Metric spaces. Distance, neighborhoods, open and closed sets. Compact metric spaces. Completeness. Contraction mapping theorem. Hilbert spaces and orthonormal sequences.

Differential equations. Cauchy problem for first order systems, existence and uniqueness of the solution. Some special first order equations. Continuation of solutions and maximal solutions. Continuous dependence on initial data. Linear differential equations and systems. Fundamental solution. Variation of constants. Flow associated with a vector field. Alpha- and omega-limit sets. Lyapunov stability of an equilibrium point. Linearization. Lyapunov functions and stability criterion.

Integral calculus. Multiple integrals, Fubini’s theorem, change of variables in multiple integrals. Polar and spherical coordinates. Surfaces and surface integrals. Gauss-Green formulas in the plane, with applications. Stokes’ theorem.

Fourier Series. Fourier coefficients, Fourier series, Bessel’s inequality. Pointwise convergence of the Fourier series of a piecewise regular function. Uniform convergence. Parseval’s identity. Gibbs’ phenomenon. Application of Fourier series to the solution of the heat and wave equation on a bounded domain.

Fourier transform. Fourier transform of a summable function, algebraic and differential properties of the transform. The Riemann-Lebesgue lemma. Transform of a convolution product and inversion of the Fourier transform. Transform of rapidly decreasing functions. Fourier transform of square summable functions and Plancherel’s theorem. Shannon’s theorem. Application of the Fourier transform to the solution of ordinary differential equations. Solutiuon of the heat and wave equation on unbounded domains.