The space R^n. Curves, line integrals.
Differential calculus for (scalar- and vector-valued) functions of several variables.
Free and constrained optimization problems.
Multiple integrals.
Vector fields, 1-forms and line integrals (of the second kind).
Parametric surfaces, surface and flow integrals.
Ordinary Differential Equations.
Function spaces. Sequences and series of functions, power series.
i) Carlo Domenico Pagani, Sandro Salsa, Analisi matematica Voll. 1 e 2 (Nuova edizione 2015), Zanichelli Ed.
(Warning: Differential calculus for functions of several variables is contained in Vol. 1)
** Alternatively:
consistently with a reference book suggested -- in the Academic Year 2017/18 -- for the preceding course "Analisi Matematica I":
ii) Nicola Fusco, Paolo Marcellini, Carlo Sbordone, Analisi matematica 2, Liguori, 2016
** Few other references, easily found in libraries:
- Franco Conti, Paolo Acquistapace, Anna Savojni, Analisi Matematica: teoria e applicazioni, McGraw-Hill, 2001 (out of print)
- Mariano Giaquinta, Giuseppe Modica, Note di analisi matematica. Funzioni di piu' variabili, Pitagora Ed., 2006
- Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976
** Exercises/Problems: plenty of material is found in libraries; a classical good reference book is, e.g.,
- Boris P. Demidovic, Esercizi e Problemi di Analisi Matematica, Editori Riuniti, 2010
** A choice of historical, educational, critical essays:
- Carl Boyer, A History of Mathematics, Wiley 1968 (in italiano, edito da Mondadori)
- Jeremy Gray, The Real and the Complex: A History of Analysis in the 19th Century, Springer International Publishing, 2015
- Enrico Giusti's work
- Umberto Bottazzini, Il flauto di Hilbert. Storia della matematica, UTET, 2005
- Gabriele Lolli, La crisalide e la farfalla. Donne e matematica, Bollati Boringhieri, 2000
- Cathy O'Neil, Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy, Penguin UK 2016
Learning Objectives
Knowledge: differential and integral calculus for functions of several variables; curves and surfaces; elements of the theory of Ordinary Differential Equations; sequences and series of functions.
Skills: autonomy in proposing, articulating and rigorously supporting arguments for the resolution of problems related to the listed subjects (see Knowledge); confident use of symbols and main results; control of errors.
Abilities/capacities: formalization of physical/mechanical problems in analytical terms; derivation of simple models in the continuous and in the discrete, and their mathematical treatment. Consolidated communication skills in writing and oral presentation. Equilibrium between autonomy in individual study and active participation in the group.
Prerequisites
Differential and integral calculus for single-variable functions. Numerical sequences and infinite series. Basics on Ordinary Differential Equations. Linear algebra and analytic geometry.
Prerequisite (formal): "Analisi Matematica I". Recommended: "Geometria".
Teaching Methods
Lessons (in the classroom), in the absence of a rigid separation between theory and practice. Spaces of collective discussion during the lessons and in the weekly 'office hours' (presumably Monday, 14:30-16:00, to be confirmed).
Exercises, problems, insights (or references to deeper insights) provided -- usually, every other week -- by means of the UniFI e-learning platform; keyword: am2fis_1819
Further information
CFU: 9 (225 hours)
Duration of the course: about 13 weeks (from September 20 to December 21, 2018), 80 hours in the classroom. Cf. https://www.fis-astro.unifi.it/vp-94-orario-delle-lezioni.html
Class schedule:
Mon. 10:45-13:30 (three hours, with two breaks) at the Blocco Aule, Polo Scientifico
Wed. 9:45-11:30 (two hours, with a break) at the Blocco Aule, Polo Scientifico
Fri. 8:45-10:30 (two hours, with a break) at the Blocco Aule, Polo Scientifico
NOTE: The 8:45-9:30 space on Wednesday is officially assigned and might be used sporadically; see classes timetable
Type of Assessment
The final examination of the course consists of a written exam including questions on key subjects (see Contents and Extended Program), and -- if admitted -- in a subsequent oral exam. The said written examination can be anticipated by partecipating to (presumably two) midterm exams. A discussion of the written report should be expected during the oral examination.
Note: according to the "Regolamento didattico di Ateneo", there are six (6) "Appelli" during the academic year; the appelli are distinct from each other: unless otherwise indicated by the professor, the oral exam cannnot be postponed to a subsequent appello.
