Course Contents :
Linear vector spaces, Basis sets. Orthogonality and completeness. Linear maps and dual space, Bra and ket notation. Inner product; Linear operators and Matrices, Hermitian and unitary operators, Normal matrices and their diagonalization, Cayley-Hamilton theorem. New vector spaces from old: Direct sum and tensor products, outer product of matrices; Examples: Vectors and Tensors in R^3, Rotation group in 2 and 3 dimensions. Spin and C2, Pauli matrices. Generators of rotations. Multiple spins and the tensor product. Hilbert space. Dirac delta function, representation and properties. Examples: L2(S1) and Fourier Series; L2(R) and Fourier transforms; Convolution in Fourier Series and Transforms; L2(S2) and spherical harmonics. Families of orthogonal polynomials as basis sets in function space, Legendre, Hermite, Laguerre, Chebyshev and Gegenbauer polynomials, Generating functions. Expansion of functions, Inversion formulas. Elements of analytic function theory, Cauchy-Riemann conditions, Cauchy's integral theorem and integral formula, Singularities-poles and essential singularities, residue theorem and contour integration. Occurrence of Laplace, Poisson, Helmholtz wave and diffusion equations in physical applications, Elementary properties of these equations and their solutions.
1. P. Dennery and A. Krzywicki, Mathematics for physicists (Dover Publications, 1996) 2. J. Hefferon, Linear Algebra (Chapters 2 and 3), (Orthogonal Press, 2014), Freely available at: http://joshua.smcvt.edu/linearalgebra/ 3. G. Arfken and H. J. Weber, Mathematical Methods for Physicists (7th Edition) (Academic Press, 2012).