ANALISI MATEMATICA II A - L

The course aims at conveying to the student the knowledge and comprehensions of the mathematical concepts in the program: sequence and series of functions, limits, derivatives and extrema of functions of several variables, differential equations and systems, Riemann theory of integration, curves and differential forms.
In particular, the learning objectives of the course, according to the Dublin descriptors, are:
1. Knowledge and understanding: The student will learn some concepts of Mathematical Analysis and will develop both computing ability and the capacity of manipulating some mathematical structures, as limits, derivatives and integrals for real functions of several real variables.
2. Applying knowledge and understanding: The student will be able to apply the acquired knowledge in the basic processes of mathematical modeling of classical problems arising from Engineering.
3. Making judgements: The student will be stimulated to autonomously deepen his/her knowledge and to carry out exercises on the topics covered by the course. Constructive discussion between students and constant discussion with the teacher will be strongly recommended so that the student will be able to critically monitor his/her own learning process.
4. Communication skills: The frequency of the lessons and the reading of the recommended books will help the student to be familiar with the rigor of the mathematical language. Through constant interaction with the teacher, the student will learn to communicate the acquired knowledge with
rigor and clarity, both in oral and written form. At the end of the course the student will have learned that mathematical language is useful for communicating clearly in the scientific field.
5. Learning skills: The student will be guided in the process of perfecting his/her study method. In particular, through suitable guided exercises, he/she will be able to independently tackle new topics, recognizing the necessary prerequisites to understand them.

The student must know all the topics of  the course  " Analisi Matematica 1" : Compute  infimjm and supremum  of a set of  real numbers; Compute the limits of sequences and functions , classify the singularities of functions ;  compute the derivatives  and the extremum points  of a function; Study the convergence of a series; compute integrals;

It is important that the student  know the basics of analytic geometry in the plane and, possibly in the space. Basics concepts of vector spaces will be useful in the theory of differential equation.

1. Sequences and series of functions. (0.5 cfu). Sequences and series of real functions. Pointwise and uniform convergence. Theorems of continuity, and of passing  to the limit under integral.  Series of real functions of one real variable.  Pointwise, absolute and  uniform convergence. Total Convergence.  Weierstrass' criterion*. Relationship among various types of convergence. Continuity Theorem and integration by series. Power series. Radius of convergence. Theorem on the radius of convergence.*  Cauchy-Hadamard Theorem*.  Abel Theorem *. Properties of the sum of a power series.  Taylor series.  Sufficient condition for  expansion as a Taylor series (equibounded derivatives).  Fundamental expansions. Concept of a Fourier series.

2. Function of several variables. (2 cfu). 

 Euclidean spaces.Functions between euclidean spaces. Algebra of functions. Composition of functions and inverse function. Limitis of functions  in euclidean spaces. Theorems which characterize the limit by sequences and restrictions. Continuous functions. Continuous functions and connection. Zeros existence theorem. Compactness and continuous functions. Heine-Borel theorem *. Weierstrass theorem. Lipschitz functions. Directional and partial derivatives of scalar functions . Differentiable functons. Necessary condtions for differentiability. First derivatives and differential. Derivability of a composition of functions. Higher order derivatives and differentials. Schwartz theorem.*. Second order Taylor formula. al primo e al secondo. Zero gradient theorem.  Local maximum and minimum for functions of several variables. Fermat theorem . Basic facts about quadratic forms and characterizations of their sign. Second order necessary condition*. Second order sufficient conditions. Absolute extremum points search-

3. DIFFERENTIAL EQUATIONS. (2 cfu). First and n order differential equation Systems of n differential equations of first order in n unknown functions. Equivalence between systems and equations*. Cauchy problem and definition of its solution. Local and global Cauchy theorem*. Sufficient condition for a function to be Lipschtz. Linear systems. Global solutions of linear systems and structure of the solution set. Wronskian matrix. Lagrange method*. Constant coefficients linear systems: construction of a base in the solution space in the case of simple eigenvalues. Linear differential equations of higher order. Euler equation. Solution methods for some specific type of differential equation: separable variable equations, homogeneous equations, Linear equations of the first order. Bernoulli equations.

4. Measure and integration. (1 cfu). Basics on the measure of  Peano-Jordan in R^n.  Riemann Integration  in R^n.   Integrability of continuous and generally continuous functions. Mean value theorem.  Reduction formulas for double and triple integrals . Change of variables*. Polar and cylindrical coordinates in the  space.

5. Curves and differential forms. (0.5 cfu). 

Curve in R^n. Simple, plane and Jordan curves. Union of curves. Regular and generally regular curves. Change of parameter. Rectifiable curves. Rectifiabilitry of regular curves*. Curvilinear abscissa. Curvilinear integral. Concept of a differential form and its curvilinear integral. Exact differential forms. Integrability criterion. Circuit integral. Closed forms. Star shaped open sets. Poincaré Theorem *. Simple connected sets. Integrability criterion of simple connected sets *. 

The proofs of the topics followed by * can be omitted

1. Di Fazio G., Zamboni P., Analisi Matematica 2, Monduzzi Editoriale.

2. Pagani C.D., Salsa S., Analisi Matematica 1, Zanichelli , seconda edizione, 2015

3. Pagani C.D., S. Salsa S., Analisi Matematica 2, Zanichelli , seconda edizione, 2016

4. Fanciullo M. S., Giacobbe A., Raciti F., Esercizi di Analisi Matematica 2, Medical Books.

The exam consists of a written test and of an optional oral test. The written test is organized in two parts. In the first part there are two definitions and two theoretical questions. In the second part there are three exercises.

To pass the exam  with the minimum mark it is necessary: to provide one of the two defintions; to solve one out of the two theoretical questions; to correctly solve one of the three exercises.

In the case that the student  completes correctly  all parts,  they will be given a mark of 26/30.After the written test, the mark can be confirmed, but the student can also ask to take an oral test on the whole program.

The theorems that can be asked in the written test will be taken from the following list:

After the oral test the new mark (higher or lower) will be given. If the oral test is considered insufficient the student will have to take again the written test.

Convergence of a power series.

Theorem on the existence of zeros of a continuous function.

Functions with zero gradient.

Definition of a double integral.

Many worked out exercises will be uploaded on Studium/Moodle