## Part A: 100 Marks

### a. Solid Earth Geophysics:

Introduction to Geophysics its branches and relationship with other sciences. Solar system, its origin, characteristics of planetary members, Earth; its rotation and figure. Age of earth & various methods of determination. Tectonics and Geodynamics, Thermal history and its characteristics. Gravity field of earth and Isostasy.

Geomagnetism, elements of earth's magnetism: Internal, External fields and their causes, Paleomagnetism, Polar wandering paths, Seafloor spreading, geophysical evidences. Elastic waves, internal structure of earth, variation of physical properties in the interior of earth.

### b. Earthquake and Engineering Seismology:

Seismology, earthquakes, focal depth, epicenter, great Indian earthquakes, Intensity and Magnitude scales, Energy of earthquakes, foreshocks, aftershocks, Elastic rebound theory, Fault plane solutions, Seismicity and Seismotectonics of India, Frequency-Magnitude relation (b values), Velocity structure, Vp/Vs studies. Elastic waves, their propagation characteristics.

Seismic ray theory for spherically and horzontally stratified earth, basic principles of Seismic Tomography and receiver function analysis, Seismic network and arrays, telemetry systems, Earthquake prediction; dilatancy theory, short-term, middle-term and long- term predictions, Seismic microzonation studies, application for engineering problems, Seismometry, Principle of electromagnetic seismograph, displacement meters, velocity meter, accelerometer, WWSSN stations, Strong motion seismograph, seismic arrays for detection of nuclear explosions, Broadband seismometry.

### c. Mathematical methods in Geophysics:

Properties of scalars, vectors and tensors, Elements of vector analysis, Gradient, Divergence and Curl, Gauss's divergence theorem, Stokes theorem, Definition of fields, Gravitational field, Newton's Law of gravitation, Gravitation potential and fields due to bodies of different geometric shapes, Electrostatic field, Coulomb's law, Electrical permittivity and dielectric constant, Basic guiding equations, Magneto static field, Origin of Magnetic field, Ampere's law, Biot and Savart's law, Geomagnetic fields, Magnetic fields due to different type of structures, Solution of Laplace equation in Cartesian Coordinate, Cylindrical Polar Coordinate and Spherical Polar Coordinate, Complex Variables in Potential theory, Green's theorem in Potential Theory.

Concept of Image in Potential Theory, Analytical continuation in Potential fields, Numerical Methods in Potential Theory. Electrical fields in geophysics, point source, continuous distribution and double layers, equipotential and line of force. Current and potential in the earth, basic concept and equations of electromagnetic, Maxwell's equations, boundary conditions, elliptic polarization, electromagnetic potential and waves, radiation from dipoles, retarded potential, near and far fields, radiation resistance, EM field of a loops of wire on half space, multi-layered media, impedance and its application.

### d. Geophysical Inversion:

Fundamental concepts of inverse theory, Basic definition of inversions with application to Geophysics. Probability, Inverses with discrete and continuous models. Forward problems versus Inverse problems. Formulation of inverse problems and their relation to a matrix problem, linear inverse problems, classification of inverse problems, least square solutions and minimum norm solution, concept of norms, concept of 'a priori' information, constrained linear least square inversion, review of matrix theory Introduction to finite difference method, forward, backward and central difference mthod.

Application of finite difference method for solving Helmholtz equation. Introduction to finite element method, various steps, simple examples showing application of finite element method. Models and data spaces, householder transformation, data resolution matrix, model resolution matrix, Eigen values and Eigen vectors, singular value decomposition (SVD), generalized inverses, Non-linear inverse problems, Gauss Newton method, steepest descent (gradient) method, Marquardt-Levenberg method, Earthquake location problem, tomography problem. Probabilistic approach of inverse problems, maximum likelihood and stochastic inverse methods, Backus-Gilbert method, Global optimization techniques, genetic algorithm, simulated annealing methods, examples of inverting geophysical data.

