B5550 - ANALISI MATEMATICA 2A

Learning outcomes

Upon completion of 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 the solution of simple practical 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

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

D. Pagani S.Salsa: Analisi Matematica 2, ed Zanichelli

Per esercizi:

S. Salsa, A. Squellati Esercizi di Anamisi matematica 2, ed. Zanichelli

P. Marcellini - C. Sbordone: Esercitazioni di Matematica, volume 2, parte prima, ed. Liguori

E. Giusti, Esercizi e complementi di Analisi Matematica, volume 2, ed. Boringhieri

Teaching methods

Lectures and exercises in the classroom.

Assessment methods

The examination consists of a preliminary written test and a written theory one before a colloquium.

The written test consists of some exercises related to the arguments of the course, it lasts two and a half hours. In order to sustain the written test the student must register at least four days before the test through AlmaEsami [https://almaesami.unibo.it/] . The maximum grade is 15. The student is admitted to the written-colloquial subsequent step, that it will be hold in a different day, if the realized score is grater or equal to 8.

The written test remains valid for the written-colloquial subsequent examination only in the same 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 time for the written test before the colloquium is 45' and it takes place for all the admitted candidates at the beginning of the day in which is scheduled the session. The maximum score is 8. After it, there will be a colloquium only if the sum between the score in the first written exam is at least 15. Both the evaluations mainly concern the theoretical aspects of the course. The student must show to know the concepts explained during the course (in particular definitions, theorems and their proofs) and how to connect with each other. The maximum score at the colloquium is 12.

How to determine the final grade: in case the score at the colloquium were les or equal to 3, then the candidate the exam has to be repeated because it is not sufficient, even if the sum of the three grades were grater or equal to 18.

In case the colloquium were graded more or equal to 4, the the final grade is obtained by the sum of the three grades. Here below some examples:

(8,7,3), then the student is rejected; (8,7,4), the student is promoted with 19.

(10,7,3), rejected; (15,8,3), rejected; (15,8,4), grade 26; (8,7,12), grade 27. (15, 0, 12), grade 27.

Teaching tools

We will use the web platform: Virtuale

 

Links to further information

https://www.unibo.it/sitoweb/fausto.ferrari/

Office hours

See the website of Fausto Ferrari