# 27993 - Mathematical Analysis T-2

## Learning outcomes

To give a good knowledge of the calculus concerning functions with several variables.

## Course contents

THE EUCLIDEAN SPACE R^n. The vector space structure, the dot product and the euclidean norm. Open, closed, bounded, compact, connected subsets of R^n.

LIMITS, CONTINUITY AND DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES.

Generalities on real and vector functions of several real variables. Definition of limit of a function and continuous function and of . The Weierstrass, zeros and Heine-Cantor's theorem and the intermediate value theorem for functions of several variables. Partial and directional derivatives. Differentiable and C^1 functions; the differential and the Jacobian matrix. The chain rule. Partial derivatives of higher order. Hessian matrix. Taylor's formula of the second order for functions of several variables. Interior and constrained local extrema for real functions of several variables.

CURVE INTEGRALS.

Curves, length of a curve, orientation. Integral of a function over a curve.

The integral of a vector field over an oriented curve. Conservative vector fields and their potentials. Work of a vector field.

MULTIPLE INTEGRALS.

Normal domains. Double and triple integrals. The reduction formula. The change of variables theorem for a double integral.Gauss-Green's formulas and Stokes'Theorem in the plane.

SURFACE INTEGRALS.

Smooth surfaces. Tangent plane and normal vector. Area of a surface. Integral of a function over a surface. The divergence theorem and the Stokes theorem.

SERIES OF NUMBERS AND OF FUNCTIONS.

Numerical series: definition, convergence, ansolute convergence. Criteria of convergence.

Power series, Taylor series, Fourier series: definition and main properties.

DIFFERENTIAL EQUATIONS. The Cauchy problem for differential equations and systems. Theorems on existence, uniqueness and continuation of solutions.

C.D. Pagani, S. Salsa: Analisi Matematica 2 (Zanichelli);

W. Rudin: Analisi Reale e Complessa (Boringhieri);

E. Giusti: Analisi Matematica 2 (Boringhieri).

S. Salsa, A. Squellati: Esercizi di Matematica volume 2 (Zanichelli);

Marcellini Sbordone: Esercitazioni di Matematica, Secondo volume (Liguori Editore);

M. Bramanti: Esercitazioni di Analisi Matematica 2, Progetto Leonardo - Esculapio (2012).

## Teaching methods

Lessons in room. complimentary lecture notes will be available on AMS Campus.

## Assessment methods

The grading is determined with a written exam. The duration time is 2 h and 30 minutes.

In the test there are both exercises and theoretical questions.

The only way to be admitted to the exam is to enroll their own name to the published lists on AlmaEsami web site.

The final grade is obtained simply converting the result obtained in the written exam.

After the results will be published on AlmaEsami they will be recorded in a week.

The student that, after the written exam, will have obtained 25 or more may ask to face an oral session. In any case the obtained result can not be modified more than two points (positively or negatively).

A grade with "lode" is possible only for the students that, having realized  29 or 30  after the written exam, ask for an oral session.

## Teaching tools

Lecture notes concerning the lessons on AMS Campus. Tutor (if assigned).

## Office hours

See the website of Fausto Ferrari

See the website of Vittorio Martino