B5550 - ANALISI MATEMATICA 2A

Learning outcomes

Concluding the course, students will have some more advanced knowledge of mathematical analysis. They will understand the fundamentals of the theory of metric spaces, differential calculus for vector functions of several real variables, and the basic aspects of ordinary differential equations. Students will be able to apply this knowledge to solve some simple  problems posed by the pure and applied sciences.

Course contents

Metric spaces: definition of metric space and linear space with norm. Sequences in metric spaces and definition of convergence in metri spaces. Definition of limit and continuity in metric spaces; basic of topology in metric spaces. Complete metric spaces. Banach Caccioppoli's theorem. Compactness in metric spaces. Heine Borel in compact subset in R^n. Weierstrass theorem and Heine-Cantor theorem in several dimensions.

Differential calculus: definition of differential function along a direction and definition of partial differential of a function in several variables. Jacobian matrix and gradient of a function. Definition of differential function in several variables and sufficient conditions for differentiability.

Local maxima and minima for a function: basic definitions. Definition of quadratic form.

Definition of quadratic form: positive (negative), indefinite, and positive (negative) semidefinite. Necessary condition for differentiability and gradient formula. Lagrange mean value theorem for scalar-valued functions of several variables; functions with zero gradient on connected open sets. Lagrange mean value type theorem for vector-valued functions of several variables. Theorem on the differential of composite functions and the chain rule. Total differential theorem. 𝐶1 functions. Schwarz's theorem. Taylor's formula for functions of class 𝐶2. Characterization of definite matrices. Fermat's theorem. Second-order necessary conditions and second-order sufficient conditions for local extrema. Dini's theorem; Inverse function theorem. Definition of diffeomorphism. Definition of a variety of dimension 𝑝 in 𝑅𝑛. Relations between varieties and their graphs. Theorem on the tangent space to a variety. Theorem on the space orthogonal to a variety. The gradient of a function (scalar) is orthogonal to the level sets. Lagrange multipliers.

Ordinary differential equations: Definition of the solution of a differential equation (or a system of differential equations) and the solution of a Cauchy problem. Definition of uniqueness of solutions. Scalar first-order linear differential equations. First-order equations with separable variables. Existence and uniqueness theorem. Statement of Peano's theorem and comments on the uniqueness of the solution. Definition of a locally Lipschitz function, and the existence and local uniqueness theorem of the Cauchy problem. Uniqueness of the solution. Extension of local solutions and maximal solution. Global existence theorem. Linear systems. Fundamental matrix and method of variation of the constant. Linear equations of order 𝑛 and their connection with linear systems. Theorem on the structure of the general integral of a linear equation of order n. Linear differential equations with constant coefficients.

Readings/Bibliography

Notes by the teachers will be available on the related web-sites.

In order to get an in-depth study of the topics of the course, students may want to consult:

E. Lanconelli: Lezioni di Analisi Matematica 2, prima parte, ed. Pitagora

N. Fusco, P. Marcellini, C. Sbordone: Analisi Matematica due, ed. Liguori

G.C. Barozzi, G. Dore, E. Obrecht: Elementi di Analisi Matematica, vol. 2, ed. Zanichelli

Exercise Textbooks:

M. Bramanti, Esercitazioni di Analisi Matematica 2, ed. Esculapio

P. Marcellini, C. Sbordone: Esercitazioni di Analisi Matematica due, parte I e parte II, ed. Zanichelli.

Teaching methods

Lectures and exercises in the classroom

Assessment methods

The examination consists of both a written and an oral exam.

The written exam consists of several exercises related to the arguments of the course. In order to participate the written test, any student must register at least three days before the test through AlmaEsami.

The written test remains valid for the oral exam in the same examination period. Namely only in January-February if the written exam has been held in January or February, or alternatively on June-July, if the written exam has been held in June-July, or alternatively on September, if the written exam has been held in September.

The oral test follows the written test; it mainly concerns the theoretical aspects of the course. Any student must exhibit his/her knowledge of the concepts provided during the course (in particular: definitions, theorems and their proofs) and how to properly connect them.

During the written exam, the use of notes, books or computer tools of any kind is not allowed. In particular, the use of mobile phones, smartwatches or earphones is not allowed.

 

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Teaching tools

Online pdf's and possibly some videos will be provided.

Office hours

See the website of Annalisa Baldi