a pair of tangents, Pole and Polar, Conjugate points and lines, Chord with a given middle point, Centre of a conic and diameter, Conjugate diameters.
- Tracing of Conic and Polar Equations
Nature of conic, Tracing of parabola, ellipse and hyperbola, Asymptotes of the hyperbola. The length and the position of axes of the conic, Polar equation of a conic when focus is at the pole, Directrices, Tracing of the conic 1/r = 1 +e cosq , Asymptotes,Equation of the chord when the vectorial angles of the extremitities are given. Equation of the tangent and the normal when the vectorial angle of the point of contact is given.
- Cylinder and Cone
Equation of a cylinder, Equation of a right circular cylinder, Equation of a cone, Equation of a cone when the vertex is at the origin. Condition for general equation of second degree to represent to a cone, Tangent plane to a cone, Conditioin of tangency, Reciprocal cone, Cone with three mutually perpendicular generators.
- Central Concoids
The standard equation, The tangent plane, Condition of tangency of plane, Section with a given centre, Locus of the mid-points of a system of parallel chords. The polar plane, Polar lines, Enveloping cone, Classification of central coicoids (Ellipsoid, Paraboloids, Elliptic paraboloid, hyperbolic paraboloid), Conjugate diametral plane and diameters of Ellipsoid, Normals on ellipsoid, Conjugate diameters of ellipsoid.
- Vector Analysis
Scalar and vector product of three vectors, Product of four vectors, Reciprocal vectors differentiation. Partial and directional derivative, Orientation Gradient, Divergence and curl.
- Differential Calculus
Unit of a function, Intermediate forms, Successive differentiation, Leibnitz theorem, Maclaurin and Taylor series expansions, Angle between radius and tangent Perpendicular from the pole on the tangent, Pedal equation, Curvature. Partial differentiation, Total differentials, Euler’s theorem on homogeneous function, Asymptotes, Concavity and Convexity, Points of inflexion, Multiple points, Tracing of curves in Cartesian and polar coordinates.
- Integral Calculus Gamma function and its properties, Cartesian parameteric and polar forms for rectification, Intrinsic equation for Cartesian and polar curves, Volume and surface of solids of revolution, Cartesian and polar forms.
Solution of algebraic and transcendental equations, The bisection and Regula Falsi method, Iteration methods namely, Newton-Raphson method, Generazlied Newton’s method, Solution of system of linear equations using direct methods such as matrix inversion method, Gauss elimination method, LU decomposition including some iteration methods namely, Jacobi and Gauss Siedel methods, The algebraic eigenvalue problems using power and Householder methods.
Newton-Gregory Forward and Backward Difference Formulae, Gauss’s, Stirling’s and Bessel’s formulae, Lagrange’s Formula, Divided Differences and their properties, Newton’s General Interpolation Formula, Inverse Interpolation Formula, Interpolation with cubic splines.
Numerical Differentiation and Integration, Numerical Differentiation of tabular and non-tabular functions including error estimations, Numerical Integration using Gauss Quadrature Formulae, Trapezoidal, Simpson’s 1/3- and 3/8-Rule, Weddle’s Rule and Newton-Cotes Formla.
Ordinary Differential Equations, Euler’s and Modified Euler’s Methods, Picard’s Method, Taylor Series Method, Runge-Kutta Methods of 2nd and 4th order, Multi-step Methods, Milne-Simpson Method, Adam Bashforth-Moulton Method, Boundary value Problems using Finite Difference Method.
Approximation and Difference Equations, Least Square Curve Fitting Procedures, Different types of approximations, Least Square Polynomial Approximation, Chebyshev Polynomials and its applications in various approximations. Difference Equations, Solution of simple difference equations, First order Homogeneous Linear Difference Equations. Higher order Homogenous Linear Difference Equations, Non homogenous Linear Difference Equations. Higher Order Homogeneous Linear Difference Equations with constant and variable Coefficients.
