A. MATHEMATICAL METHODS
Vector Analysis: Transformation properties of vectors; Differentiation and integration of vectors; Line integral, volume integral and surface integral involving vector fields; Gradient, divergence and curl of a vector field; Gauss' divergence theorem, Stokes' theorem, Green's theorem - application to simple problems; Orthogonal curvilinear co-ordinate systems, unit vectors in such systems, illustration by plane, spherical and cylindrical co-ordinate systems only.
Matrices: Hermitian adjoint and inverse of a matrix; Hermitian, orthogonal, and unitary matrices; Eigenvalue and eigenvector (for both degenerate and non-degenerate cases); Similarity transformation; digitalization of real symmetric matrices.
Ordinary Differential Equations: Solution of second order linear differential equations with constant coefficients and variable coefficients by Frobenius’ method (singularity analysis not required); Solution of Legendre and Hermite equations about x=0; Legendre and Hermite polynomials - orthonormality properties.
Partial Differential Equations: Solution by the method of separation of variables; Laplace's equation and its solution in Cartesian, spherical polar (axially symmetric problems), and cylindrical polar (`infinite cylinder' problems) coordinate systems.
Fourier Series: Fourier expansion – statement of Dirichlet’s condition, analysis of simple waveforms with Fourier series. Introduction to Fourier transforms; the Dirac-delta function and its Fourier transform; other simple examples. Vibration of stretched strings- plucked and struck cases.
B. WAVES AND OPTICS
Linear Harmonic Oscillator (LHO): LHO. Free and forced vibrations. Damping. Resonance. Sharpness of resonance. Acoustic, optical, and electrical resonances: LCR circuit as an example of the resonance condition. A pair of linearly coupled harmonic oscillators --- eigenfrequencies and normal modes.
Waves: Plane progressive wave in 1-d and 3-d. Plane wave and spherical wave solutions. Dispersion: phase velocity and group velocity.
Fermat's principle: Fermat's principle and its application on plane and curved surfaces.
Cardinal points of an optical system: Two thin lenses separated by a distance, equivalent lens, different types of magnification, Helmholtz and Lagrange's equations, paraxial approximation, introduction to matrix methods in paraxial optics – simple application.
Wave theory of light: Fourier expansion – statement of Dirichlet’s condition, analysis of simple waveforms with Fourier series. Introduction to Fourier transforms; the Dirac-delta function and its Fourier transform; other simple examples. Vibration of stretched strings- plucked and struck cases.
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