Vector analysis including curvilinear coordinates. Tensor algebra. Ordinary differential equations. Method of infinite series. Regular singularities, Frobenius method. First look at special functions. Gamma-, beta-, error functions. Airy function. Fourier series. Hilbert space, its basic properties. Sturm-Liouville theory. Orthogonal polynomials. Legendre, associated Legendre, Hermite, Laguerre etc. polynomials. Bessel functions, their properties, basic applications. Partial differential equations, their classification. Boundary conditions. Physical models with PDE. Separation of variables method, Cartesian system of coordinates. Separation of variables in cylindrical and spherical system of coordinates. Spherical functions. Fourier transform method.