(A) Mathematical Physics: [20 Lecture Periods (LP)]

1. Preliminaries: SI system of units, dimensional analysis. Plotting of functions (both cartesian and polar), Limits, Intuitive ideas about continuity and differentiability of a function. Taylor series of one variable and binomial series (statements only); Maxima and minima for functions of one variable. Calculus of functions of more than one variable: Partial derivatives, exact and inexact differentials. [5 LP]

2. Ordinary Differential Equations: First order linear differential equations and integrating factor. Linear second order homogeneous equations with constant coefficients. Simple harmonic motion as an example. [2 LP]

3. Vectors: Dot, cross, scalar triple and vector triple products of cartesian vectors (using LeviCivita symbol and summation convention). Vector differentiation. Scalar and vector fields --- gradient, divergence, curl and Laplacian (for Cartesian coordinates), solenoidal and irrotational vector field. Statement and proof of Divergence theorem and Stokes' theorem; application to simple cases. [7 LP]

4. Curvilinear coordinates: Plane polar, spherical polar and cylindrical polar coordinates: their unit vectors, role of unit vectors as basis vectors. Surface and volume element (from geometry). Line, surface and volume integrals. Form of the gradient operator in curvilinear coordinates. Velocity and acceleration of point particle in Cartesian, plane polar, spherical polar, cylindrical polar coordinates. [6 LP]

 

(B) Classical Mechanics: [30 Lecture Periods]

1. Review of Newton’s Laws: Concepts of Inertial frames; force and mass. Galilean transformations and Galilean invariance; Newton's laws of motion, principle of conservation of linear momentum, Simple problems involving motion under resistive forces. Rotational motion: Angular velocity, angular acceleration, angular momentum, torque, principle of conservation of angular momentum. [6 LP]

2. Work Kinetic Energy Theorem. Conservative Forces: Force as the gradient of a scalar field. Concept of potential and potential energy. Other equivalent definitions of a conservative force. Conservation of energy. Qualitative study of one-dimensional motion from potential energy curves. Stable and unstable equilibrium. Simple harmonic oscillation for small displacement from a stable equilibrium. [4 LP]

3. Dynamics of a system of particles: The problem of solving equation of motion; Actionreaction kind of forces and the two body problem; Reduced mass & centre of mass; Properties of the centre of mass; Effect of torque; Linear momentum, angular momentum & total energy of a system of particles. [4 LP]

4. Central force: Newton’s Law of Gravitation; Kepler’s Laws; Conservation of angular momentum, Gauss’s law for Gravitation (integral form); Gravitational potential and intensity due to uniform spherical shell, solid sphere of uniform density and infinite flat sheet. Differential equation for the path in a central force field. Motion under an inverse square force, calculation of orbits. [8 LP]

6. Scattering: Two body collision and scattering [2 LP]

7. Mechanics of Continuum: Kinematics of Moving Fluids: Idea of compressible and incompressible fluids, Equation of continuity; streamline and turbulent flow, Reynold’s number. Stokes’ law from dimensional analysis; Euler’s Equation and the special case of fluid statics. Simple applications (e.g.: Pascal’s law and Archimedes principle). Bernoulli’s Theorem. [6 LP]

8. Oscillations

9. GPM (Elasticity)