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.
LEARNING OUTCOMES: Based on lectures and exercises sessions, the aim of the course to strenghten the basis of mathematical knowledge of calculus and numerical analysis.
KNOWLEDGE AND UNDERSTANDING: Aquiring basic notions of mathematical methods, including basic notions of integral and differential calculus. Written and oral examinations at the end of the lectures.
APPLYING KNOWLEDGE AND UNDERSTANDING: The students have to prove the capacity to identify the essential elements of a physical problem and to find the corresponding mathematical methods that apply
MAKING JUDGEMENTS: The students must be able to critical analyse the mathematical problem.
COMMUNICATION SKILLS: The students must be able to present in a rigurous way the results obtained by mathematical methods.
LEARNING SKILLS: Stuedents must get a good knowledge of the mathematical methods to be used in physics.