Course program
Analisi Matematica II -- Mathematical Analysis II (Detailed Program)
The space R^n: scalar product, Euclidean norm, Cauchy-Schwarz inequality, subadditivity property. Vector product.
Curves defined by parametric equations: ..., orientation, simple and closed curves. Special plane curves: graphs of continuous functions on an interval, polar curves. Velocity vector, speed, tangent line; regular and piecewise-regular curves at times. Equivalent parameterizations. Rectifiable curves, length of a curve; formula for the length of a C^1 curve (or piecewise-regular). Curvilinear abscissa. Line integrals (of the first kind). Physical and geometric applications. The website Mathcurve.
Real-valued functions of several variables: graphs, level sets. Elements of topology in R^n: internal points, open sets, closed sets, accumulation points. Limits and continuity. Properties of continuous functions and major results: the Intermediate Zero Theorem and the Intermediate Value Theorem, the Weierstrass Theorem. Directional derivatives, partial derivatives. Linear approximation and differentiability. Tangent plane and normal vector to a graph. Gradient vector, the differential; Theorem of the Total Differential.
Derivation of compound functions: relevant cases, chain rule. Higher order derivatives, Hessian matrix, Schwarz theorem, C^k functions. Taylor's formula of the second order, higher order approximations. Relative and absolute extrema, Fermat's Theorem.
*[Quadratic forms in R^n: positive (negative) definite forms, positive (negative) semidefinite and indefinite forms. Eigenvalues test. Necessary conditions and sufficient conditions for the existence of relative maxima/minima/saddle points for a C^2 function.]
Implicit functions, the Implicit Function Theorem.
Vector-valued functions of several real variables: generality, the Jacobian matrix. Parametric surfaces. Regular surfaces, tangent plane; piecewise-regular surfaces. Examples.
Change of coordinates, local and global invertibility of transformations defined from open sets of R^n into R^n.
Constrained optimization: the Lagrange Multipliers Theorem.
The Riemann integral. Integral of step functions in a rectangle; integral of functions with compact support. Characterization of integrability. Peano-Jordan measurable sets. Sets of zero measure, examples; characterization of measurable sets. Integrable functions in bounded measurable sets. The Fubini-Tonelli Theorem. Reduction formulas for double and triple integrals. Physical/mechanical applications: barycenters and moments of inertia. Change of variables in multiple integrals. Polar coordinates; cylindrical and spherical coordinates. Improper multiple integrals. The integral of the Gauss function in R^2, R, R^n.
Vector fields, differential 1-forms. Work done by a vector field along a curve, line integrals (of the second kind). Conservative vector fields (exact forms) and their potentials (antiderivatives). Characterization of exact forms. Necessary and/or sufficient conditions: closed forms and irrotational vector fields, the curl operator; convex, star-shaped, simply connected sets. Examples in R^n, n=2,3. Locally exact forms. Gauss-Green formulas in the plane.
Area of a surface. Surface and flux integrals. The divergence Theorem. Stokes' Theorem.
Function spaces. Complete metric spaces. The contraction mapping Theorem. Ordinary Differential Equations (ODE). Order and type of an equation, linear vs nonlinear. The Cauchy problem. Theorem of (local) existence and uniqueness; Lipschitz condition; the Peano phenomenon. Global existence, the sub-linearity requirement.
First order linear ODE: general integral and representation formula for the solutions to the associated Cauchy problems (see Analisi Mat. I course). Linear ODE of order n: algebraic structure of the space of the solutions to the homogeneous equation. The case of constant coefficients. Non-homogeneous equations, particular solutions: variation of parameters and method of undetermined coefficients. Euler equations. First order nonlinear ODE: separable equations (see Analisi Mat. I course), homogeneous equations; exact equations, integrating factors.
Sequences of functions. Pointwise and uniform convergence. Passage to the limit under the integral sign; other implications of uniform convergence. Function series, Weierstrass criterion, total and uniform convergence. Power series, radius of convergence. Absolute, total, uniform convergence. Taylor series and analytic functions.
Note: The lessons contents are gradually rendered explicit and available in the e-learning platform as well as directly in Professor Bucci's personal page; see http://www.dma. unifi.it/~fbucci (Teaching Activity, current Academic Year)