Ordinary Differential Equations:
First and second order linear differential equations.
Partial Differential Equations:
Introduction to important PDEs in Physics (wave equation, diffusion equation, Poisson’s equation, Schrodinger’s equation), general form of solution, general and particular solutions (first order, inhomogeneous, second order), characteristics and existence of solutions, uniqueness of solutions, separation of variables in Cartesian coordinates, superposition of separated solutions, separation of variables in curvilinear coordinates, special functions, integral transform methods, Green’s functions .
Complex Analysis:
Review (polar form of complex numbers and de Moivre’s theorem, complex logarithms and powers), functions of a complex variable, Cauchy-Riemann conditions, power series in a complex variable and analytic continuation with examples, multi-valued functions and branch cuts, singularities and zeroes of complex functions, complex integration, Cauchy’s theorem, Cauchy’s integral formula, Laurent series and residues, residue integration theorem, definite integrals using contour integration. Vector analysis: Review of basics properties, Vector in 3-D spaces, Coordinates transformations, Rotations in R3 , Differential vectors operators, Vector integrations, Integral theorem,