27993 - Mathematical Analysis T-2

Learning outcomes

The student knows the basic concepts and the main properties of functions of several real variables, in particular differential and integral calculus.

Course contents

The essential prerequisites of the course are the knowledge of all the topics covered in the course of Mathematical Analysis T1, as well as of numerous topics covered in the course of Geometry and Algebra T (vector spaces, linear transformations, matrices, determinants, analytic geometry in the plane and in space) .

 

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

 

LIMITS, CONTINUITY AND DIFFERENTIAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES. Real and vector functions of several real variables: generalities. Definition of continuous function and limit. Weierstrass theorems, of intermediate values for functions of several variables. Definition of partial derivative and directional derivative. Differentiable functions and functions of class C^1; the differential and the Jacobian matrix. The theorem on the differentiability of a compound function. Partial derivatives of higher order. Second-order Taylor formula for functions of several variables. Relative extrema for real functions of several free real variables.

 

MULTIPLE INTEGRAL. Definition of double Riemann integral on limited and measurable sets. Properties of the double integral. Reduction theorems on rectangles and simple sets. The change of variables theorem. Triple integrals: extension of the definitions and theorems on double integrals. Notes on generalized double integrals. Curvilinear and surface integrals. Regular and piecewise regular curves, length of a curve, integral of a function on a curve. The integral of a vector field on an oriented curve. Conservative vector fields and their potentials. The Green-Gauss theorem. Regular and piecewise regular surfaces in R^3, area of a surface, integral of a function on a surface. Flux of a vector field through an oriented surface. Divergence and Stokes theorems. Solenoid fields and vector potential.

 

DIFFERENTIAL EQUATIONS. The Cauchy problem for differential equations and systems. Theorems of existence, uniqueness and prolongability. Solving methods for non-linear equations with separable variables and for linear equations of the first order. Space of solutions of a homogeneous and non-homogeneous linear differential equation of order n. Solving non-homogeneous linear second-order differential equations with the similarity method.

Readings/Bibliography

Any Analysis 2 textbook will suffice.

Anyway, for the theoretical preparation, we will provide 

our lecture notes, thanks to the internet platform Virtuale.

All the books we suggest here will be in italian,

but sure there are plenty of very good books in English.

 

 

-Theory-

Bramanti-Pagani-Salsa, Analisi Matematica 2, Zanichelli.

or

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

or

Giusti, Analisi Matematica 2, Bollati Boringhieri.

 

 

-Practice-

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

or 

P.Marcellini, C. Sbordone: Esercitazioni di Analisi Matematica Due (prima e Seconda Parte) ed. Zanichelli.

Teaching methods

The course is structured in classroom lectures which illustrate the fundamental concepts relating to the properties of real functions of several real variables, vector fields in R^n and nonlinear differential equations. The lessons are always integrated with examples and counterexamples related to the fundamental concepts illustrated. In addition, numerous exercises are carried out in the classroom.

Assessment methods

The preparation of the students will be verified through an exercise test and a theoretical one to be carried out in sequence. The student must first carry out the exercise test and, if she/he obtains a mark equal to or greater than 16, he/she will be admitted to the theory test. The theoretical test will be aimed at verifying the understanding of the definitions and theorems of the course and their proofs. With the theoretical exam, the student will be able to raise the overall grade by a maximum of 6 points but may also lower it or be rejected.

Teaching tools

Tutoring (when assigned by the faculty).

Main homepage of the Course:

https://virtuale.unibo.it/course

 

Office Hours:

Have a look at the website of Matteo Franca [https://www.unibo.it/sitoweb/matteo.franca4]

Have a look at the website of Simone Ciani [https://www.unibo.it/sitoweb/simone.ciani3]

Office hours

See the website of Simone Ciani

See the website of Matteo Franca