Metric and normed spaces. Banach and Hilbert spaces. Completeness. Functional sequences and functional series; types of convergence. Power series. Differential and integral multivariable calculus. Applications to (differential) geometry and optimization. Itroduction to the ordinary differential equations.
Carlo Domenico Pagani, Sandro Salsa, Analisi matematica Voll. 1 e 2 (Nuova edizione 2015), Zanichelli Ed. (Attenzione: il calcolo differenziale in piu' variabili e' nel Vol. 1).
Mariano Giaquinta, Giuseppe Modica, Note di analisi matematica. Funzioni di piu' variabili, Pitagora Ed., 2006
Learning Objectives
To teach students the main notions and results of differential and integral calculus for the functions of several variables. Provide with the knowledge of main methods of solution of ordinary differential equations, in particular the ones, which appear in physical models. Students are intriduced into a basic techniques of functional alysis, getting knowledge of the concepts of metric and normed spaces, completeness, Banach and Hilbert spaces. The notions are applied to the study of functional sequences and functional series.
Prerequisites
Programs of the courses of Analysis I and Geometry.
Teaching Methods
Lectures and exercises in classes and online.
Type of Assessment
Written test consisting in the solution of exercises and oral examination.
Course program
Metric and normed spaces, neighborhoods, open and closed sets, continuous maps between metric spaces. Cauchy sequences in metric spaces, completeness, Banach spaces.
Compact spaces. arzela-ascoli theorem. Completeness of a compact space. Fixed point theotem for contractions.
Functional series, pointwise, unifiorm and total convergence. Excersises. Power series; radius of convergence. Hadamard formula. Regularity for the power series and analytic functions. Taylor formula and Taylor series. Exercises for calculus the series coiefficients and the sum.
Function from R^N to R. Continuity. Directional and partial derivatives.
Differentiability of a function in R^N. Gradiente. Gradient as a direction of steepest ascent. Formula of total derivative. High-order derivatives. Schwartz theorem.
Derfinition of smooth curve and its properties. Theorem on a drerivation of a composition along the curve.
Taylor formula in R^N with a rest term in Lagrange and Peano form.
Hesse matrix.
Fermat theorem and its application to the search of the maxima and minima. Second order neccesaary/sufficient conditions for extremum.
Positive/negative definite and semidefinite matrices. Aplications to the maxima and minima.
Homogeneous functions and their properties. derivatives of the homogeneous functions and Euler formula.
Functions f:R^N ->R^M and their properties. Jacobi matrix. Differential of a composition.
Multiple Riemann integrals. Double integrals of bounded functions over elementary rectangles (bidimensional intervals). Definition and properties. integrability of the continuous functions. Reduction formula on a rectangle.
2D integrals of the bounded functions over boiunded sets. Peano-Jordan measurable sets and zero measure sets. Reduction formulae for elementary domains.
Global and local invertibility of the map T:R^2->R^2
Jacobi matrix and invertibility.
Theorem on the change of variables in a double integral (formulation and an idea of the proof).
Examples: polar coordinates. Theorem on the iintegral average. Examples of the change of variables.
Symmetries of the plane and even and odd functions with respect to a symmetry. Applications to the calculation of double integrals.
Pysical and mechanical applications: surface density, mass and barycentre of a plane domain. Moments of inertia.
Elliptic coordinates for calculation of double integrals. Triple Riemann integrals on 3D intervals. Reduction formula. Integrals over simple and regular domains. Integrazione per stratti e per fili.
Change of variables for triple integrals: cylindrical and spherical coordinates. Examples: calculation of the volume of a sphere.
Regolar parametric surfaces: tangent plane and normal vector.
Area of a surface, normal vector. Surface integrals of continuous functions. Orientable surface and example of Moebius sheet.
Flow of a vector field through a parametric surface.
First fundamental form of a surface and its properties.
Invariance of a surface integral with resoect to a change of parameterization.
Divergence theorem (formulation and comments).
Border of a surface and its orientation. Application of the divergence theorem to the calculation of the flow over a surface with a board. Dependece of the flow on the orientation. Formulation of the Stokes theorem. Stokes theorem and Green's formula. Correspondence between the vector fields and trhe differential forms. Orientation of the plane domains and of their boards. Green's formula applied to the study of closed differential forms in connected plane domains.
Differential equations. ODE in implicit and normal forms. High-order ODE and its representation of a system of the forst-order ODE.
Equation with separable variables, radioactive decay, Malthus growth, logostic equation, SIR model for an epidemics.
Cauchy problem, existence and uniqueness theorem; scheme of the proof.
Extendibility of solutions. Lack of uniqueness.
Linear homogeneous and non-homogeneous ODE and their properties. (Linear) space of solutions of the homogeneous linear equation. Solution of non-homogeneous linear equation by variation of constants method.
Bernulli equation and its properties.
Implicit differential equations.
Clairaut equation. Envelope of a family of solutions.
Particular types of the second-order ODE; reduction of the order. Linear second-order ODE . Linear dependence/independence of solutions, Wronskian; Abel's theorem.
Second-order linear ODE with constant coefficients. Search of particular solutions by variation of constants method and by the method of undetermined coefficients.
Parametric curves: definition and properties. Regular and piecewise regular curves; closed curves. Equivalent curves. Oriented and non oriented paths. Tangent vector.
Curve length. Curve integral; path dependence.
Implicit functions. Dini theorem in dimension 2. Successive derivatives and Taylor formula for implicit function.
Dini theorem in dimension 3 and in general form.
Constrained maxima and minima with constraints in parametric and explicit form. Lagrange multipliers rule.
Convex sets. Exact forms on convex sets. Corollaries for closed forms, which are "locally exact".
Differential 2-forms and exterior derivatives of differential forms. Formulation of De Rham's theorem as generalization of Newton-Leibniz and Stokes theorem.
Closed and exact differential forms on star haped domains.
Differential equations and differential forms. Method of solution of exact differential equations. Integrating factor.