Scalars and Vectors: Scalars, Vectors, Vector Products, Vector Spaces, linear independence, basis, curvilinear coordinates, Tensors.Applications: Two body problem, Center of Mass and Relative coordinates – Vector Integration and Differentiation: Gradient, Divergence and Curl, Line Integrals, Surface integrals, volume integrals, Greens theorem, Stokes Theorem.Applications: Force, Work and Potentials. Path integrals – Matrix Algebra: Matrices, Rank, Determinants, Eigenvalues, Eigenvectors, System of Equations. Applications: Slater Determinants, Huckel MO theory – 1st order ODEs: Differential Equations of 1st order. Separation of variables, integrating factor, exact differentials, system of ODEs. Applications: Reaction rates.
2nd order differential equations: 2nd order differential equations with constant coefficients, general solution, particular solution, power series method. Applications: Angular Momentum Eigenfunctions for a single particle – Integral Transforms: Sturm-Liouville problem, basis functions, Fourier and Laplace transforms, dirac-delta functions. Applications: Power spectrum, momentum and position basis – Group Theory Basics: Group, subgroup, group multiplication table, symmetry operations, Great Orthogonality Theorem, Character Tables. Applications: This section is done with illustration of actual calculations – Symmetry adapted linear combinations, molecular motions, selection rules.Applications: This section is done with illustration of actual calculations – Data Analysis, Interpolation, least square fitting, asymptotic analysis, error estimates, random numbers, correlations. Applications: Fluctuations, noise, signal processing, scattering experiments – Numerical Methods: Taylor series, numerical differentiation and integration, matrix diagonalization methods.Applications: Perturbation method for radiation-matter interaction
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