Mathematical Methods in Physics

Coordinate transformation in N-dimensional space: Contravariant and covariant tensor, Jacobian ,  pseudo tensors , Algebra of tensors, Metric tensors, Associated tensors,  Christoffel symbols, transformation of Christoffel symbols.

Equation of Geodesic, Covariant differentiation, Ricci's theorem, Divergence, Curl and Laplacian in tensor form, Stress and Strain tensors, Hooke's law in tensor form, Lorentz Covariance of Maxwell equation.

Group of transformations, (Example: symmetry transformations of a square), Generators of a finite group, Normal subgroup, Direct product of groups, Isomorphism and Homomorphism, Representation theory of finite groups, Invariant subspace and reducible representations, irreducible representations, Crystallo-graphic point groups, Irreducible representation of  C_4ν, Translation group and the reciprocal lattice.

Fourier Transforms: Development of the Fourier integral from the Fourier series, Fourier and inverse Fourier transform, Fourier transform of Derivatives, Solution of wave equation as an application, Convolution theorem, intensity in terms of spectral density for quasi-monochromatic EM waves, momentum representation, Application of Fourier Transform to Diffraction Theory, Diffraction  pattern of single  and double slits.

Laplace transforms and their properties, Laplace transform of derivatives and integrals of Laplace transform, Laplace convolution theorem, Impulsive function , Application of Laplace transform in solving linear differential equations with constant coefficient, with variable coefficient and linear partial differential equation.