Exact different equations, Integrating factors, change of variables. Total differential equations, Differential equations of first order but not of first degree, Equations solvable for x,y,q Equations of the first degree in x and y, Clairaut’s equations. Linear differential equations of order n, Homogenous and non-homogenous linear differential equations of order n with constant coefficients, Different forms of particular integrals, Linear differential equations with non-constant coefficients, Reduction of order method. The Cauchy-Euler’s equation of order n, Legendre’s linear equation.
Methods of undertermined coefficients and variation of paprameters, Series solution of differential equations, Frobenius method, Bessel and Legendre differential equations and their series solutions.
Solution of a system of linear differential equation with constant coefficients, An equivalent triangular system, Degenerate case Laplace transforms, Linearity of Laplace transforms, First shifting property, Transforms of Degenerate case Laplace transforms, Linearity of Laplace transforms, First shifting property, Convolution and periodic function theorems, Solution of linear differential equations using Laplace transforms.
Formation and solution of a partial differential equations easily integrable, Linear (Lagrange’s) and nonlinear partial differential equation of first order, Charpit’s methods, Homogenous partial differential equations with constant coefficients.
Non-homogenous partial differential equations with constant coefficients, Classification of second order linear partial differential equations, Method of separation of variables, Fourier series, Even and odd functions and their Fourier series, Change of interval, Vibration of stretched string, One and two dimensional heat flow, Laplace equation in Cartesian form.
The variation of a functional and Euler’s equations, Externals, Functional depending on a unknown functionals, Functionals’ depending on higher derivatives, Variational problems in parametric form, Isoparametric problem, Canaonical form of Euler’s equation, Functionals depending on functions of several variables.
Binary operations, Definition of group with examples and elementary properties, Subgroups. Order of group, Statement of Lagrange’s theorem. Homomorphism of groups, Kernel of Homomorphism, Defintion of isomorphism, Introduction to rings, subrings and fields with examples and elementary properties. Vector spaces. Subspaces, sum of subspaces, Span of a set, Linear dependence and independence, dimension and basis, dimension theorem, Linear transformation and its properties, range and kernel of a linear transformation, rank and nullity of linear transformation, Rank-nullity theorem, inverse of linear transformation. The space L(u,v) of linear transformation and its dimension, composition of linear transformation, matrix associated with a linear transformation. Linear transformation associated with matrix, matrix as a liner transformation and its rank and nullity. Elementary row operations and row reduced echelon form, inverse of a matrix. Application of matrices to a system of linear equations, Eigen vectors and characteristics equation of matrix, Cayley-Hamilton theorem and its use in finding inverse of matrix.
Statistics : Probability Theory, Descriptive Statistics, Sample Surveys and Design of Experiment, Statistical Inference
Computer Science : Fundamentals of Computers and Problem solving, Business Data Processing, Data Structure and Structured Programming, Systems Programming and Operating System
- Partial Differentiation Functions of several variables, Partial derivatives, implicit functions, Limits and continuity, derivatives, composite functions, differential functions, homogenous functions, Euler’s theorem, Higher derivatives, Simultaneous equations, Jacobians, inverse of transformation. Dependent and independent variables, Differential and directional derivatives, Taylor’s theorem, Jacobian’s of implicit functions, Inverse of transformations, Change of variable, functional dependence and equality of cross derivatives.
- Application of Partial Differentiation
Maximum and minima for functions of two variables, sufficient conditions, Functions of three variables, Quadratic form, Relative extrema, Lagrange’s multipliers, one relation between two variables. One relation among three variables. Two relations among three variables, Envelopes and Evolutes of families of plane curves.
- Multiple Integrals
Definition, Properties and evolution of double as well as triple integrals, related results, Iterated integrals and change in order of integration.
- Line and Surface Integrals
Definition, Properties and evaluation of double as well as triple integrals, relted results, Iterated integrals and change in order of integration.
- Line and Surface Integrals
Definition, Properties and evaluation of line as well as surface integrals, related problems, Green’s theorem and deductions, Stoke’s theorem.