27993 - Mathematical Analysis T-2

Learning outcomes

Improvement of the basic mathematical tools to solve standard problems arising from applications (series, curves, several types of integrals, differential equations).

Course contents

PREREQUISITES

The knowledge of all the topics explained in Analisi Matematica T1, as well as many topics of Geometria e Algebra T (linear spaces, linear transformations, determinants, analytic geometry in the plane and in the space) are key prerequisites for this course (Analisi Matematica T2)

COURSE CONTENTS

SERIES. Numerical series. Definition of a convergent series. Absolute convergence of a series. Convergence criteria for positive series: ratio criterion and asymptotic criterion. Sign changing series, Leibnitz Criterion

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 a continuous function and of limit of a function. The Weierstrass 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. Taylor's formula of the second order for functions of several variables. Interior (and hints on constrained) local extrema for real functions of several variables.

MULTIPLE INTEGRALS.

Definition of Riemann double integral for functions defined on a normal domain. Properties of the double integral. Double integrals on normal domains computed by iterated integrals. The change of variables theorem for a double integral. Generalizations to triple integrals. Hints on double   integrals on unbounded domains

CURVE AND SURFACE INTEGRALS.

Smooth and piecewise smooth curves, length of a curve, integral of a function over a curve. The integral of a vector field over an oriented curve. Conservative vector fields and their potentials. The Green-Gauss theorem. Smooth and piecewise smooth surfaces in R^3, area of a surface, integral of a function over a surface. The flux of a vector field through an oriented surface. The divergence theorem and the Stokes theorem.

DIFFERENTIAL EQUATIONS.

The Cauchy problem for differential equations and systems. Theorems on existence, uniqueness and continuation of solutions. Methods of solutions for nonlinear differential equation with separable variables and linear differential equations

Readings/Bibliography

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

or

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

An exercise book on functions of several real variables, such as, for example:

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.

However the on-line material found in iol should be enough for the theorical part but probably not for the exercises

Teaching methods

The course consists of lessons describing the fundamental concepts of real and vector functions of several real variables. Lessons are completed with examples and counterexamples illuminating the theoretical content. Futhermore a lot of exercises are solved in the classroom.

Assessment methods

The assessment is divided in 2 tests and the student needs to pass the first test to be admitted to the second.

The first test will be a written exam lasting between 2 hours and 2 hours and a half, in which the student will solve some exercises using the tools learned in the course. This exercise test is passed with a grade greater or equal to 15/30.

The second test will be a theorical one, meant to verify the comprehension of the definitions, the theorems and their proofs. The student may take this exam in a different session with respect to the one in which he passed the exercise test, but before 6 weeks.

Through this theorical exam he may gain up to 7 points but the grade may also decrease and the student may also fail it.

If the student fails the theorical test he may retry it once without repeating the exercise exam, but if he fails the theorical test twice he must repeat also the exercise exam.

To avoid too crowded exams in the summer exams, in order to be admitted to the exercises test, the student should also pass a multple choice test with simple exercises which will last 10-15 minutes.

Teaching tools

On-line material (on IOL: INSEGNAMENTI ONLINE [https://iol.unibo.it/].).

Links to further information

https://virtuale.unibo.it/course/view.php?id=5146

Office hours

See the website of Matteo Franca