First and second order ODE's, series solutions, second-order self-adjoint operators. Second order PDE's, multidimensional Fourier transformations, Green's functions and their eigenfunction expansions, curl, divergence and the Laplacian in curvilinear coordinates, separation of variables, spherical Bessel functions and spherical harmonics, solution of boundary value problems. <>Calculus of variations and variational methods. Integral equations and their kernels, Schmidt-Hilbert theory.