## Part B: 100 Marks

### a. Mathematical Methods of Physics:

Dimensional analysis, Vector algebra and vector calculus, Linear algebra, matrices, Cayley-Hamilton Theorem. Eigen values and eigenvectors. Linear ordinary differential equations of first & second order, Special functions (Hermite, Bessel, Laguerre and Legendre functions). Fourier series, Fourier and Laplace transforms. Elements of complex analysis, analytic functions; Taylor & Laurent series; poles, residues and evaluation of integrals. Elementary probability theory, random variables, binomial, Poisson and normal distributions. Central limit theorem. Green's function.

Partial differential equations (Laplace, wave and heat equations in two and three dimensions). Elements of computational techniques: root of functions, interpolation, and extrapolation, integration by trapezoid and Simpson's rule, solution of first order differential equation using Runge-Kutta method. Finite difference methods. Tensors.

Introductory group theory: SU (2), O (3). Complex Variables, Beta, Gamma functions and special functions, Laplace Transform & Fourier series, Fourier Transforms, Introductory methods of Numerical analysis. Operators and their properties.

### b. Thermodynamics and Statistical Physics:

Laws of thermodynamics and their consequences; Thermodynamic potentials, Maxwell relations; Chemical potential, phase equilibria; Phase space, micro- and macrostates; Micro canonical, canonical and grand-canonical ensembles and partition functions; Free Energy and connection with thermodynamic quantities; First and second-order phase transitions; Classical and quantum statistics, ideal Fermi and Bose gases; Principle of detailed balance; Blackbody radiation and Planck's distribution law; Bose-Einstein condensation; Random walk and Brownian motion; Introduction to non equilibrium processes; Diffusion equation.

### c. Electrodynamics:

Gauss Theorem, Poison's equation, Laplace's equation, solution to Laplace's equation in Cartesian coordinates, spherical, cylindrical coordinates, use of Laplace's equation in the solutions of electrostatic problems. Ampere's circuital law, magnetic vector potential, displacement current, Faraday's law of electromagnetic induction.

Maxwell's equations, differential and integral forms, physical significance of Maxwell's equations. Wave equation, plane electromagnetic waves in free space, in non conducting isotropic medium, in conducting medium, electromagnetic vector an scalar potentials, uniqueness of electromagnetic potentials and concept of gauge, Lorentz gauge, Columb gauge, charged particles in electric and magnetic fields, charged particles in uniform electric field, charged particle in homogeneous mag netic fields, charged particles in simultaneous electric and magnetic fields, charged particles in non homogeneous magnetic fields. Lienard - Wiechert potentials, electromagnetic fields from Lienard - Wiechert potentials of a moving charge, electromagnetic fields of a uniformly moving charge, radiation due to non-relativistic charges, radiation damping, Abrahama-Lorentz formula, Cherenkov radiation, radiation due to oscillatory electric dipole, radiation due to small current element.

Condition for plasma existence, occurrence of plasma, magneto hydrodynamics, plasma waves. Transformation of electromagnetic potentials, Lorentz condition in covariant form, invariance or covariance of Maxwell field equations in terms of 4 vectors, electromagnetic field tensor, Lorentz transformation of electric and magnetic fields.

### d. Introductory Atmospheric and Space Physics:

The Neutral atmosphere, atmospheric nomenclature, the Hydrostatic equation, geopotential height, expansion and contraction, fundamental forces in the atmosphere, apparent forces, atmospheric composition, solar radiation interaction with the neutral atmosphere, climate change. Electromagnetic radiation and propagation of Waves: EM Radiation, fundamentals of EM waves, effects of environment, Antennas-basic considerations, types of antennas. Propagation of Waves: ground wave, sky wave, and space wave propagation, troposcatter communication and extra terrestrial communication. The Ionosphere, morphology of ionosphere, the D, E and F-regions, chemistry of the ionosphere, ionospheric parameters, E and F region anomalies and irregularities in the ionosphere. Global Positioning Systems (GPS)-basic concepts, overview of GPS system, augmentation services, GPS system segment, GPS signal characteristics, GPS errors, multi path effects, GPS performance, satellite navigation system and applications.