27993 - Mathematical Analysis T-2

Learning outcomes

To provide students with a solid methodological and operational understanding of the fundamental concepts and techniques of differential and integral calculus for functions of several variables.

Course contents

Module 2

EUCLIDEAN SPACE 𝑅 𝑛 Rn

The structure of a vector space, inner product, and Euclidean norm. Open, closed, bounded, compact, and connected subsets of 𝑅 𝑛 Rn.

DIFFERENTIAL EQUATIONS

Linear differential equations and separable differential equations. The Cauchy initial value problem for differential equations and systems of differential equations. Existence, uniqueness, and continuation theorems.

LINE INTEGRALS

Curves. Arc length of a curve. Oriented curves. Line integral of a scalar function. Work of a vector field.

Module 1

CONTINUITY AND DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES

Real-valued and vector-valued functions of several real variables: general concepts. Limits of functions. Continuous functions. Partial derivatives and directional derivatives. Differentiable functions. Jacobian matrix. Differentiability of composite functions.

Higher-order partial derivatives. Hessian matrix. Second-order Taylor formula for functions of several variables. Unconstrained and constrained local extrema.

DOUBLE AND TRIPLE INTEGRALS

Normal domains. Double and triple integrals. Reduction formulas (iterated integrals). Change of variables in double and triple integrals. Green–Gauss formulas and Green's theorem in the plane.

VECTOR FIELDS

Vector fields: definition. Conservative and irrotational vector fields.

SURFACES AND SURFACE INTEGRALS

Regular surfaces. Tangent plane and unit normal vector. Surface area. Surface integrals. The Divergence Theorem and Stokes' Theorem.

 

Readings/Bibliography

One of the following books

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

Fusco-Marcellini-Sbordone: Analisi Matematica Due, Liguori Editore.

M. Bramanti, C. D. Pagani, S. Salsa, Analisi matematica 2. Ed. Zanichelli.

V. Barutello, M. Conti, D. Ferrario, S. Terracini, G. Verzina: Analisi Matematica vol. 2, ed. Apogeo

M. Bertsch, R. Dal Passo, L. Giacomelli: Analisi Matematica, seconda edizione, Mc Graw Hill

 

 

Bramanti M.: Esercitazioni di Analisi Matematica 2 , Ed. Esculapio

Teaching methods

The course consists of in-person lectures that introduce the fundamental concepts concerning the properties of real-valued functions of several real variables, as well as linear and separable differential equations. The lectures are consistently complemented by examples and counterexamples illustrating the key concepts presented. In addition, a substantial number of exercises are worked through during class.

Assessment methods

The assessment is based on an examination consisting of two parts:

• A written exam, lasting two and a half hours, consisting of problem-solving exercises.
• A theory exam, consisting of written theoretical questions followed by an oral discussion with the instructor. The theory exam may be taken in a subsequent examination sitting, provided that the written exam has already been passed and both parts are completed within the same examination session (June–July–September or January–February).

The final mark is awarded on a 30-point grading scale and is determined by the overall performance in both parts of the examination.

Teaching tools

Tutoring (if assigned)

During the course, PDF exercise sheets will be made available on the Virtuale platform.

These exercise sheets are an essential resource for preparing for the written examination

Office hours

See the website of Giovanna Citti

See the website of Nicola Abatangelo