To give the understanding of Differential equations and their uses in Physics, Introduction to special functions, Fourier Series, Fourier Transforms, Solution of Boundary value problems and their uses.

Group Theory and Representations for finite groups:

Transformations, groups definitions and examples, subgroups and Cayley’s theorem, cosets and Lagrange’s theorem, conjugate classes, invariant subgroups, factor groups, homomorphism, direct products, mappings, linear operators, matrix representations, similarity transformation and equivalent matrix representations, group representations, equivalent representations and characters, construction of representations and addition of representations, invariance of functions and operators, unitary spaces and Hermitian matrices, operators: adjoint, self-adjoint, unitary, Hilbert space, reducibility of representations, Schur’s lemmas, orthogonality relations,    group  algebra, expansion of  functions in  basis  of  irreducible representations, Kronecker product, symmetrized and anti-symmetrized representations, adjoint and complex-conjugate representations, real representations, Clebsch-Gordan series and coefficients, applications of these ideas to classification of spectral terms, perturbation theory and coupled systems

Tensor Analysis:

Vector calculus (differentiation, integration, space curves, multi-variable vectors, surfaces, scalar and vector fields, gradient, divergence and curl, cylindrical and spherical coordinates, general curvilinear coordinates), change of basis, Cartesian tensor as a  geometrical object, order/rank  of   a   tensor,  tensor  algebra,  quotient  law,   pseudotensors, Kronecker delta and Levi cevita, dual tensors, physical applications, integral theorems for tensors, non-Cartesian tensors, general coordinate transformations and tensors, relative tensors, Christoffel symbols, covariant differentiation, vector operators in tensor form,   absolute derivatives along curves, geodesics.