Mathematical Analysis II A - L

The course aims at conveying to the student the knowledge and comprehension 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 and Lebesgue 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 various concepts of Mathematical
Analysis and will develop both computing ability and the capacity of manipulating some
mathematical structures, such as limits, derivatives and integrals of real functions of
more real variables.
2. Applying knowledge and understanding: The student will be able to apply the acquired
knowledge to 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 acquire 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  infimum 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. (2 cfu). Real sequences of functions of one real variable. Pointwise and uniform convergence. Characterization of uniform convergence through the suprema sequence. Cauchy test of pointwise and uniform convergence. Limits exchange theorem*, continuity theorem, derivability theorem *, passage of limit under integral sign theorem. Series of real functions of one real variable. Pointwise and uniform convergence. Cauchy test. Absolute and total convergence. Weierstrass test. Comparison among various type of convergence. Theorems of: continuity, derivation and integration by series. Power series. Radius of convergence and related theorem. Cauchy-Hadamard theorem. Abel theorem*. Properties of the sum function of a power series. Taylor series. Conditions for the Taylor expansion. Important expansions (sinus, cosinus, exp, etc.). Fourier series. Sufficient conditions for the Fourier expansion*.

 2. FUNCTIONS 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. Uniform continuity. Cantor 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.  Zero gradient theorem. Homogeneous functions and Euler 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. Implicit functions and implicit function theorem (by Dini) for scalar functions of two variables*. Scalar and vector implict functions of several variables and related Dini theorems*.

 3. DIFFERENTIAL EQUATIONS. (2 cfu). First and n order differential equation Systems of n differential equations of first order in n unknown functions. 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. (2 cfu). Basic facts about Lebesgue measure in R^n. Elementary measure of intervals and multi-intervals. Measure of bounded open and closed sets. Measurability for bounded and unbounded sets. Properties: countable additivity numerabile additivity*, monotonicity, upper and lower continuity*, subtractivity . Measurable functions. Basics on the Lebesgue integration theory in R^n: Integration of bounded functions on measurable set of bounded measure. Mean value theorem. Integration of arbitrary measurable functions defined on measurable sets . Geometric meaning of the integral. Integrability tests. Passage of limit under integral sign. Theorem of B.Levi*, Theorem of i Lebesgue*. Integration by series. Method of the invading sets*. Theorem of differentiation under integral sign*. Fubini theorem*. Tonelli theorem*. Reduction formulas for double and triple integrals. Change of variables in integrals*. Polar coordinates in the plane, Spherical and cylindrical coordinates in the space. Comparison between Riemann and Lebesgue integrals*.

5. CURVES AND DIFFERENTIAL FORMS. (1 cfu). Curves 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 *. Regular domains, Green formulas. *.Exact differential equations.

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.

After the oral test the new mark (higher or lower) will be given.

Convergence of a power series.

Theorem on the existence of zeros of a continuous function.

Functions with zero gradient.

Definition of a double integral.